.file "libm_tan.s" // Copyright (C) 2000, 2001, Intel Corporation // All rights reserved. // // Contributed 2/2/2000 by John Harrison, Ted Kubaska, Bob Norin, Shane Story, // and Ping Tak Peter Tang of the Computational Software Lab, Intel Corporation. // // Redistribution and use in source and binary forms, with or without // modification, are permitted provided that the following conditions are // met: // // * Redistributions of source code must retain the above copyright // notice, this list of conditions and the following disclaimer. // // * Redistributions in binary form must reproduce the above copyright // notice, this list of conditions and the following disclaimer in the // documentation and/or other materials provided with the distribution. // // * The name of Intel Corporation may not be used to endorse or promote // products derived from this software without specific prior written // permission. // // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS // CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, // EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY // OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING // NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. // // Intel Corporation is the author of this code, and requests that all // problem reports or change requests be submitted to it directly at // http://developer.intel.com/opensource. // // ********************************************************************* // // History: // 02/02/00 Initial Version // 4/04/00 Unwind support added // 12/28/00 Fixed false invalid flags // // ********************************************************************* // // Function: tan(x) = tangent(x), for double precision x values // // ********************************************************************* // // Accuracy: Very accurate for double-precision values // // ********************************************************************* // // Resources Used: // // Floating-Point Registers: f8 (Input and Return Value) // f9-f15 // f32-f112 // // General Purpose Registers: // r32-r48 // r49-r50 (Used to pass arguments to pi_by_2 reduce routine) // // Predicate Registers: p6-p15 // // ********************************************************************* // // IEEE Special Conditions: // // Denormal fault raised on denormal inputs // Overflow exceptions do not occur // Underflow exceptions raised when appropriate for tan // (No specialized error handling for this routine) // Inexact raised when appropriate by algorithm // // tan(SNaN) = QNaN // tan(QNaN) = QNaN // tan(inf) = QNaN // tan(+/-0) = +/-0 // // ********************************************************************* // // Mathematical Description // // We consider the computation of FPTAN of Arg. Now, given // // Arg = N pi/2 + alpha, |alpha| <= pi/4, // // basic mathematical relationship shows that // // tan( Arg ) = tan( alpha ) if N is even; // = -cot( alpha ) otherwise. // // The value of alpha is obtained by argument reduction and // represented by two working precision numbers r and c where // // alpha = r + c accurately. // // The reduction method is described in a previous write up. // The argument reduction scheme identifies 4 cases. For Cases 2 // and 4, because |alpha| is small, tan(r+c) and -cot(r+c) can be // computed very easily by 2 or 3 terms of the Taylor series // expansion as follows: // // Case 2: // ------- // // tan(r + c) = r + c + r^3/3 ...accurately // -cot(r + c) = -1/(r+c) + r/3 ...accurately // // Case 4: // ------- // // tan(r + c) = r + c + r^3/3 + 2r^5/15 ...accurately // -cot(r + c) = -1/(r+c) + r/3 + r^3/45 ...accurately // // // The only cases left are Cases 1 and 3 of the argument reduction // procedure. These two cases will be merged since after the // argument is reduced in either cases, we have the reduced argument // represented as r + c and that the magnitude |r + c| is not small // enough to allow the usage of a very short approximation. // // The greatest challenge of this task is that the second terms of // the Taylor series for tan(r) and -cot(r) // // r + r^3/3 + 2 r^5/15 + ... // // and // // -1/r + r/3 + r^3/45 + ... // // are not very small when |r| is close to pi/4 and the rounding // errors will be a concern if simple polynomial accumulation is // used. When |r| < 2^(-2), however, the second terms will be small // enough (5 bits or so of right shift) that a normal Horner // recurrence suffices. Hence there are two cases that we consider // in the accurate computation of tan(r) and cot(r), |r| <= pi/4. // // Case small_r: |r| < 2^(-2) // -------------------------- // // Since Arg = N pi/4 + r + c accurately, we have // // tan(Arg) = tan(r+c) for N even, // = -cot(r+c) otherwise. // // Here for this case, both tan(r) and -cot(r) can be approximated // by simple polynomials: // // tan(r) = r + P1_1 r^3 + P1_2 r^5 + ... + P1_9 r^19 // -cot(r) = -1/r + Q1_1 r + Q1_2 r^3 + ... + Q1_7 r^13 // // accurately. Since |r| is relatively small, tan(r+c) and // -cot(r+c) can be accurately approximated by replacing r with // r+c only in the first two terms of the corresponding polynomials. // // Note that P1_1 (and Q1_1 for that matter) approximates 1/3 to // almost 64 sig. bits, thus // // P1_1 (r+c)^3 = P1_1 r^3 + c * r^2 accurately. // // Hence, // // tan(r+c) = r + P1_1 r^3 + P1_2 r^5 + ... + P1_9 r^19 // + c*(1 + r^2) // // -cot(r+c) = -1/(r+c) + Q1_1 r + Q1_2 r^3 + ... + Q1_7 r^13 // + Q1_1*c // // // Case normal_r: 2^(-2) <= |r| <= pi/4 // ------------------------------------ // // This case is more likely than the previous one if one considers // r to be uniformly distributed in [-pi/4 pi/4]. // // The required calculation is either // // tan(r + c) = tan(r) + correction, or // -cot(r + c) = -cot(r) + correction. // // Specifically, // // tan(r + c) = tan(r) + c tan'(r) + O(c^2) // = tan(r) + c sec^2(r) + O(c^2) // = tan(r) + c SEC_sq ...accurately // as long as SEC_sq approximates sec^2(r) // to, say, 5 bits or so. // // Similarly, // // -cot(r + c) = -cot(r) - c cot'(r) + O(c^2) // = -cot(r) + c csc^2(r) + O(c^2) // = -cot(r) + c CSC_sq ...accurately // as long as CSC_sq approximates csc^2(r) // to, say, 5 bits or so. // // We therefore concentrate on accurately calculating tan(r) and // cot(r) for a working-precision number r, |r| <= pi/4 to within // 0.1% or so. // // We will employ a table-driven approach. Let // // r = sgn_r * 2^k * 1.b_1 b_2 ... b_5 ... b_63 // = sgn_r * ( B + x ) // // where // // B = 2^k * 1.b_1 b_2 ... b_5 1 // x = |r| - B // // Now, // tan(B) + tan(x) // tan( B + x ) = ------------------------ // 1 - tan(B)*tan(x) // // / \ // | tan(B) + tan(x) | // = tan(B) + | ------------------------ - tan(B) | // | 1 - tan(B)*tan(x) | // \ / // // sec^2(B) * tan(x) // = tan(B) + ------------------------ // 1 - tan(B)*tan(x) // // (1/[sin(B)*cos(B)]) * tan(x) // = tan(B) + -------------------------------- // cot(B) - tan(x) // // // Clearly, the values of tan(B), cot(B) and 1/(sin(B)*cos(B)) are // calculated beforehand and stored in a table. Since // // |x| <= 2^k * 2^(-6) <= 2^(-7) (because k = -1, -2) // // a very short polynomial will be sufficient to approximate tan(x) // accurately. The details involved in computing the last expression // will be given in the next section on algorithm description. // // // Now, we turn to the case where cot( B + x ) is needed. // // // 1 - tan(B)*tan(x) // cot( B + x ) = ------------------------ // tan(B) + tan(x) // // / \ // | 1 - tan(B)*tan(x) | // = cot(B) + | ----------------------- - cot(B) | // | tan(B) + tan(x) | // \ / // // [tan(B) + cot(B)] * tan(x) // = cot(B) - ---------------------------- // tan(B) + tan(x) // // (1/[sin(B)*cos(B)]) * tan(x) // = cot(B) - -------------------------------- // tan(B) + tan(x) // // // Note that the values of tan(B), cot(B) and 1/(sin(B)*cos(B)) that // are needed are the same set of values needed in the previous // case. // // Finally, we can put all the ingredients together as follows: // // Arg = N * pi/2 + r + c ...accurately // // tan(Arg) = tan(r) + correction if N is even; // = -cot(r) + correction otherwise. // // For Cases 2 and 4, // // Case 2: // tan(Arg) = tan(r + c) = r + c + r^3/3 N even // = -cot(r + c) = -1/(r+c) + r/3 N odd // Case 4: // tan(Arg) = tan(r + c) = r + c + r^3/3 + 2r^5/15 N even // = -cot(r + c) = -1/(r+c) + r/3 + r^3/45 N odd // // // For Cases 1 and 3, // // Case small_r: |r| < 2^(-2) // // tan(Arg) = r + P1_1 r^3 + P1_2 r^5 + ... + P1_9 r^19 // + c*(1 + r^2) N even // // = -1/(r+c) + Q1_1 r + Q1_2 r^3 + ... + Q1_7 r^13 // + Q1_1*c N odd // // Case normal_r: 2^(-2) <= |r| <= pi/4 // // tan(Arg) = tan(r) + c * sec^2(r) N even // = -cot(r) + c * csc^2(r) otherwise // // For N even, // // tan(Arg) = tan(r) + c*sec^2(r) // = tan( sgn_r * (B+x) ) + c * sec^2(|r|) // = sgn_r * ( tan(B+x) + sgn_r*c*sec^2(|r|) ) // = sgn_r * ( tan(B+x) + sgn_r*c*sec^2(B) ) // // since B approximates |r| to 2^(-6) in relative accuracy. // // / (1/[sin(B)*cos(B)]) * tan(x) // tan(Arg) = sgn_r * | tan(B) + -------------------------------- // \ cot(B) - tan(x) // \ // + CORR | // / // where // // CORR = sgn_r*c*tan(B)*SC_inv(B); SC_inv(B) = 1/(sin(B)*cos(B)). // // For N odd, // // tan(Arg) = -cot(r) + c*csc^2(r) // = -cot( sgn_r * (B+x) ) + c * csc^2(|r|) // = sgn_r * ( -cot(B+x) + sgn_r*c*csc^2(|r|) ) // = sgn_r * ( -cot(B+x) + sgn_r*c*csc^2(B) ) // // since B approximates |r| to 2^(-6) in relative accuracy. // // / (1/[sin(B)*cos(B)]) * tan(x) // tan(Arg) = sgn_r * | -cot(B) + -------------------------------- // \ tan(B) + tan(x) // \ // + CORR | // / // where // // CORR = sgn_r*c*cot(B)*SC_inv(B); SC_inv(B) = 1/(sin(B)*cos(B)). // // // The actual algorithm prescribes how all the mathematical formulas // are calculated. // // // 2. Algorithmic Description // ========================== // // 2.1 Computation for Cases 2 and 4. // ---------------------------------- // // For Case 2, we use two-term polynomials. // // For N even, // // rsq := r * r // Result := c + r * rsq * P1_1 // Result := r + Result ...in user-defined rounding // // For N odd, // S_hi := -frcpa(r) ...8 bits // S_hi := S_hi + S_hi*(1 + S_hi*r) ...16 bits // S_hi := S_hi + S_hi*(1 + S_hi*r) ...32 bits // S_hi := S_hi + S_hi*(1 + S_hi*r) ...64 bits // S_lo := S_hi*( (1 + S_hi*r) + S_hi*c ) // ...S_hi + S_lo is -1/(r+c) to extra precision // S_lo := S_lo + Q1_1*r // // Result := S_hi + S_lo ...in user-defined rounding // // For Case 4, we use three-term polynomials // // For N even, // // rsq := r * r // Result := c + r * rsq * (P1_1 + rsq * P1_2) // Result := r + Result ...in user-defined rounding // // For N odd, // S_hi := -frcpa(r) ...8 bits // S_hi := S_hi + S_hi*(1 + S_hi*r) ...16 bits // S_hi := S_hi + S_hi*(1 + S_hi*r) ...32 bits // S_hi := S_hi + S_hi*(1 + S_hi*r) ...64 bits // S_lo := S_hi*( (1 + S_hi*r) + S_hi*c ) // ...S_hi + S_lo is -1/(r+c) to extra precision // rsq := r * r // P := Q1_1 + rsq*Q1_2 // S_lo := S_lo + r*P // // Result := S_hi + S_lo ...in user-defined rounding // // // Note that the coefficients P1_1, P1_2, Q1_1, and Q1_2 are // the same as those used in the small_r case of Cases 1 and 3 // below. // // // 2.2 Computation for Cases 1 and 3. // ---------------------------------- // This is further divided into the case of small_r, // where |r| < 2^(-2), and the case of normal_r, where |r| lies between // 2^(-2) and pi/4. // // Algorithm for the case of small_r // --------------------------------- // // For N even, // rsq := r * r // Poly1 := rsq*(P1_1 + rsq*(P1_2 + rsq*P1_3)) // r_to_the_8 := rsq * rsq // r_to_the_8 := r_to_the_8 * r_to_the_8 // Poly2 := P1_4 + rsq*(P1_5 + rsq*(P1_6 + ... rsq*P1_9)) // CORR := c * ( 1 + rsq ) // Poly := Poly1 + r_to_the_8*Poly2 // Result := r*Poly + CORR // Result := r + Result ...in user-defined rounding // ...note that Poly1 and r_to_the_8 can be computed in parallel // ...with Poly2 (Poly1 is intentionally set to be much // ...shorter than Poly2 so that r_to_the_8 and CORR can be hidden) // // For N odd, // S_hi := -frcpa(r) ...8 bits // S_hi := S_hi + S_hi*(1 + S_hi*r) ...16 bits // S_hi := S_hi + S_hi*(1 + S_hi*r) ...32 bits // S_hi := S_hi + S_hi*(1 + S_hi*r) ...64 bits // S_lo := S_hi*( (1 + S_hi*r) + S_hi*c ) // ...S_hi + S_lo is -1/(r+c) to extra precision // S_lo := S_lo + Q1_1*c // // ...S_hi and S_lo are computed in parallel with // ...the following // rsq := r*r // P := Q1_1 + rsq*(Q1_2 + rsq*(Q1_3 + ... + rsq*Q1_7)) // // Result := r*P + S_lo // Result := S_hi + Result ...in user-defined rounding // // // Algorithm for the case of normal_r // ---------------------------------- // // Here, we first consider the computation of tan( r + c ). As // presented in the previous section, // // tan( r + c ) = tan(r) + c * sec^2(r) // = sgn_r * [ tan(B+x) + CORR ] // CORR = sgn_r * c * tan(B) * 1/[sin(B)*cos(B)] // // because sec^2(r) = sec^(|r|), and B approximate |r| to 6.5 bits. // // tan( r + c ) = // / (1/[sin(B)*cos(B)]) * tan(x) // sgn_r * | tan(B) + -------------------------------- + // \ cot(B) - tan(x) // \ // CORR | // / // // The values of tan(B), cot(B) and 1/(sin(B)*cos(B)) are // calculated beforehand and stored in a table. Specifically, // the table values are // // tan(B) as T_hi + T_lo; // cot(B) as C_hi + C_lo; // 1/[sin(B)*cos(B)] as SC_inv // // T_hi, C_hi are in double-precision memory format; // T_lo, C_lo are in single-precision memory format; // SC_inv is in extended-precision memory format. // // The value of tan(x) will be approximated by a short polynomial of // the form // // tan(x) as x + x * P, where // P = x^2 * (P2_1 + x^2 * (P2_2 + x^2 * P2_3)) // // Because |x| <= 2^(-7), cot(B) - x approximates cot(B) - tan(x) // to a relative accuracy better than 2^(-20). Thus, a good // initial guess of 1/( cot(B) - tan(x) ) to initiate the iterative // division is: // // 1/(cot(B) - tan(x)) is approximately // 1/(cot(B) - x) is // tan(B)/(1 - x*tan(B)) is approximately // T_hi / ( 1 - T_hi * x ) is approximately // // T_hi * [ 1 + (Thi * x) + (T_hi * x)^2 ] // // The calculation of tan(r+c) therefore proceed as follows: // // Tx := T_hi * x // xsq := x * x // // V_hi := T_hi*(1 + Tx*(1 + Tx)) // P := xsq * (P1_1 + xsq*(P1_2 + xsq*P1_3)) // ...V_hi serves as an initial guess of 1/(cot(B) - tan(x)) // ...good to about 20 bits of accuracy // // tanx := x + x*P // D := C_hi - tanx // ...D is a double precision denominator: cot(B) - tan(x) // // V_hi := V_hi + V_hi*(1 - V_hi*D) // ....V_hi approximates 1/(cot(B)-tan(x)) to 40 bits // // V_lo := V_hi * ( [ (1 - V_hi*C_hi) + V_hi*tanx ] // - V_hi*C_lo ) ...observe all order // ...V_hi + V_lo approximates 1/(cot(B) - tan(x)) // ...to extra accuracy // // ... SC_inv(B) * (x + x*P) // ... tan(B) + ------------------------- + CORR // ... cot(B) - (x + x*P) // ... // ... = tan(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR // ... // // Sx := SC_inv * x // CORR := sgn_r * c * SC_inv * T_hi // // ...put the ingredients together to compute // ... SC_inv(B) * (x + x*P) // ... tan(B) + ------------------------- + CORR // ... cot(B) - (x + x*P) // ... // ... = tan(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR // ... // ... = T_hi + T_lo + CORR + // ... Sx * V_hi + Sx * V_lo + Sx * P *(V_hi + V_lo) // // CORR := CORR + T_lo // tail := V_lo + P*(V_hi + V_lo) // tail := Sx * tail + CORR // tail := Sx * V_hi + tail // T_hi := sgn_r * T_hi // // ...T_hi + sgn_r*tail now approximate // ...sgn_r*(tan(B+x) + CORR) accurately // // Result := T_hi + sgn_r*tail ...in user-defined // ...rounding control // ...It is crucial that independent paths be fully // ...exploited for performance's sake. // // // Next, we consider the computation of -cot( r + c ). As // presented in the previous section, // // -cot( r + c ) = -cot(r) + c * csc^2(r) // = sgn_r * [ -cot(B+x) + CORR ] // CORR = sgn_r * c * cot(B) * 1/[sin(B)*cos(B)] // // because csc^2(r) = csc^(|r|), and B approximate |r| to 6.5 bits. // // -cot( r + c ) = // / (1/[sin(B)*cos(B)]) * tan(x) // sgn_r * | -cot(B) + -------------------------------- + // \ tan(B) + tan(x) // \ // CORR | // / // // The values of tan(B), cot(B) and 1/(sin(B)*cos(B)) are // calculated beforehand and stored in a table. Specifically, // the table values are // // tan(B) as T_hi + T_lo; // cot(B) as C_hi + C_lo; // 1/[sin(B)*cos(B)] as SC_inv // // T_hi, C_hi are in double-precision memory format; // T_lo, C_lo are in single-precision memory format; // SC_inv is in extended-precision memory format. // // The value of tan(x) will be approximated by a short polynomial of // the form // // tan(x) as x + x * P, where // P = x^2 * (P2_1 + x^2 * (P2_2 + x^2 * P2_3)) // // Because |x| <= 2^(-7), tan(B) + x approximates tan(B) + tan(x) // to a relative accuracy better than 2^(-18). Thus, a good // initial guess of 1/( tan(B) + tan(x) ) to initiate the iterative // division is: // // 1/(tan(B) + tan(x)) is approximately // 1/(tan(B) + x) is // cot(B)/(1 + x*cot(B)) is approximately // C_hi / ( 1 + C_hi * x ) is approximately // // C_hi * [ 1 - (C_hi * x) + (C_hi * x)^2 ] // // The calculation of -cot(r+c) therefore proceed as follows: // // Cx := C_hi * x // xsq := x * x // // V_hi := C_hi*(1 - Cx*(1 - Cx)) // P := xsq * (P1_1 + xsq*(P1_2 + xsq*P1_3)) // ...V_hi serves as an initial guess of 1/(tan(B) + tan(x)) // ...good to about 18 bits of accuracy // // tanx := x + x*P // D := T_hi + tanx // ...D is a double precision denominator: tan(B) + tan(x) // // V_hi := V_hi + V_hi*(1 - V_hi*D) // ....V_hi approximates 1/(tan(B)+tan(x)) to 40 bits // // V_lo := V_hi * ( [ (1 - V_hi*T_hi) - V_hi*tanx ] // - V_hi*T_lo ) ...observe all order // ...V_hi + V_lo approximates 1/(tan(B) + tan(x)) // ...to extra accuracy // // ... SC_inv(B) * (x + x*P) // ... -cot(B) + ------------------------- + CORR // ... tan(B) + (x + x*P) // ... // ... =-cot(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR // ... // // Sx := SC_inv * x // CORR := sgn_r * c * SC_inv * C_hi // // ...put the ingredients together to compute // ... SC_inv(B) * (x + x*P) // ... -cot(B) + ------------------------- + CORR // ... tan(B) + (x + x*P) // ... // ... =-cot(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR // ... // ... =-C_hi - C_lo + CORR + // ... Sx * V_hi + Sx * V_lo + Sx * P *(V_hi + V_lo) // // CORR := CORR - C_lo // tail := V_lo + P*(V_hi + V_lo) // tail := Sx * tail + CORR // tail := Sx * V_hi + tail // C_hi := -sgn_r * C_hi // // ...C_hi + sgn_r*tail now approximates // ...sgn_r*(-cot(B+x) + CORR) accurately // // Result := C_hi + sgn_r*tail in user-defined rounding control // ...It is crucial that independent paths be fully // ...exploited for performance's sake. // // 3. Implementation Notes // ======================= // // Table entries T_hi, T_lo; C_hi, C_lo; SC_inv // // Recall that 2^(-2) <= |r| <= pi/4; // // r = sgn_r * 2^k * 1.b_1 b_2 ... b_63 // // and // // B = 2^k * 1.b_1 b_2 b_3 b_4 b_5 1 // // Thus, for k = -2, possible values of B are // // B = 2^(-2) * ( 1 + index/32 + 1/64 ), // index ranges from 0 to 31 // // For k = -1, however, since |r| <= pi/4 = 0.78... // possible values of B are // // B = 2^(-1) * ( 1 + index/32 + 1/64 ) // index ranges from 0 to 19. // // #include "libm_support.h" #ifdef _LIBC .rodata #else .data #endif .align 128 TAN_BASE_CONSTANTS: .type TAN_BASE_CONSTANTS, @object data4 0x4B800000, 0xCB800000, 0x38800000, 0xB8800000 // two**24, -two**24 // two**-14, -two**-14 data4 0x4E44152A, 0xA2F9836E, 0x00003FFE, 0x00000000 // two_by_pi data4 0xCE81B9F1, 0xC84D32B0, 0x00004016, 0x00000000 // P_0 data4 0x2168C235, 0xC90FDAA2, 0x00003FFF, 0x00000000 // P_1 data4 0xFC8F8CBB, 0xECE675D1, 0x0000BFBD, 0x00000000 // P_2 data4 0xACC19C60, 0xB7ED8FBB, 0x0000BF7C, 0x00000000 // P_3 data4 0x5F000000, 0xDF000000, 0x00000000, 0x00000000 // two_to_63, -two_to_63 data4 0x6EC6B45A, 0xA397E504, 0x00003FE7, 0x00000000 // Inv_P_0 data4 0xDBD171A1, 0x8D848E89, 0x0000BFBF, 0x00000000 // d_1 data4 0x18A66F8E, 0xD5394C36, 0x0000BF7C, 0x00000000 // d_2 data4 0x2168C234, 0xC90FDAA2, 0x00003FFE, 0x00000000 // PI_BY_4 data4 0x2168C234, 0xC90FDAA2, 0x0000BFFE, 0x00000000 // MPI_BY_4 data4 0x3E800000, 0xBE800000, 0x00000000, 0x00000000 // two**-2, -two**-2 data4 0x2F000000, 0xAF000000, 0x00000000, 0x00000000 // two**-33, -two**-33 data4 0xAAAAAABD, 0xAAAAAAAA, 0x00003FFD, 0x00000000 // P1_1 data4 0x88882E6A, 0x88888888, 0x00003FFC, 0x00000000 // P1_2 data4 0x0F0177B6, 0xDD0DD0DD, 0x00003FFA, 0x00000000 // P1_3 data4 0x646B8C6D, 0xB327A440, 0x00003FF9, 0x00000000 // P1_4 data4 0x1D5F7D20, 0x91371B25, 0x00003FF8, 0x00000000 // P1_5 data4 0x61C67914, 0xEB69A5F1, 0x00003FF6, 0x00000000 // P1_6 data4 0x019318D2, 0xBEDD37BE, 0x00003FF5, 0x00000000 // P1_7 data4 0x3C794015, 0x9979B146, 0x00003FF4, 0x00000000 // P1_8 data4 0x8C6EB58A, 0x8EBD21A3, 0x00003FF3, 0x00000000 // P1_9 data4 0xAAAAAAB4, 0xAAAAAAAA, 0x00003FFD, 0x00000000 // Q1_1 data4 0x0B5FC93E, 0xB60B60B6, 0x00003FF9, 0x00000000 // Q1_2 data4 0x0C9BBFBF, 0x8AB355E0, 0x00003FF6, 0x00000000 // Q1_3 data4 0xCBEE3D4C, 0xDDEBBC89, 0x00003FF2, 0x00000000 // Q1_4 data4 0x5F80BBB6, 0xB3548A68, 0x00003FEF, 0x00000000 // Q1_5 data4 0x4CED5BF1, 0x91362560, 0x00003FEC, 0x00000000 // Q1_6 data4 0x8EE92A83, 0xF189D95A, 0x00003FE8, 0x00000000 // Q1_7 data4 0xAAAB362F, 0xAAAAAAAA, 0x00003FFD, 0x00000000 // P2_1 data4 0xE97A6097, 0x88888886, 0x00003FFC, 0x00000000 // P2_2 data4 0x25E716A1, 0xDD108EE0, 0x00003FFA, 0x00000000 // P2_3 // // Entries T_hi double-precision memory format // Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64) // Entries T_lo single-precision memory format // Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64) // data4 0x62400794, 0x3FD09BC3, 0x23A05C32, 0x00000000 data4 0xDFFBC074, 0x3FD124A9, 0x240078B2, 0x00000000 data4 0x5BD4920F, 0x3FD1AE23, 0x23826B8E, 0x00000000 data4 0x15E2701D, 0x3FD23835, 0x22D31154, 0x00000000 data4 0x63739C2D, 0x3FD2C2E4, 0x2265C9E2, 0x00000000 data4 0xAFEEA48B, 0x3FD34E36, 0x245C05EB, 0x00000000 data4 0x7DBB35D1, 0x3FD3DA31, 0x24749F2D, 0x00000000 data4 0x67321619, 0x3FD466DA, 0x2462CECE, 0x00000000 data4 0x1F94A4D5, 0x3FD4F437, 0x246D0DF1, 0x00000000 data4 0x740C3E6D, 0x3FD5824D, 0x240A85B5, 0x00000000 data4 0x4CB1E73D, 0x3FD61123, 0x23F96E33, 0x00000000 data4 0xAD9EA64B, 0x3FD6A0BE, 0x247C5393, 0x00000000 data4 0xB804FD01, 0x3FD73125, 0x241F3B29, 0x00000000 data4 0xAB53EE83, 0x3FD7C25E, 0x2479989B, 0x00000000 data4 0xE6640EED, 0x3FD8546F, 0x23B343BC, 0x00000000 data4 0xE8AF1892, 0x3FD8E75F, 0x241454D1, 0x00000000 data4 0x53928BDA, 0x3FD97B35, 0x238613D9, 0x00000000 data4 0xEB9DE4DE, 0x3FDA0FF6, 0x22859FA7, 0x00000000 data4 0x99ECF92D, 0x3FDAA5AB, 0x237A6D06, 0x00000000 data4 0x6D8F1796, 0x3FDB3C5A, 0x23952F6C, 0x00000000 data4 0x9CFB8BE4, 0x3FDBD40A, 0x2280FC95, 0x00000000 data4 0x87943100, 0x3FDC6CC3, 0x245D2EC0, 0x00000000 data4 0xB736C500, 0x3FDD068C, 0x23C4AD7D, 0x00000000 data4 0xE1DDBC31, 0x3FDDA16D, 0x23D076E6, 0x00000000 data4 0xEB515A93, 0x3FDE3D6E, 0x244809A6, 0x00000000 data4 0xE6E9E5F1, 0x3FDEDA97, 0x220856C8, 0x00000000 data4 0x1963CE69, 0x3FDF78F1, 0x244BE993, 0x00000000 data4 0x7D635BCE, 0x3FE00C41, 0x23D21799, 0x00000000 data4 0x1C302CD3, 0x3FE05CAB, 0x248A1B1D, 0x00000000 data4 0xDB6A1FA0, 0x3FE0ADB9, 0x23D53E33, 0x00000000 data4 0x4A20BA81, 0x3FE0FF72, 0x24DB9ED5, 0x00000000 data4 0x153FA6F5, 0x3FE151D9, 0x24E9E451, 0x00000000 // // Entries T_hi double-precision memory format // Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64) // Entries T_lo single-precision memory format // Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64) // data4 0xBA1BE39E, 0x3FE1CEC4, 0x24B60F9E, 0x00000000 data4 0x5ABD9B2D, 0x3FE277E4, 0x248C2474, 0x00000000 data4 0x0272B110, 0x3FE32418, 0x247B8311, 0x00000000 data4 0x890E2DF0, 0x3FE3D38B, 0x24C55751, 0x00000000 data4 0x46236871, 0x3FE4866D, 0x24E5BC34, 0x00000000 data4 0x45E044B0, 0x3FE53CEE, 0x24001BA4, 0x00000000 data4 0x82EC06E4, 0x3FE5F742, 0x24B973DC, 0x00000000 data4 0x25DF43F9, 0x3FE6B5A1, 0x24895440, 0x00000000 data4 0xCAFD348C, 0x3FE77844, 0x240021CA, 0x00000000 data4 0xCEED6B92, 0x3FE83F6B, 0x24C45372, 0x00000000 data4 0xA34F3665, 0x3FE90B58, 0x240DAD33, 0x00000000 data4 0x2C1E56B4, 0x3FE9DC52, 0x24F846CE, 0x00000000 data4 0x27041578, 0x3FEAB2A4, 0x2323FB6E, 0x00000000 data4 0x9DD8C373, 0x3FEB8E9F, 0x24B3090B, 0x00000000 data4 0x65C9AA7B, 0x3FEC709B, 0x2449F611, 0x00000000 data4 0xACCF8435, 0x3FED58F4, 0x23616A7E, 0x00000000 data4 0x97635082, 0x3FEE480F, 0x24C2FEAE, 0x00000000 data4 0xF0ACC544, 0x3FEF3E57, 0x242CE964, 0x00000000 data4 0xF7E06E4B, 0x3FF01E20, 0x2480D3EE, 0x00000000 data4 0x8A798A69, 0x3FF0A125, 0x24DB8967, 0x00000000 // // Entries C_hi double-precision memory format // Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64) // Entries C_lo single-precision memory format // Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64) // data4 0xE63EFBD0, 0x400ED3E2, 0x259D94D4, 0x00000000 data4 0xC515DAB5, 0x400DDDB4, 0x245F0537, 0x00000000 data4 0xBE19A79F, 0x400CF57A, 0x25D4EA9F, 0x00000000 data4 0xD15298ED, 0x400C1A06, 0x24AE40A0, 0x00000000 data4 0x164B2708, 0x400B4A4C, 0x25A5AAB6, 0x00000000 data4 0x5285B068, 0x400A855A, 0x25524F18, 0x00000000 data4 0x3FFA549F, 0x4009CA5A, 0x24C999C0, 0x00000000 data4 0x646AF623, 0x4009188A, 0x254FD801, 0x00000000 data4 0x6084D0E7, 0x40086F3C, 0x2560F5FD, 0x00000000 data4 0xA29A76EE, 0x4007CDD2, 0x255B9D19, 0x00000000 data4 0x6C8ECA95, 0x400733BE, 0x25CB021B, 0x00000000 data4 0x1F8DDC52, 0x4006A07E, 0x24AB4722, 0x00000000 data4 0xC298AD58, 0x4006139B, 0x252764E2, 0x00000000 data4 0xBAD7164B, 0x40058CAB, 0x24DAF5DB, 0x00000000 data4 0xAE31A5D3, 0x40050B4B, 0x25EA20F4, 0x00000000 data4 0x89F85A8A, 0x40048F21, 0x2583A3E8, 0x00000000 data4 0xA862380D, 0x400417DA, 0x25DCC4CC, 0x00000000 data4 0x1088FCFE, 0x4003A52B, 0x2430A492, 0x00000000 data4 0xCD3527D5, 0x400336CC, 0x255F77CF, 0x00000000 data4 0x5760766D, 0x4002CC7F, 0x25DA0BDA, 0x00000000 data4 0x11CE02E3, 0x40026607, 0x256FF4A2, 0x00000000 data4 0xD37BBE04, 0x4002032C, 0x25208AED, 0x00000000 data4 0x7F050775, 0x4001A3BD, 0x24B72DD6, 0x00000000 data4 0xA554848A, 0x40014789, 0x24AB4DAA, 0x00000000 data4 0x323E81B7, 0x4000EE65, 0x2584C440, 0x00000000 data4 0x21CF1293, 0x40009827, 0x25C9428D, 0x00000000 data4 0x3D415EEB, 0x400044A9, 0x25DC8482, 0x00000000 data4 0xBD72C577, 0x3FFFE78F, 0x257F5070, 0x00000000 data4 0x75EFD28E, 0x3FFF4AC3, 0x23EBBF7A, 0x00000000 data4 0x60B52DDE, 0x3FFEB2AF, 0x22EECA07, 0x00000000 data4 0x35204180, 0x3FFE1F19, 0x24191079, 0x00000000 data4 0x54F7E60A, 0x3FFD8FCA, 0x248D3058, 0x00000000 // // Entries C_hi double-precision memory format // Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64) // Entries C_lo single-precision memory format // Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64) // data4 0x79F6FADE, 0x3FFCC06A, 0x239C7886, 0x00000000 data4 0x891662A6, 0x3FFBB91F, 0x250BD191, 0x00000000 data4 0x529F155D, 0x3FFABFB6, 0x256CC3E6, 0x00000000 data4 0x2E964AE9, 0x3FF9D300, 0x250843E3, 0x00000000 data4 0x89DCB383, 0x3FF8F1EF, 0x2277C87E, 0x00000000 data4 0x7C87DBD6, 0x3FF81B93, 0x256DA6CF, 0x00000000 data4 0x1042EDE4, 0x3FF74F14, 0x2573D28A, 0x00000000 data4 0x1784B360, 0x3FF68BAF, 0x242E489A, 0x00000000 data4 0x7C923C4C, 0x3FF5D0B5, 0x2532D940, 0x00000000 data4 0xF418EF20, 0x3FF51D88, 0x253C7DD6, 0x00000000 data4 0x02F88DAE, 0x3FF4719A, 0x23DB59BF, 0x00000000 data4 0x49DA0788, 0x3FF3CC66, 0x252B4756, 0x00000000 data4 0x0B980DB8, 0x3FF32D77, 0x23FE585F, 0x00000000 data4 0xE56C987A, 0x3FF2945F, 0x25378A63, 0x00000000 data4 0xB16523F6, 0x3FF200BD, 0x247BB2E0, 0x00000000 data4 0x8CE27778, 0x3FF17235, 0x24446538, 0x00000000 data4 0xFDEFE692, 0x3FF0E873, 0x2514638F, 0x00000000 data4 0x33154062, 0x3FF0632C, 0x24A7FC27, 0x00000000 data4 0xB3EF115F, 0x3FEFC42E, 0x248FD0FE, 0x00000000 data4 0x135D26F6, 0x3FEEC9E8, 0x2385C719, 0x00000000 // // Entries SC_inv in Swapped IEEE format (extended) // Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64) // data4 0x1BF30C9E, 0x839D6D4A, 0x00004001, 0x00000000 data4 0x554B0EB0, 0x80092804, 0x00004001, 0x00000000 data4 0xA1CF0DE9, 0xF959F94C, 0x00004000, 0x00000000 data4 0x77378677, 0xF3086BA0, 0x00004000, 0x00000000 data4 0xCCD4723C, 0xED154515, 0x00004000, 0x00000000 data4 0x1C27CF25, 0xE7790944, 0x00004000, 0x00000000 data4 0x8DDACB88, 0xE22D037D, 0x00004000, 0x00000000 data4 0x89C73522, 0xDD2B2D8A, 0x00004000, 0x00000000 data4 0xBB2C1171, 0xD86E1A23, 0x00004000, 0x00000000 data4 0xDFF5E0F9, 0xD3F0E288, 0x00004000, 0x00000000 data4 0x283BEBD5, 0xCFAF16B1, 0x00004000, 0x00000000 data4 0x0D88DD53, 0xCBA4AFAA, 0x00004000, 0x00000000 data4 0xCA67C43D, 0xC7CE03CC, 0x00004000, 0x00000000 data4 0x0CA0DDB0, 0xC427BC82, 0x00004000, 0x00000000 data4 0xF13D8CAB, 0xC0AECD57, 0x00004000, 0x00000000 data4 0x71ECE6B1, 0xBD606C38, 0x00004000, 0x00000000 data4 0xA44C4929, 0xBA3A0A96, 0x00004000, 0x00000000 data4 0xE5CCCEC1, 0xB7394F6F, 0x00004000, 0x00000000 data4 0x9637D8BC, 0xB45C1203, 0x00004000, 0x00000000 data4 0x92CB051B, 0xB1A05528, 0x00004000, 0x00000000 data4 0x6BA2FFD0, 0xAF04432B, 0x00004000, 0x00000000 data4 0x7221235F, 0xAC862A23, 0x00004000, 0x00000000 data4 0x5F00A9D1, 0xAA2478AF, 0x00004000, 0x00000000 data4 0x81E082BF, 0xA7DDBB0C, 0x00004000, 0x00000000 data4 0x45684FEE, 0xA5B0987D, 0x00004000, 0x00000000 data4 0x627A8F53, 0xA39BD0F5, 0x00004000, 0x00000000 data4 0x6EC5C8B0, 0xA19E3B03, 0x00004000, 0x00000000 data4 0x91CD7C66, 0x9FB6C1F0, 0x00004000, 0x00000000 data4 0x1FA3DF8A, 0x9DE46410, 0x00004000, 0x00000000 data4 0xA8F6B888, 0x9C263139, 0x00004000, 0x00000000 data4 0xC27B0450, 0x9A7B4968, 0x00004000, 0x00000000 data4 0x5EE614EE, 0x98E2DB7E, 0x00004000, 0x00000000 // // Entries SC_inv in Swapped IEEE format (extended) // Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64) // data4 0x13B2B5BA, 0x969F335C, 0x00004000, 0x00000000 data4 0xD4C0F548, 0x93D446D9, 0x00004000, 0x00000000 data4 0x61B798AF, 0x9147094F, 0x00004000, 0x00000000 data4 0x758787AC, 0x8EF317CC, 0x00004000, 0x00000000 data4 0xB99EEFDB, 0x8CD498B3, 0x00004000, 0x00000000 data4 0xDFF8BC37, 0x8AE82A7D, 0x00004000, 0x00000000 data4 0xE3C55D42, 0x892AD546, 0x00004000, 0x00000000 data4 0xD15573C1, 0x8799FEA9, 0x00004000, 0x00000000 data4 0x435A4B4C, 0x86335F88, 0x00004000, 0x00000000 data4 0x3E93A87B, 0x84F4FB6E, 0x00004000, 0x00000000 data4 0x80A382FB, 0x83DD1952, 0x00004000, 0x00000000 data4 0xA4CB8C9E, 0x82EA3D7F, 0x00004000, 0x00000000 data4 0x6861D0A8, 0x821B247C, 0x00004000, 0x00000000 data4 0x63E8D244, 0x816EBED1, 0x00004000, 0x00000000 data4 0x27E4CFC6, 0x80E42D91, 0x00004000, 0x00000000 data4 0x28E64AFD, 0x807ABF8D, 0x00004000, 0x00000000 data4 0x863B4FD8, 0x8031EF26, 0x00004000, 0x00000000 data4 0xAE8C11FD, 0x800960AD, 0x00004000, 0x00000000 data4 0x5FDBEC21, 0x8000E147, 0x00004000, 0x00000000 data4 0xA07791FA, 0x80186650, 0x00004000, 0x00000000 Arg = f8 Result = f8 fp_tmp = f9 U_2 = f10 rsq = f11 C_hi = f12 C_lo = f13 T_hi = f14 T_lo = f15 N_0 = f32 d_1 = f33 MPI_BY_4 = f34 tail = f35 tanx = f36 Cx = f37 Sx = f38 sgn_r = f39 CORR = f40 P = f41 D = f42 ArgPrime = f43 P_0 = f44 P2_1 = f45 P2_2 = f46 P2_3 = f47 P1_1 = f45 P1_2 = f46 P1_3 = f47 P1_4 = f48 P1_5 = f49 P1_6 = f50 P1_7 = f51 P1_8 = f52 P1_9 = f53 TWO_TO_63 = f54 NEGTWO_TO_63 = f55 x = f56 xsq = f57 Tx = f58 Tx1 = f59 Set = f60 poly1 = f61 poly2 = f62 Poly = f63 Poly1 = f64 Poly2 = f65 r_to_the_8 = f66 B = f67 SC_inv = f68 Pos_r = f69 N_0_fix = f70 PI_BY_4 = f71 NEGTWO_TO_NEG2 = f72 TWO_TO_24 = f73 TWO_TO_NEG14 = f74 TWO_TO_NEG33 = f75 NEGTWO_TO_24 = f76 NEGTWO_TO_NEG14 = f76 NEGTWO_TO_NEG33 = f77 two_by_PI = f78 N = f79 N_fix = f80 P_1 = f81 P_2 = f82 P_3 = f83 s_val = f84 w = f85 c = f86 r = f87 Z = f88 A = f89 a = f90 t = f91 U_1 = f92 d_2 = f93 TWO_TO_NEG2 = f94 Q1_1 = f95 Q1_2 = f96 Q1_3 = f97 Q1_4 = f98 Q1_5 = f99 Q1_6 = f100 Q1_7 = f101 Q1_8 = f102 S_hi = f103 S_lo = f104 V_hi = f105 V_lo = f106 U_hi = f107 U_lo = f108 U_hiabs = f109 V_hiabs = f110 V = f111 Inv_P_0 = f112 GR_SAVE_B0 = r33 GR_SAVE_GP = r34 GR_SAVE_PFS = r35 delta1 = r36 table_ptr1 = r37 table_ptr2 = r38 i_0 = r39 i_1 = r40 N_fix_gr = r41 N_inc = r42 exp_Arg = r43 exp_r = r44 sig_r = r45 lookup = r46 table_offset = r47 Create_B = r48 gr_tmp = r49 GR_Parameter_X = r49 GR_Parameter_r = r50 .global __libm_tan .section .text .proc __libm_tan __libm_tan: { .mfi alloc r32 = ar.pfs, 0,17,2,0 (p0) fclass.m.unc p6,p0 = Arg, 0x1E7 addl gr_tmp = -1,r0 } ;; { .mfi nop.m 999 (p0) fclass.nm.unc p7,p0 = Arg, 0x1FF nop.i 999 } ;; { .mfi (p0) addl table_ptr1 = @ltoff(TAN_BASE_CONSTANTS), gp nop.f 999 nop.i 999 } ;; { .mmi ld8 table_ptr1 = [table_ptr1] setf.sig fp_tmp = gr_tmp // Make a constant so fmpy produces inexact nop.i 999 } ;; // // Check for NatVals, Infs , NaNs, and Zeros // Check for everything - if false, then must be pseudo-zero // or pseudo-nan. // Local table pointer // { .mbb (p0) add table_ptr2 = 96, table_ptr1 (p6) br.cond.spnt __libm_TAN_SPECIAL (p7) br.cond.spnt __libm_TAN_SPECIAL ;; } // // Point to Inv_P_0 // Branch out to deal with unsupporteds and special values. // { .mmf (p0) ldfs TWO_TO_24 = [table_ptr1],4 (p0) ldfs TWO_TO_63 = [table_ptr2],4 // // Load -2**24, load -2**63. // (p0) fcmp.eq.s0 p0, p6 = Arg, f1 ;; } { .mfi (p0) ldfs NEGTWO_TO_63 = [table_ptr2],12 (p0) fnorm.s1 Arg = Arg nop.i 999 } // // Load 2**24, Load 2**63. // { .mmi (p0) ldfs NEGTWO_TO_24 = [table_ptr1],12 ;; // // Do fcmp to generate Denormal exception // - can't do FNORM (will generate Underflow when U is unmasked!) // Normalize input argument. // (p0) ldfe two_by_PI = [table_ptr1],16 nop.i 999 } { .mmi (p0) ldfe Inv_P_0 = [table_ptr2],16 ;; (p0) ldfe d_1 = [table_ptr2],16 nop.i 999 } // // Decide about the paths to take: // PR_1 and PR_3 set if -2**24 < Arg < 2**24 - CASE 1 OR 2 // OTHERWISE - CASE 3 OR 4 // Load inverse of P_0 . // Set PR_6 if Arg <= -2**63 // Are there any Infs, NaNs, or zeros? // { .mmi (p0) ldfe P_0 = [table_ptr1],16 ;; (p0) ldfe d_2 = [table_ptr2],16 nop.i 999 } // // Set PR_8 if Arg <= -2**24 // Set PR_6 if Arg >= 2**63 // { .mmi (p0) ldfe P_1 = [table_ptr1],16 ;; (p0) ldfe PI_BY_4 = [table_ptr2],16 nop.i 999 } // // Set PR_8 if Arg >= 2**24 // { .mmi (p0) ldfe P_2 = [table_ptr1],16 ;; (p0) ldfe MPI_BY_4 = [table_ptr2],16 nop.i 999 } // // Load P_2 and PI_BY_4 // { .mfi (p0) ldfe P_3 = [table_ptr1],16 nop.f 999 nop.i 999 ;; } { .mfi nop.m 999 (p0) fcmp.le.unc.s1 p6,p7 = Arg,NEGTWO_TO_63 nop.i 999 } { .mfi nop.m 999 (p0) fcmp.le.unc.s1 p8,p9 = Arg,NEGTWO_TO_24 nop.i 999 ;; } { .mfi nop.m 999 (p7) fcmp.ge.s1 p6,p0 = Arg,TWO_TO_63 nop.i 999 } { .mfi nop.m 999 (p9) fcmp.ge.s1 p8,p0 = Arg,TWO_TO_24 nop.i 999 ;; } { .mib nop.m 999 nop.i 999 // // Load P_3 and -PI_BY_4 // (p6) br.cond.spnt TAN_ARG_TOO_LARGE ;; } { .mib nop.m 999 nop.i 999 // // Load 2**(-2). // Load -2**(-2). // Branch out if we have a special argument. // Branch out if the magnitude of the input argument is too large // - do this branch before the next. // (p8) br.cond.spnt TAN_LARGER_ARG ;; } // // Branch to Cases 3 or 4 if Arg <= -2**24 or Arg >= 2**24 // { .mfi (p0) ldfs TWO_TO_NEG2 = [table_ptr2],4 // ARGUMENT REDUCTION CODE - CASE 1 and 2 // Load 2**(-2). // Load -2**(-2). (p0) fmpy.s1 N = Arg,two_by_PI nop.i 999 ;; } { .mfi (p0) ldfs NEGTWO_TO_NEG2 = [table_ptr2],12 // // N = Arg * 2/pi // (p0) fcmp.lt.unc.s1 p8,p9= Arg,PI_BY_4 nop.i 999 ;; } { .mfi nop.m 999 // // if Arg < pi/4, set PR_8. // (p8) fcmp.gt.s1 p8,p9= Arg,MPI_BY_4 nop.i 999 ;; } // // Case 1: Is |r| < 2**(-2). // Arg is the same as r in this case. // r = Arg // c = 0 // { .mfi (p8) mov N_fix_gr = r0 // // if Arg > -pi/4, reset PR_8. // Select the case when |Arg| < pi/4 - set PR[8] = true. // Else Select the case when |Arg| >= pi/4 - set PR[9] = true. // (p0) fcvt.fx.s1 N_fix = N nop.i 999 ;; } { .mfi nop.m 999 // // Grab the integer part of N . // (p8) mov r = Arg nop.i 999 } { .mfi nop.m 999 (p8) mov c = f0 nop.i 999 ;; } { .mfi nop.m 999 (p8) fcmp.lt.unc.s1 p10, p11 = Arg, TWO_TO_NEG2 nop.i 999 ;; } { .mfi nop.m 999 (p10) fcmp.gt.s1 p10,p0 = Arg, NEGTWO_TO_NEG2 nop.i 999 ;; } { .mfi nop.m 999 // // Case 2: Place integer part of N in GP register. // (p9) fcvt.xf N = N_fix nop.i 999 ;; } { .mib (p9) getf.sig N_fix_gr = N_fix nop.i 999 // // Case 2: Convert integer N_fix back to normalized floating-point value. // (p10) br.cond.spnt TAN_SMALL_R ;; } { .mib nop.m 999 nop.i 999 (p8) br.cond.sptk TAN_NORMAL_R ;; } // // Case 1: PR_3 is only affected when PR_1 is set. // { .mmi (p9) ldfs TWO_TO_NEG33 = [table_ptr2], 4 ;; // // Case 2: Load 2**(-33). // (p9) ldfs NEGTWO_TO_NEG33 = [table_ptr2], 4 nop.i 999 ;; } { .mfi nop.m 999 // // Case 2: Load -2**(-33). // (p9) fnma.s1 s_val = N, P_1, Arg nop.i 999 } { .mfi nop.m 999 (p9) fmpy.s1 w = N, P_2 nop.i 999 ;; } { .mfi nop.m 999 // // Case 2: w = N * P_2 // Case 2: s_val = -N * P_1 + Arg // (p0) fcmp.lt.unc.s1 p9,p8 = s_val, TWO_TO_NEG33 nop.i 999 ;; } { .mfi nop.m 999 // // Decide between case_1 and case_2 reduce: // (p9) fcmp.gt.s1 p9, p8 = s_val, NEGTWO_TO_NEG33 nop.i 999 ;; } { .mfi nop.m 999 // // Case 1_reduce: s <= -2**(-33) or s >= 2**(-33) // Case 2_reduce: -2**(-33) < s < 2**(-33) // (p8) fsub.s1 r = s_val, w nop.i 999 } { .mfi nop.m 999 (p9) fmpy.s1 w = N, P_3 nop.i 999 ;; } { .mfi nop.m 999 (p9) fma.s1 U_1 = N, P_2, w nop.i 999 } { .mfi nop.m 999 // // Case 1_reduce: Is |r| < 2**(-2), if so set PR_10 // else set PR_11. // (p8) fsub.s1 c = s_val, r nop.i 999 ;; } { .mfi nop.m 999 // // Case 1_reduce: r = s + w (change sign) // Case 2_reduce: w = N * P_3 (change sign) // (p8) fcmp.lt.unc.s1 p10, p11 = r, TWO_TO_NEG2 nop.i 999 ;; } { .mfi nop.m 999 (p10) fcmp.gt.s1 p10, p11 = r, NEGTWO_TO_NEG2 nop.i 999 ;; } { .mfi nop.m 999 (p9) fsub.s1 r = s_val, U_1 nop.i 999 } { .mfi nop.m 999 // // Case 1_reduce: c is complete here. // c = c + w (w has not been negated.) // Case 2_reduce: r is complete here - continue to calculate c . // r = s - U_1 // (p9) fms.s1 U_2 = N, P_2, U_1 nop.i 999 ;; } { .mfi nop.m 999 // // Case 1_reduce: c = s - r // Case 2_reduce: U_1 = N * P_2 + w // (p8) fsub.s1 c = c, w nop.i 999 ;; } { .mfi nop.m 999 (p9) fsub.s1 s_val = s_val, r nop.i 999 } { .mfb nop.m 999 // // Case 2_reduce: // U_2 = N * P_2 - U_1 // Not needed until later. // (p9) fadd.s1 U_2 = U_2, w // // Case 2_reduce: // s = s - r // U_2 = U_2 + w // (p10) br.cond.spnt TAN_SMALL_R ;; } { .mib nop.m 999 nop.i 999 (p11) br.cond.sptk TAN_NORMAL_R ;; } { .mii nop.m 999 // // Case 2_reduce: // c = c - U_2 // c is complete here // Argument reduction ends here. // (p9) extr.u i_1 = N_fix_gr, 0, 1 ;; (p9) cmp.eq.unc p11, p12 = 0x0000,i_1 ;; } { .mfi nop.m 999 // // Is i_1 even or odd? // if i_1 == 0, set p11, else set p12. // (p11) fmpy.s1 rsq = r, Z nop.i 999 ;; } { .mfi nop.m 999 (p12) frcpa.s1 S_hi,p0 = f1, r nop.i 999 } // // Case 1: Branch to SMALL_R or NORMAL_R. // Case 1 is done now. // { .mfi (p9) addl table_ptr1 = @ltoff(TAN_BASE_CONSTANTS), gp (p9) fsub.s1 c = s_val, U_1 nop.i 999 ;; } ;; { .mmi (p9) ld8 table_ptr1 = [table_ptr1] nop.m 999 nop.i 999 } ;; { .mmi (p9) add table_ptr1 = 224, table_ptr1 ;; (p9) ldfe P1_1 = [table_ptr1],144 nop.i 999 ;; } // // Get [i_1] - lsb of N_fix_gr . // Load P1_1 and point to Q1_1 . // { .mfi (p9) ldfe Q1_1 = [table_ptr1] , 0 // // N even: rsq = r * Z // N odd: S_hi = frcpa(r) // (p12) fmerge.ns S_hi = S_hi, S_hi nop.i 999 } { .mfi nop.m 999 // // Case 2_reduce: // c = s - U_1 // (p9) fsub.s1 c = c, U_2 nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 poly1 = S_hi, r, f1 nop.i 999 ;; } { .mfi nop.m 999 // // N odd: Change sign of S_hi // (p11) fmpy.s1 rsq = rsq, P1_1 nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 S_hi = S_hi, poly1, S_hi nop.i 999 ;; } { .mfi nop.m 999 // // N even: rsq = rsq * P1_1 // N odd: poly1 = 1.0 + S_hi * r 16 bits partial account for necessary // (p11) fma.s1 Result = r, rsq, c nop.i 999 ;; } { .mfi nop.m 999 // // N even: Result = c + r * rsq // N odd: S_hi = S_hi + S_hi*poly1 16 bits account for necessary // (p12) fma.s1 poly1 = S_hi, r, f1 nop.i 999 ;; } { .mfi nop.m 999 // // N even: Result = Result + r // N odd: poly1 = 1.0 + S_hi * r 32 bits partial // (p11) fadd.s0 Result = r, Result nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 S_hi = S_hi, poly1, S_hi nop.i 999 ;; } { .mfi nop.m 999 // // N even: Result1 = Result + r // N odd: S_hi = S_hi * poly1 + S_hi 32 bits // (p12) fma.s1 poly1 = S_hi, r, f1 nop.i 999 ;; } { .mfi nop.m 999 // // N odd: poly1 = S_hi * r + 1.0 64 bits partial // (p12) fma.s1 S_hi = S_hi, poly1, S_hi nop.i 999 ;; } { .mfi nop.m 999 // // N odd: poly1 = S_hi * poly + 1.0 64 bits // (p12) fma.s1 poly1 = S_hi, r, f1 nop.i 999 ;; } { .mfi nop.m 999 // // N odd: poly1 = S_hi * r + 1.0 // (p12) fma.s1 poly1 = S_hi, c, poly1 nop.i 999 ;; } { .mfi nop.m 999 // // N odd: poly1 = S_hi * c + poly1 // (p12) fmpy.s1 S_lo = S_hi, poly1 nop.i 999 ;; } { .mfi nop.m 999 // // N odd: S_lo = S_hi * poly1 // (p12) fma.s1 S_lo = Q1_1, r, S_lo nop.i 999 } { .mfi nop.m 999 // // N odd: Result = S_hi + S_lo // (p0) fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact nop.i 999 ;; } { .mfb nop.m 999 // // N odd: S_lo = S_lo + Q1_1 * r // (p12) fadd.s0 Result = S_hi, S_lo (p0) br.ret.sptk b0 ;; } TAN_LARGER_ARG: { .mmf (p0) addl table_ptr1 = @ltoff(TAN_BASE_CONSTANTS), gp nop.m 999 (p0) fmpy.s1 N_0 = Arg, Inv_P_0 } ;; // // ARGUMENT REDUCTION CODE - CASE 3 and 4 // // // Adjust table_ptr1 to beginning of table. // N_0 = Arg * Inv_P_0 // { .mmi (p0) ld8 table_ptr1 = [table_ptr1] nop.m 999 nop.i 999 } ;; { .mmi (p0) add table_ptr1 = 8, table_ptr1 ;; // // Point to 2*-14 // (p0) ldfs TWO_TO_NEG14 = [table_ptr1], 4 nop.i 999 ;; } // // Load 2**(-14). // { .mmi (p0) ldfs NEGTWO_TO_NEG14 = [table_ptr1], 180 ;; // // N_0_fix = integer part of N_0 . // Adjust table_ptr1 to beginning of table. // (p0) ldfs TWO_TO_NEG2 = [table_ptr1], 4 nop.i 999 ;; } // // Make N_0 the integer part. // { .mfi (p0) ldfs NEGTWO_TO_NEG2 = [table_ptr1] // // Load -2**(-14). // (p0) fcvt.fx.s1 N_0_fix = N_0 nop.i 999 ;; } { .mfi nop.m 999 (p0) fcvt.xf N_0 = N_0_fix nop.i 999 ;; } { .mfi nop.m 999 (p0) fnma.s1 ArgPrime = N_0, P_0, Arg nop.i 999 } { .mfi nop.m 999 (p0) fmpy.s1 w = N_0, d_1 nop.i 999 ;; } { .mfi nop.m 999 // // ArgPrime = -N_0 * P_0 + Arg // w = N_0 * d_1 // (p0) fmpy.s1 N = ArgPrime, two_by_PI nop.i 999 ;; } { .mfi nop.m 999 // // N = ArgPrime * 2/pi // (p0) fcvt.fx.s1 N_fix = N nop.i 999 ;; } { .mfi nop.m 999 // // N_fix is the integer part. // (p0) fcvt.xf N = N_fix nop.i 999 ;; } { .mfi (p0) getf.sig N_fix_gr = N_fix nop.f 999 nop.i 999 ;; } { .mfi nop.m 999 // // N is the integer part of the reduced-reduced argument. // Put the integer in a GP register. // (p0) fnma.s1 s_val = N, P_1, ArgPrime nop.i 999 } { .mfi nop.m 999 (p0) fnma.s1 w = N, P_2, w nop.i 999 ;; } { .mfi nop.m 999 // // s_val = -N*P_1 + ArgPrime // w = -N*P_2 + w // (p0) fcmp.lt.unc.s1 p11, p10 = s_val, TWO_TO_NEG14 nop.i 999 ;; } { .mfi nop.m 999 (p11) fcmp.gt.s1 p11, p10 = s_val, NEGTWO_TO_NEG14 nop.i 999 ;; } { .mfi nop.m 999 // // Case 3: r = s_val + w (Z complete) // Case 4: U_hi = N_0 * d_1 // (p10) fmpy.s1 V_hi = N, P_2 nop.i 999 } { .mfi nop.m 999 (p11) fmpy.s1 U_hi = N_0, d_1 nop.i 999 ;; } { .mfi nop.m 999 // // Case 3: r = s_val + w (Z complete) // Case 4: U_hi = N_0 * d_1 // (p11) fmpy.s1 V_hi = N, P_2 nop.i 999 } { .mfi nop.m 999 (p11) fmpy.s1 U_hi = N_0, d_1 nop.i 999 ;; } { .mfi nop.m 999 // // Decide between case 3 and 4: // Case 3: s <= -2**(-14) or s >= 2**(-14) // Case 4: -2**(-14) < s < 2**(-14) // (p10) fadd.s1 r = s_val, w nop.i 999 } { .mfi nop.m 999 (p11) fmpy.s1 w = N, P_3 nop.i 999 ;; } { .mfi nop.m 999 // // Case 4: We need abs of both U_hi and V_hi - dont // worry about switched sign of V_hi . // (p11) fsub.s1 A = U_hi, V_hi nop.i 999 } { .mfi nop.m 999 // // Case 4: A = U_hi + V_hi // Note: Worry about switched sign of V_hi, so subtract instead of add. // (p11) fnma.s1 V_lo = N, P_2, V_hi nop.i 999 ;; } { .mfi nop.m 999 (p11) fms.s1 U_lo = N_0, d_1, U_hi nop.i 999 ;; } { .mfi nop.m 999 (p11) fabs V_hiabs = V_hi nop.i 999 } { .mfi nop.m 999 // // Case 4: V_hi = N * P_2 // w = N * P_3 // Note the product does not include the (-) as in the writeup // so (-) missing for V_hi and w . (p10) fadd.s1 r = s_val, w nop.i 999 ;; } { .mfi nop.m 999 // // Case 3: c = s_val - r // Case 4: U_lo = N_0 * d_1 - U_hi // (p11) fabs U_hiabs = U_hi nop.i 999 } { .mfi nop.m 999 (p11) fmpy.s1 w = N, P_3 nop.i 999 ;; } { .mfi nop.m 999 // // Case 4: Set P_12 if U_hiabs >= V_hiabs // (p11) fadd.s1 C_hi = s_val, A nop.i 999 ;; } { .mfi nop.m 999 // // Case 4: C_hi = s_val + A // (p11) fadd.s1 t = U_lo, V_lo nop.i 999 ;; } { .mfi nop.m 999 // // Case 3: Is |r| < 2**(-2), if so set PR_7 // else set PR_8. // Case 3: If PR_7 is set, prepare to branch to Small_R. // Case 3: If PR_8 is set, prepare to branch to Normal_R. // (p10) fsub.s1 c = s_val, r nop.i 999 ;; } { .mfi nop.m 999 // // Case 3: c = (s - r) + w (c complete) // (p11) fcmp.ge.unc.s1 p12, p13 = U_hiabs, V_hiabs nop.i 999 } { .mfi nop.m 999 (p11) fms.s1 w = N_0, d_2, w nop.i 999 ;; } { .mfi nop.m 999 // // Case 4: V_hi = N * P_2 // w = N * P_3 // Note the product does not include the (-) as in the writeup // so (-) missing for V_hi and w . // (p10) fcmp.lt.unc.s1 p14, p15 = r, TWO_TO_NEG2 nop.i 999 ;; } { .mfi nop.m 999 (p14) fcmp.gt.s1 p14, p15 = r, NEGTWO_TO_NEG2 nop.i 999 ;; } { .mfb nop.m 999 // // Case 4: V_lo = -N * P_2 - V_hi (U_hi is in place of V_hi in writeup) // Note: the (-) is still missing for V_hi . // Case 4: w = w + N_0 * d_2 // Note: the (-) is now incorporated in w . // (p10) fadd.s1 c = c, w // // Case 4: t = U_lo + V_lo // Note: remember V_lo should be (-), subtract instead of add. NO // (p14) br.cond.spnt TAN_SMALL_R ;; } { .mib nop.m 999 nop.i 999 (p15) br.cond.spnt TAN_NORMAL_R ;; } { .mfi nop.m 999 // // Case 3: Vector off when |r| < 2**(-2). Recall that PR_3 will be true. // The remaining stuff is for Case 4. // (p12) fsub.s1 a = U_hi, A (p11) extr.u i_1 = N_fix_gr, 0, 1 ;; } { .mfi nop.m 999 // // Case 4: C_lo = s_val - C_hi // (p11) fadd.s1 t = t, w nop.i 999 } { .mfi nop.m 999 (p13) fadd.s1 a = V_hi, A nop.i 999 ;; } // // Case 4: a = U_hi - A // a = V_hi - A (do an add to account for missing (-) on V_hi // { .mfi (p11) addl table_ptr1 = @ltoff(TAN_BASE_CONSTANTS), gp (p11) fsub.s1 C_lo = s_val, C_hi nop.i 999 } ;; { .mmi (p11) ld8 table_ptr1 = [table_ptr1] nop.m 999 nop.i 999 } ;; // // Case 4: a = (U_hi - A) + V_hi // a = (V_hi - A) + U_hi // In each case account for negative missing form V_hi . // // // Case 4: C_lo = (s_val - C_hi) + A // { .mmi (p11) add table_ptr1 = 224, table_ptr1 ;; (p11) ldfe P1_1 = [table_ptr1], 16 nop.i 999 ;; } { .mfi (p11) ldfe P1_2 = [table_ptr1], 128 // // Case 4: w = U_lo + V_lo + w // (p12) fsub.s1 a = a, V_hi nop.i 999 ;; } // // Case 4: r = C_hi + C_lo // { .mfi (p11) ldfe Q1_1 = [table_ptr1], 16 (p11) fadd.s1 C_lo = C_lo, A nop.i 999 ;; } // // Case 4: c = C_hi - r // Get [i_1] - lsb of N_fix_gr. // { .mfi (p11) ldfe Q1_2 = [table_ptr1], 16 nop.f 999 nop.i 999 ;; } { .mfi nop.m 999 (p13) fsub.s1 a = U_hi, a nop.i 999 ;; } { .mfi nop.m 999 (p11) fadd.s1 t = t, a nop.i 999 ;; } { .mfi nop.m 999 // // Case 4: t = t + a // (p11) fadd.s1 C_lo = C_lo, t nop.i 999 ;; } { .mfi nop.m 999 // // Case 4: C_lo = C_lo + t // (p11) fadd.s1 r = C_hi, C_lo nop.i 999 ;; } { .mfi nop.m 999 (p11) fsub.s1 c = C_hi, r nop.i 999 } { .mfi nop.m 999 // // Case 4: c = c + C_lo finished. // Is i_1 even or odd? // if i_1 == 0, set PR_4, else set PR_5. // // r and c have been computed. // We known whether this is the sine or cosine routine. // Make sure ftz mode is set - should be automatic when using wre (p0) fmpy.s1 rsq = r, r nop.i 999 ;; } { .mfi nop.m 999 (p11) fadd.s1 c = c , C_lo (p11) cmp.eq.unc p11, p12 = 0x0000, i_1 ;; } { .mfi nop.m 999 (p12) frcpa.s1 S_hi, p0 = f1, r nop.i 999 } { .mfi nop.m 999 // // N odd: Change sign of S_hi // (p11) fma.s1 Result = rsq, P1_2, P1_1 nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 P = rsq, Q1_2, Q1_1 nop.i 999 } { .mfi nop.m 999 // // N odd: Result = S_hi + S_lo (User supplied rounding mode for C1) // (p0) fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact nop.i 999 ;; } { .mfi nop.m 999 // // N even: rsq = r * r // N odd: S_hi = frcpa(r) // (p12) fmerge.ns S_hi = S_hi, S_hi nop.i 999 } { .mfi nop.m 999 // // N even: rsq = rsq * P1_2 + P1_1 // N odd: poly1 = 1.0 + S_hi * r 16 bits partial account for necessary // (p11) fmpy.s1 Result = rsq, Result nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 poly1 = S_hi, r,f1 nop.i 999 } { .mfi nop.m 999 // // N even: Result = Result * rsq // N odd: S_hi = S_hi + S_hi*poly1 16 bits account for necessary // (p11) fma.s1 Result = r, Result, c nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 S_hi = S_hi, poly1, S_hi nop.i 999 } { .mfi nop.m 999 // // N odd: S_hi = S_hi * poly1 + S_hi 32 bits // (p11) fadd.s0 Result= r, Result nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 poly1 = S_hi, r, f1 nop.i 999 ;; } { .mfi nop.m 999 // // N even: Result = Result * r + c // N odd: poly1 = 1.0 + S_hi * r 32 bits partial // (p12) fma.s1 S_hi = S_hi, poly1, S_hi nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 poly1 = S_hi, r, f1 nop.i 999 ;; } { .mfi nop.m 999 // // N even: Result1 = Result + r (Rounding mode S0) // N odd: poly1 = S_hi * r + 1.0 64 bits partial // (p12) fma.s1 S_hi = S_hi, poly1, S_hi nop.i 999 ;; } { .mfi nop.m 999 // // N odd: poly1 = S_hi * poly + S_hi 64 bits // (p12) fma.s1 poly1 = S_hi, r, f1 nop.i 999 ;; } { .mfi nop.m 999 // // N odd: poly1 = S_hi * r + 1.0 // (p12) fma.s1 poly1 = S_hi, c, poly1 nop.i 999 ;; } { .mfi nop.m 999 // // N odd: poly1 = S_hi * c + poly1 // (p12) fmpy.s1 S_lo = S_hi, poly1 nop.i 999 ;; } { .mfi nop.m 999 // // N odd: S_lo = S_hi * poly1 // (p12) fma.s1 S_lo = P, r, S_lo nop.i 999 ;; } { .mfb nop.m 999 // // N odd: S_lo = S_lo + r * P // (p12) fadd.s0 Result = S_hi, S_lo (p0) br.ret.sptk b0 ;; } TAN_SMALL_R: { .mii nop.m 999 (p0) extr.u i_1 = N_fix_gr, 0, 1 ;; (p0) cmp.eq.unc p11, p12 = 0x0000, i_1 } { .mfi nop.m 999 (p0) fmpy.s1 rsq = r, r nop.i 999 ;; } { .mfi nop.m 999 (p12) frcpa.s1 S_hi, p0 = f1, r nop.i 999 } { .mfi (p0) addl table_ptr1 = @ltoff(TAN_BASE_CONSTANTS), gp nop.f 999 nop.i 999 } ;; { .mmi (p0) ld8 table_ptr1 = [table_ptr1] nop.m 999 nop.i 999 } ;; // ***************************************************************** // ***************************************************************** // ***************************************************************** { .mmi (p0) add table_ptr1 = 224, table_ptr1 ;; (p0) ldfe P1_1 = [table_ptr1], 16 nop.i 999 ;; } // r and c have been computed. // We known whether this is the sine or cosine routine. // Make sure ftz mode is set - should be automatic when using wre // |r| < 2**(-2) { .mfi (p0) ldfe P1_2 = [table_ptr1], 16 (p11) fmpy.s1 r_to_the_8 = rsq, rsq nop.i 999 ;; } // // Set table_ptr1 to beginning of constant table. // Get [i_1] - lsb of N_fix_gr. // { .mfi (p0) ldfe P1_3 = [table_ptr1], 96 // // N even: rsq = r * r // N odd: S_hi = frcpa(r) // (p12) fmerge.ns S_hi = S_hi, S_hi nop.i 999 ;; } // // Is i_1 even or odd? // if i_1 == 0, set PR_11. // if i_1 != 0, set PR_12. // { .mfi (p11) ldfe P1_9 = [table_ptr1], -16 // // N even: Poly2 = P1_7 + Poly2 * rsq // N odd: poly2 = Q1_5 + poly2 * rsq // (p11) fadd.s1 CORR = rsq, f1 nop.i 999 ;; } { .mmi (p11) ldfe P1_8 = [table_ptr1], -16 ;; // // N even: Poly1 = P1_2 + P1_3 * rsq // N odd: poly1 = 1.0 + S_hi * r // 16 bits partial account for necessary (-1) // (p11) ldfe P1_7 = [table_ptr1], -16 nop.i 999 ;; } // // N even: Poly1 = P1_1 + Poly1 * rsq // N odd: S_hi = S_hi + S_hi * poly1) 16 bits account for necessary // { .mfi (p11) ldfe P1_6 = [table_ptr1], -16 // // N even: Poly2 = P1_5 + Poly2 * rsq // N odd: poly2 = Q1_3 + poly2 * rsq // (p11) fmpy.s1 r_to_the_8 = r_to_the_8, r_to_the_8 nop.i 999 ;; } // // N even: Poly1 = Poly1 * rsq // N odd: poly1 = 1.0 + S_hi * r 32 bits partial // { .mfi (p11) ldfe P1_5 = [table_ptr1], -16 (p12) fma.s1 poly1 = S_hi, r, f1 nop.i 999 ;; } // // N even: CORR = CORR * c // N odd: S_hi = S_hi * poly1 + S_hi 32 bits // // // N even: Poly2 = P1_6 + Poly2 * rsq // N odd: poly2 = Q1_4 + poly2 * rsq // { .mmf (p0) addl table_ptr2 = @ltoff(TAN_BASE_CONSTANTS), gp (p11) ldfe P1_4 = [table_ptr1], -16 (p11) fmpy.s1 CORR = CORR, c } ;; { .mmi (p0) ld8 table_ptr2 = [table_ptr2] nop.m 999 nop.i 999 } ;; { .mii (p0) add table_ptr2 = 464, table_ptr2 nop.i 999 ;; nop.i 999 } { .mfi nop.m 999 (p11) fma.s1 Poly1 = P1_3, rsq, P1_2 nop.i 999 ;; } { .mfi (p0) ldfe Q1_7 = [table_ptr2], -16 (p12) fma.s1 S_hi = S_hi, poly1, S_hi nop.i 999 ;; } { .mfi (p0) ldfe Q1_6 = [table_ptr2], -16 (p11) fma.s1 Poly2 = P1_9, rsq, P1_8 nop.i 999 ;; } { .mmi (p0) ldfe Q1_5 = [table_ptr2], -16 ;; (p12) ldfe Q1_4 = [table_ptr2], -16 nop.i 999 ;; } { .mfi (p12) ldfe Q1_3 = [table_ptr2], -16 // // N even: Poly2 = P1_8 + P1_9 * rsq // N odd: poly2 = Q1_6 + Q1_7 * rsq // (p11) fma.s1 Poly1 = Poly1, rsq, P1_1 nop.i 999 ;; } { .mfi (p12) ldfe Q1_2 = [table_ptr2], -16 (p12) fma.s1 poly1 = S_hi, r, f1 nop.i 999 ;; } { .mfi (p12) ldfe Q1_1 = [table_ptr2], -16 (p11) fma.s1 Poly2 = Poly2, rsq, P1_7 nop.i 999 ;; } { .mfi nop.m 999 // // N even: CORR = rsq + 1 // N even: r_to_the_8 = rsq * rsq // (p11) fmpy.s1 Poly1 = Poly1, rsq nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 S_hi = S_hi, poly1, S_hi nop.i 999 } { .mfi nop.m 999 (p12) fma.s1 poly2 = Q1_7, rsq, Q1_6 nop.i 999 ;; } { .mfi nop.m 999 (p11) fma.s1 Poly2 = Poly2, rsq, P1_6 nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 poly1 = S_hi, r, f1 nop.i 999 } { .mfi nop.m 999 (p12) fma.s1 poly2 = poly2, rsq, Q1_5 nop.i 999 ;; } { .mfi nop.m 999 (p11) fma.s1 Poly2= Poly2, rsq, P1_5 nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 S_hi = S_hi, poly1, S_hi nop.i 999 } { .mfi nop.m 999 (p12) fma.s1 poly2 = poly2, rsq, Q1_4 nop.i 999 ;; } { .mfi nop.m 999 // // N even: r_to_the_8 = r_to_the_8 * r_to_the_8 // N odd: poly1 = S_hi * r + 1.0 64 bits partial // (p11) fma.s1 Poly2 = Poly2, rsq, P1_4 nop.i 999 ;; } { .mfi nop.m 999 // // N even: Result = CORR + Poly * r // N odd: P = Q1_1 + poly2 * rsq // (p12) fma.s1 poly1 = S_hi, r, f1 nop.i 999 } { .mfi nop.m 999 (p12) fma.s1 poly2 = poly2, rsq, Q1_3 nop.i 999 ;; } { .mfi nop.m 999 // // N even: Poly2 = P1_4 + Poly2 * rsq // N odd: poly2 = Q1_2 + poly2 * rsq // (p11) fma.s1 Poly = Poly2, r_to_the_8, Poly1 nop.i 999 ;; } { .mfi nop.m 999 (p12) fma.s1 poly1 = S_hi, c, poly1 nop.i 999 } { .mfi nop.m 999 (p12) fma.s1 poly2 = poly2, rsq, Q1_2 nop.i 999 ;; } { .mfi nop.m 999 // // N even: Poly = Poly1 + Poly2 * r_to_the_8 // N odd: S_hi = S_hi * poly1 + S_hi 64 bits // (p11) fma.s1 Result = Poly, r, CORR nop.i 999 ;; } { .mfi nop.m 999 // // N even: Result = r + Result (User supplied rounding mode) // N odd: poly1 = S_hi * c + poly1 // (p12) fmpy.s1 S_lo = S_hi, poly1 nop.i 999 } { .mfi nop.m 999 (p12) fma.s1 P = poly2, rsq, Q1_1 nop.i 999 ;; } { .mfi nop.m 999 // // N odd: poly1 = S_hi * r + 1.0 // (p11) fadd.s0 Result = Result, r nop.i 999 ;; } { .mfi nop.m 999 // // N odd: S_lo = S_hi * poly1 // (p12) fma.s1 S_lo = Q1_1, c, S_lo nop.i 999 } { .mfi nop.m 999 // // N odd: Result = Result + S_hi (user supplied rounding mode) // (p0) fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact nop.i 999 ;; } { .mfi nop.m 999 // // N odd: S_lo = Q1_1 * c + S_lo // (p12) fma.s1 Result = P, r, S_lo nop.i 999 ;; } { .mfb nop.m 999 // // N odd: Result = S_lo + r * P // (p12) fadd.s0 Result = Result, S_hi (p0) br.ret.sptk b0 ;; } TAN_NORMAL_R: { .mfi (p0) getf.sig sig_r = r // ******************************************************************* // ******************************************************************* // ******************************************************************* // // r and c have been computed. // Make sure ftz mode is set - should be automatic when using wre // // // Get [i_1] - lsb of N_fix_gr alone. // (p0) fmerge.s Pos_r = f1, r (p0) extr.u i_1 = N_fix_gr, 0, 1 ;; } { .mfi nop.m 999 (p0) fmerge.s sgn_r = r, f1 (p0) cmp.eq.unc p11, p12 = 0x0000, i_1 ;; } { .mfi nop.m 999 nop.f 999 (p0) extr.u lookup = sig_r, 58, 5 } { .mlx nop.m 999 (p0) movl Create_B = 0x8200000000000000 ;; } { .mfi (p0) addl table_ptr1 = @ltoff(TAN_BASE_CONSTANTS), gp nop.f 999 (p0) dep Create_B = lookup, Create_B, 58, 5 } ;; // // Get [i_1] - lsb of N_fix_gr alone. // Pos_r = abs (r) // { .mmi ld8 table_ptr1 = [table_ptr1] nop.m 999 nop.i 999 } ;; { .mmi nop.m 999 (p0) setf.sig B = Create_B // // Set table_ptr1 and table_ptr2 to base address of // constant table. // (p0) add table_ptr1 = 480, table_ptr1 ;; } { .mmb nop.m 999 // // Is i_1 or i_0 == 0 ? // Create the constant 1 00000 1000000000000000000000... // (p0) ldfe P2_1 = [table_ptr1], 16 nop.b 999 } { .mmi nop.m 999 ;; (p0) getf.exp exp_r = Pos_r nop.i 999 } // // Get r's exponent // Get r's significand // { .mmi (p0) ldfe P2_2 = [table_ptr1], 16 ;; // // Get the 5 bits or r for the lookup. 1.xxxxx .... // from sig_r. // Grab lsb of exp of B // (p0) ldfe P2_3 = [table_ptr1], 16 nop.i 999 ;; } { .mii nop.m 999 (p0) andcm table_offset = 0x0001, exp_r ;; (p0) shl table_offset = table_offset, 9 ;; } { .mii nop.m 999 // // Deposit 0 00000 1000000000000000000000... on // 1 xxxxx yyyyyyyyyyyyyyyyyyyyyy..., // getting rid of the ys. // Is B = 2** -2 or B= 2** -1? If 2**-1, then // we want an offset of 512 for table addressing. // (p0) shladd table_offset = lookup, 4, table_offset ;; // // B = ........ 1xxxxx 1000000000000000000... // (p0) add table_ptr1 = table_ptr1, table_offset ;; } { .mmb nop.m 999 // // B = ........ 1xxxxx 1000000000000000000... // Convert B so it has the same exponent as Pos_r // (p0) ldfd T_hi = [table_ptr1], 8 nop.b 999 ;; } // // x = |r| - B // Load T_hi. // Load C_hi. // { .mmf (p0) addl table_ptr2 = @ltoff(TAN_BASE_CONSTANTS), gp (p0) ldfs T_lo = [table_ptr1] (p0) fmerge.se B = Pos_r, B } ;; { .mmi ld8 table_ptr2 = [table_ptr2] nop.m 999 nop.i 999 } ;; { .mii (p0) add table_ptr2 = 1360, table_ptr2 nop.i 999 ;; (p0) add table_ptr2 = table_ptr2, table_offset ;; } { .mfi (p0) ldfd C_hi = [table_ptr2], 8 (p0) fsub.s1 x = Pos_r, B nop.i 999 ;; } { .mii (p0) ldfs C_lo = [table_ptr2],255 nop.i 999 ;; // // xsq = x * x // N even: Tx = T_hi * x // Load T_lo. // Load C_lo - increment pointer to get SC_inv // - cant get all the way, do an add later. // (p0) add table_ptr2 = 569, table_ptr2 ;; } // // N even: Tx1 = Tx + 1 // N odd: Cx1 = 1 - Cx // { .mfi (p0) ldfe SC_inv = [table_ptr2], 0 nop.f 999 nop.i 999 ;; } { .mfi nop.m 999 (p0) fmpy.s1 xsq = x, x nop.i 999 } { .mfi nop.m 999 (p11) fmpy.s1 Tx = T_hi, x nop.i 999 ;; } { .mfi nop.m 999 (p12) fmpy.s1 Cx = C_hi, x nop.i 999 ;; } { .mfi nop.m 999 // // N odd: Cx = C_hi * x // (p0) fma.s1 P = P2_3, xsq, P2_2 nop.i 999 } { .mfi nop.m 999 // // N even and odd: P = P2_3 + P2_2 * xsq // (p11) fadd.s1 Tx1 = Tx, f1 nop.i 999 ;; } { .mfi nop.m 999 // // N even: D = C_hi - tanx // N odd: D = T_hi + tanx // (p11) fmpy.s1 CORR = SC_inv, T_hi nop.i 999 } { .mfi nop.m 999 (p0) fmpy.s1 Sx = SC_inv, x nop.i 999 ;; } { .mfi nop.m 999 (p12) fmpy.s1 CORR = SC_inv, C_hi nop.i 999 ;; } { .mfi nop.m 999 (p12) fsub.s1 V_hi = f1, Cx nop.i 999 ;; } { .mfi nop.m 999 (p0) fma.s1 P = P, xsq, P2_1 nop.i 999 } { .mfi nop.m 999 // // N even and odd: P = P2_1 + P * xsq // (p11) fma.s1 V_hi = Tx, Tx1, f1 nop.i 999 ;; } { .mfi nop.m 999 // // N even: Result = sgn_r * tail + T_hi (user rounding mode for C1) // N odd: Result = sgn_r * tail + C_hi (user rounding mode for C1) // (p0) fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact nop.i 999 ;; } { .mfi nop.m 999 (p0) fmpy.s1 CORR = CORR, c nop.i 999 ;; } { .mfi nop.m 999 (p12) fnma.s1 V_hi = Cx,V_hi,f1 nop.i 999 ;; } { .mfi nop.m 999 // // N even: V_hi = Tx * Tx1 + 1 // N odd: Cx1 = 1 - Cx * Cx1 // (p0) fmpy.s1 P = P, xsq nop.i 999 } { .mfi nop.m 999 // // N even and odd: P = P * xsq // (p11) fmpy.s1 V_hi = V_hi, T_hi nop.i 999 ;; } { .mfi nop.m 999 // // N even and odd: tail = P * tail + V_lo // (p11) fmpy.s1 T_hi = sgn_r, T_hi nop.i 999 ;; } { .mfi nop.m 999 (p0) fmpy.s1 CORR = CORR, sgn_r nop.i 999 ;; } { .mfi nop.m 999 (p12) fmpy.s1 V_hi = V_hi,C_hi nop.i 999 ;; } { .mfi nop.m 999 // // N even: V_hi = T_hi * V_hi // N odd: V_hi = C_hi * V_hi // (p0) fma.s1 tanx = P, x, x nop.i 999 } { .mfi nop.m 999 (p12) fnmpy.s1 C_hi = sgn_r, C_hi nop.i 999 ;; } { .mfi nop.m 999 // // N even: V_lo = 1 - V_hi + C_hi // N odd: V_lo = 1 - V_hi + T_hi // (p11) fadd.s1 CORR = CORR, T_lo nop.i 999 } { .mfi nop.m 999 (p12) fsub.s1 CORR = CORR, C_lo nop.i 999 ;; } { .mfi nop.m 999 // // N even and odd: tanx = x + x * P // N even and odd: Sx = SC_inv * x // (p11) fsub.s1 D = C_hi, tanx nop.i 999 } { .mfi nop.m 999 (p12) fadd.s1 D = T_hi, tanx nop.i 999 ;; } { .mfi nop.m 999 // // N odd: CORR = SC_inv * C_hi // N even: CORR = SC_inv * T_hi // (p0) fnma.s1 D = V_hi, D, f1 nop.i 999 ;; } { .mfi nop.m 999 // // N even and odd: D = 1 - V_hi * D // N even and odd: CORR = CORR * c // (p0) fma.s1 V_hi = V_hi, D, V_hi nop.i 999 ;; } { .mfi nop.m 999 // // N even and odd: V_hi = V_hi + V_hi * D // N even and odd: CORR = sgn_r * CORR // (p11) fnma.s1 V_lo = V_hi, C_hi, f1 nop.i 999 } { .mfi nop.m 999 (p12) fnma.s1 V_lo = V_hi, T_hi, f1 nop.i 999 ;; } { .mfi nop.m 999 // // N even: CORR = COOR + T_lo // N odd: CORR = CORR - C_lo // (p11) fma.s1 V_lo = tanx, V_hi, V_lo nop.i 999 } { .mfi nop.m 999 (p12) fnma.s1 V_lo = tanx, V_hi, V_lo nop.i 999 ;; } { .mfi nop.m 999 // // N even: V_lo = V_lo + V_hi * tanx // N odd: V_lo = V_lo - V_hi * tanx // (p11) fnma.s1 V_lo = C_lo, V_hi, V_lo nop.i 999 } { .mfi nop.m 999 (p12) fnma.s1 V_lo = T_lo, V_hi, V_lo nop.i 999 ;; } { .mfi nop.m 999 // // N even: V_lo = V_lo - V_hi * C_lo // N odd: V_lo = V_lo - V_hi * T_lo // (p0) fmpy.s1 V_lo = V_hi, V_lo nop.i 999 ;; } { .mfi nop.m 999 // // N even and odd: V_lo = V_lo * V_hi // (p0) fadd.s1 tail = V_hi, V_lo nop.i 999 ;; } { .mfi nop.m 999 // // N even and odd: tail = V_hi + V_lo // (p0) fma.s1 tail = tail, P, V_lo nop.i 999 ;; } { .mfi nop.m 999 // // N even: T_hi = sgn_r * T_hi // N odd : C_hi = -sgn_r * C_hi // (p0) fma.s1 tail = tail, Sx, CORR nop.i 999 ;; } { .mfi nop.m 999 // // N even and odd: tail = Sx * tail + CORR // (p0) fma.s1 tail = V_hi, Sx, tail nop.i 999 ;; } { .mfi nop.m 999 // // N even an odd: tail = Sx * V_hi + tail // (p11) fma.s0 Result = sgn_r, tail, T_hi nop.i 999 } { .mfb nop.m 999 (p12) fma.s0 Result = sgn_r, tail, C_hi (p0) br.ret.sptk b0 ;; } .endp __libm_tan ASM_SIZE_DIRECTIVE(__libm_tan) // ******************************************************************* // ******************************************************************* // ******************************************************************* // // Special Code to handle very large argument case. // Call int pi_by_2_reduce(&x,&r) // for |arguments| >= 2**63 // (Arg or x) is in f8 // Address to save r and c as double // (1) (2) (3) (call) (4) // sp -> + psp -> + psp -> + sp -> + // | | | | // | r50 ->| <- r50 f0 ->| r50 -> | -> c // | | | | // sp-32 -> | <- r50 f0 ->| f0 ->| <- r50 r49 -> | -> r // | | | | // | r49 ->| <- r49 Arg ->| <- r49 | -> x // | | | | // sp -64 ->| sp -64 ->| sp -64 ->| | // // save pfs save b0 restore gp // save gp restore b0 // restore pfs .proc __libm_callout __libm_callout: TAN_ARG_TOO_LARGE: .prologue // (1) { .mfi add GR_Parameter_r =-32,sp // Parameter: r address nop.f 0 .save ar.pfs,GR_SAVE_PFS mov GR_SAVE_PFS=ar.pfs // Save ar.pfs } { .mfi .fframe 64 add sp=-64,sp // Create new stack nop.f 0 mov GR_SAVE_GP=gp // Save gp };; // (2) { .mmi stfe [GR_Parameter_r ] = f0,16 // Clear Parameter r on stack add GR_Parameter_X = 16,sp // Parameter x address .save b0, GR_SAVE_B0 mov GR_SAVE_B0=b0 // Save b0 };; // (3) .body { .mib stfe [GR_Parameter_r ] = f0,-16 // Clear Parameter c on stack nop.i 0 nop.b 0 } { .mib stfe [GR_Parameter_X] = Arg // Store Parameter x on stack nop.i 0 (p0) br.call.sptk b0=__libm_pi_by_2_reduce# } ;; // (4) { .mmi mov gp = GR_SAVE_GP // Restore gp (p0) mov N_fix_gr = r8 nop.i 999 } ;; { .mmi (p0) ldfe Arg =[GR_Parameter_X],16 (p0) ldfs TWO_TO_NEG2 = [table_ptr2],4 nop.i 999 } ;; { .mmb (p0) ldfe r =[GR_Parameter_r ],16 (p0) ldfs NEGTWO_TO_NEG2 = [table_ptr2],4 nop.b 999 ;; } { .mfi (p0) ldfe c =[GR_Parameter_r ] nop.f 999 nop.i 999 ;; } { .mfi nop.m 999 // // Is |r| < 2**(-2) // (p0) fcmp.lt.unc.s1 p6, p0 = r, TWO_TO_NEG2 mov b0 = GR_SAVE_B0 // Restore return address } ;; { .mfi nop.m 999 (p6) fcmp.gt.unc.s1 p6, p0 = r, NEGTWO_TO_NEG2 mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs } ;; { .mbb .restore sp add sp = 64,sp // Restore stack pointer (p6) br.cond.spnt TAN_SMALL_R (p0) br.cond.sptk TAN_NORMAL_R } ;; .endp __libm_callout ASM_SIZE_DIRECTIVE(__libm_callout) .proc __libm_TAN_SPECIAL __libm_TAN_SPECIAL: // // Code for NaNs, Unsupporteds, Infs, or +/- zero ? // Invalid raised for Infs and SNaNs. // { .mfb nop.m 999 (p0) fmpy.s0 Arg = Arg, f0 (p0) br.ret.sptk b0 } .endp __libm_TAN_SPECIAL ASM_SIZE_DIRECTIVE(__libm_TAN_SPECIAL) .type __libm_pi_by_2_reduce#,@function .global __libm_pi_by_2_reduce#