.file "asin.s" // Copyright (c) 2000 - 2003 Intel Corporation // All rights reserved. // // Contributed 2000 by the Intel Numerics Group, 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://www.intel.com/software/products/opensource/libraries/num.htm. // History //============================================================== // 02/02/00 Initial version // 08/17/00 New and much faster algorithm. // 08/31/00 Avoided bank conflicts on loads, shortened |x|=1 path, // fixed mfb split issue stalls. // 12/19/00 Fixed small arg cases to force inexact, or inexact and underflow. // 08/02/02 New and much faster algorithm II // 02/06/03 Reordered header: .section, .global, .proc, .align // Description //========================================= // The asin function computes the principal value of the arc sine of x. // asin(0) returns 0, asin(1) returns pi/2, asin(-1) returns -pi/2. // A doman error occurs for arguments not in the range [-1,+1]. // // The asin function returns the arc sine in the range [-pi/2, +pi/2] radians. // // There are 8 paths: // 1. x = +/-0.0 // Return asin(x) = +/-0.0 // // 2. 0.0 < |x| < 0.625 // Return asin(x) = x + x^3 *PolA(x^2) // where PolA(x^2) = A3 + A5*x^2 + A7*x^4 +...+ A35*x^32 // // 3. 0.625 <=|x| < 1.0 // Return asin(x) = sign(x) * ( Pi/2 - sqrt(R) * PolB(R)) // Where R = 1 - |x|, // PolB(R) = B0 + B1*R + B2*R^2 +...+B12*R^12 // // sqrt(R) is approximated using the following sequence: // y0 = (1 + eps)/sqrt(R) - initial approximation by frsqrta, // |eps| < 2^(-8) // Then 3 iterations are used to refine the result: // H0 = 0.5*y0 // S0 = R*y0 // // d0 = 0.5 - H0*S0 // H1 = H0 + d0*H0 // S1 = S0 + d0*S0 // // d1 = 0.5 - H1*S1 // H2 = H1 + d0*H1 // S2 = S1 + d0*S1 // // d2 = 0.5 - H2*S2 // S3 = S3 + d2*S3 // // S3 approximates sqrt(R) with enough accuracy for this algorithm // // So, the result should be reconstracted as follows: // asin(x) = sign(x) * (Pi/2 - S3*PolB(R)) // // But for optimization perposes the reconstruction step is slightly // changed: // asin(x) = sign(x)*(Pi/2 - PolB(R)*S2) + sign(x)*d2*S2*PolB(R) // // 4. |x| = 1.0 // Return asin(x) = sign(x)*Pi/2 // // 5. 1.0 < |x| <= +INF // A doman error occurs for arguments not in the range [-1,+1] // // 6. x = [S,Q]NaN // Return asin(x) = QNaN // // 7. x is denormal // Return asin(x) = x + x^3, // // 8. x is unnormal // Normalize input in f8 and return to the very beginning of the function // // Registers used //============================================================== // Floating Point registers used: // f8, input, output // f6, f7, f9 -> f15, f32 -> f63 // General registers used: // r3, r21 -> r31, r32 -> r38 // Predicate registers used: // p0, p6 -> p14 // // Assembly macros //========================================= // integer registers used // scratch rTblAddr = r3 rPiBy2Ptr = r21 rTmpPtr3 = r22 rDenoBound = r23 rOne = r24 rAbsXBits = r25 rHalf = r26 r0625 = r27 rSign = r28 rXBits = r29 rTmpPtr2 = r30 rTmpPtr1 = r31 // stacked GR_SAVE_PFS = r32 GR_SAVE_B0 = r33 GR_SAVE_GP = r34 GR_Parameter_X = r35 GR_Parameter_Y = r36 GR_Parameter_RESULT = r37 GR_Parameter_TAG = r38 // floating point registers used FR_X = f10 FR_Y = f1 FR_RESULT = f8 // scratch fXSqr = f6 fXCube = f7 fXQuadr = f9 f1pX = f10 f1mX = f11 f1pXRcp = f12 f1mXRcp = f13 fH = f14 fS = f15 // stacked fA3 = f32 fB1 = f32 fA5 = f33 fB2 = f33 fA7 = f34 fPiBy2 = f34 fA9 = f35 fA11 = f36 fB10 = f35 fB11 = f36 fA13 = f37 fA15 = f38 fB4 = f37 fB5 = f38 fA17 = f39 fA19 = f40 fB6 = f39 fB7 = f40 fA21 = f41 fA23 = f42 fB3 = f41 fB8 = f42 fA25 = f43 fA27 = f44 fB9 = f43 fB12 = f44 fA29 = f45 fA31 = f46 fA33 = f47 fA35 = f48 fBaseP = f49 fB0 = f50 fSignedS = f51 fD = f52 fHalf = f53 fR = f54 fCloseTo1Pol = f55 fSignX = f56 fDenoBound = f57 fNormX = f58 fX8 = f59 fRSqr = f60 fRQuadr = f61 fR8 = f62 fX16 = f63 // Data tables //============================================================== RODATA .align 16 LOCAL_OBJECT_START(asin_base_range_table) // Ai: Polynomial coefficients for the asin(x), |x| < .625000 // Bi: Polynomial coefficients for the asin(x), |x| > .625000 data8 0xBFDAAB56C01AE468 //A29 data8 0x3FE1C470B76A5B2B //A31 data8 0xBFDC5FF82A0C4205 //A33 data8 0x3FC71FD88BFE93F0 //A35 data8 0xB504F333F9DE6487, 0x00003FFF //B0 data8 0xAAAAAAAAAAAAFC18, 0x00003FFC //A3 data8 0x3F9F1C71BC4A7823 //A9 data8 0x3F96E8BBAAB216B2 //A11 data8 0x3F91C4CA1F9F8A98 //A13 data8 0x3F8C9DDCEDEBE7A6 //A15 data8 0x3F877784442B1516 //A17 data8 0x3F859C0491802BA2 //A19 data8 0x9999999998C88B8F, 0x00003FFB //A5 data8 0x3F6BD7A9A660BF5E //A21 data8 0x3F9FC1659340419D //A23 data8 0xB6DB6DB798149BDF, 0x00003FFA //A7 data8 0xBFB3EF18964D3ED3 //A25 data8 0x3FCD285315542CF2 //A27 data8 0xF15BEEEFF7D2966A, 0x00003FFB //B1 data8 0x3EF0DDA376D10FB3 //B10 data8 0xBEB83CAFE05EBAC9 //B11 data8 0x3F65FFB67B513644 //B4 data8 0x3F5032FBB86A4501 //B5 data8 0x3F392162276C7CBA //B6 data8 0x3F2435949FD98BDF //B7 data8 0xD93923D7FA08341C, 0x00003FF9 //B2 data8 0x3F802995B6D90BDB //B3 data8 0x3F10DF86B341A63F //B8 data8 0xC90FDAA22168C235, 0x00003FFF // Pi/2 data8 0x3EFA3EBD6B0ECB9D //B9 data8 0x3EDE18BA080E9098 //B12 LOCAL_OBJECT_END(asin_base_range_table) .section .text GLOBAL_LIBM_ENTRY(asin) asin_unnormal_back: { .mfi getf.d rXBits = f8 // grab bits of input value // set p12 = 1 if x is a NaN, denormal, or zero fclass.m p12, p0 = f8, 0xcf adds rSign = 1, r0 } { .mfi addl rTblAddr = @ltoff(asin_base_range_table),gp // 1 - x = 1 - |x| for positive x fms.s1 f1mX = f1, f1, f8 addl rHalf = 0xFFFE, r0 // exponent of 1/2 } ;; { .mfi addl r0625 = 0x3FE4, r0 // high 16 bits of 0.625 // set p8 = 1 if x < 0 fcmp.lt.s1 p8, p9 = f8, f0 shl rSign = rSign, 63 // sign bit } { .mfi // point to the beginning of the table ld8 rTblAddr = [rTblAddr] // 1 + x = 1 - |x| for negative x fma.s1 f1pX = f1, f1, f8 adds rOne = 0x3FF, r0 } ;; { .mfi andcm rAbsXBits = rXBits, rSign // bits of |x| fmerge.s fSignX = f8, f1 // signum(x) shl r0625 = r0625, 48 // bits of DP representation of 0.625 } { .mfb setf.exp fHalf = rHalf // load A2 to FP reg fma.s1 fXSqr = f8, f8, f0 // x^2 // branch on special path if x is a NaN, denormal, or zero (p12) br.cond.spnt asin_special } ;; { .mfi adds rPiBy2Ptr = 272, rTblAddr nop.f 0 shl rOne = rOne, 52 // bits of 1.0 } { .mfi adds rTmpPtr1 = 16, rTblAddr nop.f 0 // set p6 = 1 if |x| < 0.625 cmp.lt p6, p7 = rAbsXBits, r0625 } ;; { .mfi ldfpd fA29, fA31 = [rTblAddr] // A29, fA31 // 1 - x = 1 - |x| for positive x (p9) fms.s1 fR = f1, f1, f8 // point to coefficient of "near 1" polynomial (p7) adds rTmpPtr2 = 176, rTblAddr } { .mfi ldfpd fA33, fA35 = [rTmpPtr1], 16 // A33, fA35 // 1 + x = 1 - |x| for negative x (p8) fma.s1 fR = f1, f1, f8 (p6) adds rTmpPtr2 = 48, rTblAddr } ;; { .mfi ldfe fB0 = [rTmpPtr1], 16 // B0 nop.f 0 nop.i 0 } { .mib adds rTmpPtr3 = 16, rTmpPtr2 // set p10 = 1 if |x| = 1.0 cmp.eq p10, p0 = rAbsXBits, rOne // branch on special path for |x| = 1.0 (p10) br.cond.spnt asin_abs_1 } ;; { .mfi ldfe fA3 = [rTmpPtr2], 48 // A3 or B1 nop.f 0 adds rTmpPtr1 = 64, rTmpPtr3 } { .mib ldfpd fA9, fA11 = [rTmpPtr3], 16 // A9, A11 or B10, B11 // set p11 = 1 if |x| > 1.0 cmp.gt p11, p0 = rAbsXBits, rOne // branch on special path for |x| > 1.0 (p11) br.cond.spnt asin_abs_gt_1 } ;; { .mfi ldfpd fA17, fA19 = [rTmpPtr2], 16 // A17, A19 or B6, B7 // initial approximation of 1 / sqrt(1 - x) frsqrta.s1 f1mXRcp, p0 = f1mX nop.i 0 } { .mfi ldfpd fA13, fA15 = [rTmpPtr3] // A13, A15 or B4, B5 fma.s1 fXCube = fXSqr, f8, f0 // x^3 nop.i 0 } ;; { .mfi ldfe fA5 = [rTmpPtr2], 48 // A5 or B2 // initial approximation of 1 / sqrt(1 + x) frsqrta.s1 f1pXRcp, p0 = f1pX nop.i 0 } { .mfi ldfpd fA21, fA23 = [rTmpPtr1], 16 // A21, A23 or B3, B8 fma.s1 fXQuadr = fXSqr, fXSqr, f0 // x^4 nop.i 0 } ;; { .mfi ldfe fA7 = [rTmpPtr1] // A7 or Pi/2 fma.s1 fRSqr = fR, fR, f0 // R^2 nop.i 0 } { .mfb ldfpd fA25, fA27 = [rTmpPtr2] // A25, A27 or B9, B12 nop.f 0 (p6) br.cond.spnt asin_base_range; } ;; { .mfi nop.m 0 (p9) fma.s1 fH = fHalf, f1mXRcp, f0 // H0 for x > 0 nop.i 0 } { .mfi nop.m 0 (p9) fma.s1 fS = f1mX, f1mXRcp, f0 // S0 for x > 0 nop.i 0 } ;; { .mfi nop.m 0 (p8) fma.s1 fH = fHalf, f1pXRcp, f0 // H0 for x < 0 nop.i 0 } { .mfi nop.m 0 (p8) fma.s1 fS = f1pX, f1pXRcp, f0 // S0 for x > 0 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fRQuadr = fRSqr, fRSqr, f0 // R^4 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fB11 = fB11, fR, fB10 nop.i 0 } { .mfi nop.m 0 fma.s1 fB1 = fB1, fR, fB0 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fB5 = fB5, fR, fB4 nop.i 0 } { .mfi nop.m 0 fma.s1 fB7 = fB7, fR, fB6 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fB3 = fB3, fR, fB2 nop.i 0 } ;; { .mfi nop.m 0 fnma.s1 fD = fH, fS, fHalf // d0 = 1/2 - H0*S0 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fR8 = fRQuadr, fRQuadr, f0 // R^4 nop.i 0 } { .mfi nop.m 0 fma.s1 fB9 = fB9, fR, fB8 nop.i 0 } ;; {.mfi nop.m 0 fma.s1 fB12 = fB12, fRSqr, fB11 nop.i 0 } {.mfi nop.m 0 fma.s1 fB7 = fB7, fRSqr, fB5 nop.i 0 } ;; {.mfi nop.m 0 fma.s1 fB3 = fB3, fRSqr, fB1 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fH = fH, fD, fH // H1 = H0 + H0*d0 nop.i 0 } { .mfi nop.m 0 fma.s1 fS = fS, fD, fS // S1 = S0 + S0*d0 nop.i 0 } ;; {.mfi nop.m 0 fma.s1 fPiBy2 = fPiBy2, fSignX, f0 // signum(x)*Pi/2 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fB12 = fB12, fRSqr, fB9 nop.i 0 } { .mfi nop.m 0 fma.s1 fB7 = fB7, fRQuadr, fB3 nop.i 0 } ;; {.mfi nop.m 0 fnma.s1 fD = fH, fS, fHalf // d1 = 1/2 - H1*S1 nop.i 0 } { .mfi nop.m 0 fnma.s1 fSignedS = fSignX, fS, f0 // -signum(x)*S1 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fCloseTo1Pol = fB12, fR8, fB7 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fH = fH, fD, fH // H2 = H1 + H1*d1 nop.i 0 } { .mfi nop.m 0 fma.s1 fS = fS, fD, fS // S2 = S1 + S1*d1 nop.i 0 } ;; { .mfi nop.m 0 // -signum(x)* S2 = -signum(x)*(S1 + S1*d1) fma.s1 fSignedS = fSignedS, fD, fSignedS nop.i 0 } ;; {.mfi nop.m 0 fnma.s1 fD = fH, fS, fHalf // d2 = 1/2 - H2*S2 nop.i 0 } ;; { .mfi nop.m 0 // signum(x)*(Pi/2 - PolB*S2) fma.s1 fPiBy2 = fSignedS, fCloseTo1Pol, fPiBy2 nop.i 0 } { .mfi nop.m 0 // -signum(x)*PolB * S2 fma.s1 fCloseTo1Pol = fSignedS, fCloseTo1Pol, f0 nop.i 0 } ;; { .mfb nop.m 0 // final result for 0.625 <= |x| < 1 fma.d.s0 f8 = fCloseTo1Pol, fD, fPiBy2 // exit here for 0.625 <= |x| < 1 br.ret.sptk b0 } ;; // here if |x| < 0.625 .align 32 asin_base_range: { .mfi nop.m 0 fma.s1 fA33 = fA33, fXSqr, fA31 nop.i 0 } { .mfi nop.m 0 fma.s1 fA15 = fA15, fXSqr, fA13 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fA29 = fA29, fXSqr, fA27 nop.i 0 } { .mfi nop.m 0 fma.s1 fA25 = fA25, fXSqr, fA23 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fA21 = fA21, fXSqr, fA19 nop.i 0 } { .mfi nop.m 0 fma.s1 fA9 = fA9, fXSqr, fA7 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fA5 = fA5, fXSqr, fA3 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fA35 = fA35, fXQuadr, fA33 nop.i 0 } { .mfi nop.m 0 fma.s1 fA17 = fA17, fXQuadr, fA15 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fX8 = fXQuadr, fXQuadr, f0 // x^8 nop.i 0 } { .mfi nop.m 0 fma.s1 fA25 = fA25, fXQuadr, fA21 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fA9 = fA9, fXQuadr, fA5 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fA35 = fA35, fXQuadr, fA29 nop.i 0 } { .mfi nop.m 0 fma.s1 fA17 = fA17, fXSqr, fA11 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fX16 = fX8, fX8, f0 // x^16 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fA35 = fA35, fX8, fA25 nop.i 0 } { .mfi nop.m 0 fma.s1 fA17 = fA17, fX8, fA9 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fBaseP = fA35, fX16, fA17 nop.i 0 } ;; { .mfb nop.m 0 // final result for |x| < 0.625 fma.d.s0 f8 = fBaseP, fXCube, f8 // exit here for |x| < 0.625 path br.ret.sptk b0 } ;; // here if |x| = 1 // asin(x) = sign(x) * Pi/2 .align 32 asin_abs_1: { .mfi ldfe fPiBy2 = [rPiBy2Ptr] // Pi/2 nop.f 0 nop.i 0 } ;; {.mfb nop.m 0 // result for |x| = 1.0 fma.d.s0 f8 = fPiBy2, fSignX, f0 // exit here for |x| = 1.0 br.ret.sptk b0 } ;; // here if x is a NaN, denormal, or zero .align 32 asin_special: { .mfi nop.m 0 // set p12 = 1 if x is a NaN fclass.m p12, p0 = f8, 0xc3 nop.i 0 } { .mlx nop.m 0 // smallest positive DP normalized number movl rDenoBound = 0x0010000000000000 } ;; { .mfi nop.m 0 // set p13 = 1 if x = 0.0 fclass.m p13, p0 = f8, 0x07 nop.i 0 } { .mfi nop.m 0 fnorm.s1 fNormX = f8 nop.i 0 } ;; { .mfb // load smallest normal to FP reg setf.d fDenoBound = rDenoBound // answer if x is a NaN (p12) fma.d.s0 f8 = f8,f1,f0 // exit here if x is a NaN (p12) br.ret.spnt b0 } ;; { .mfb nop.m 0 nop.f 0 // exit here if x = 0.0 (p13) br.ret.spnt b0 } ;; // if we still here then x is denormal or unnormal { .mfi nop.m 0 // absolute value of normalized x fmerge.s fNormX = f1, fNormX nop.i 0 } ;; { .mfi nop.m 0 // set p14 = 1 if normalized x is greater than or // equal to the smallest denormalized value // So, if p14 is set to 1 it means that we deal with // unnormal rather than with "true" denormal fcmp.ge.s1 p14, p0 = fNormX, fDenoBound nop.i 0 } ;; { .mfi nop.m 0 (p14) fcmp.eq.s0 p6, p0 = f8, f0 // Set D flag if x unnormal nop.i 0 } { .mfb nop.m 0 // normalize unnormal input (p14) fnorm.s1 f8 = f8 // return to the main path (p14) br.cond.sptk asin_unnormal_back } ;; // if we still here it means that input is "true" denormal { .mfb nop.m 0 // final result if x is denormal fma.d.s0 f8 = f8, fXSqr, f8 // exit here if x is denormal br.ret.sptk b0 } ;; // here if |x| > 1.0 // error handler should be called .align 32 asin_abs_gt_1: { .mfi alloc r32 = ar.pfs, 0, 3, 4, 0 // get some registers fmerge.s FR_X = f8,f8 nop.i 0 } { .mfb mov GR_Parameter_TAG = 61 // error code frcpa.s0 FR_RESULT, p0 = f0,f0 // call error handler routine br.cond.sptk __libm_error_region } ;; GLOBAL_LIBM_END(asin) LOCAL_LIBM_ENTRY(__libm_error_region) .prologue { .mfi add GR_Parameter_Y=-32,sp // Parameter 2 value 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 };; { .mmi stfd [GR_Parameter_Y] = FR_Y,16 // STORE Parameter 2 on stack add GR_Parameter_X = 16,sp // Parameter 1 address .save b0, GR_SAVE_B0 mov GR_SAVE_B0=b0 // Save b0 };; .body { .mib stfd [GR_Parameter_X] = FR_X // STORE Parameter 1 on stack add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address nop.b 0 } { .mib stfd [GR_Parameter_Y] = FR_RESULT // STORE Parameter 3 on stack add GR_Parameter_Y = -16,GR_Parameter_Y br.call.sptk b0=__libm_error_support# // Call error handling function };; { .mmi add GR_Parameter_RESULT = 48,sp nop.m 0 nop.i 0 };; { .mmi ldfd f8 = [GR_Parameter_RESULT] // Get return result off stack .restore sp add sp = 64,sp // Restore stack pointer mov b0 = GR_SAVE_B0 // Restore return address };; { .mib mov gp = GR_SAVE_GP // Restore gp mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs br.ret.sptk b0 // Return };; LOCAL_LIBM_END(__libm_error_region) .type __libm_error_support#,@function .global __libm_error_support#