.file "asinhl.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: // 09/04/01 Initial version // 09/13/01 Performance improved, symmetry problems fixed // 10/10/01 Performance improved, split issues removed // 12/11/01 Changed huges_logp to not be global // 05/20/02 Cleaned up namespace and sf0 syntax // 02/10/03 Reordered header: .section, .global, .proc, .align; // used data8 for long double table values // //********************************************************************* // // API //============================================================== // long double asinhl(long double); // // Overview of operation //============================================================== // // There are 6 paths: // 1. x = 0, [S,Q]Nan or +/-INF // Return asinhl(x) = x + x; // // 2. x = + denormal // Return asinhl(x) = x - x^2; // // 3. x = - denormal // Return asinhl(x) = x + x^2; // // 4. 'Near 0': max denormal < |x| < 1/128 // Return asinhl(x) = sign(x)*(x+x^3*(c3+x^2*(c5+x^2*(c7+x^2*(c9))))); // // 5. 'Huges': |x| > 2^63 // Return asinhl(x) = sign(x)*(logl(2*x)); // // 6. 'Main path': 1/128 < |x| < 2^63 // b_hi + b_lo = x + sqrt(x^2 + 1); // asinhl(x) = sign(x)*(log_special(b_hi, b_lo)); // // Algorithm description //============================================================== // // Main path algorithm // ( thanks to Peter Markstein for the idea of sqrt(x^2+1) computation! ) // ************************************************************************* // // There are 3 parts of x+sqrt(x^2+1) computation: // // 1) p2 = (p2_hi+p2_lo) = x^2+1 obtaining // ------------------------------------ // p2_hi = x2_hi + 1, where x2_hi = x * x; // p2_lo = x2_lo + p1_lo, where // x2_lo = FMS(x*x-x2_hi), // p1_lo = (1 - p2_hi) + x2_hi; // // 2) g = (g_hi+g_lo) = sqrt(p2) = sqrt(p2_hi+p2_lo) // ---------------------------------------------- // r = invsqrt(p2_hi) (8-bit reciprocal square root approximation); // g = p2_hi * r (first 8 bit-approximation of sqrt); // // h = 0.5 * r; // e = 0.5 - g * h; // g = g * e + g (second 16 bit-approximation of sqrt); // // h = h * e + h; // e = 0.5 - g * h; // g = g * e + g (third 32 bit-approximation of sqrt); // // h = h * e + h; // e = 0.5 - g * h; // g_hi = g * e + g (fourth 64 bit-approximation of sqrt); // // Remainder computation: // h = h * e + h; // d = (p2_hi - g_hi * g_hi) + p2_lo; // g_lo = d * h; // // 3) b = (b_hi + b_lo) = x + g, where g = (g_hi + g_lo) = sqrt(x^2+1) // ------------------------------------------------------------------- // b_hi = (g_hi + x) + gl; // b_lo = (g_hi - b_hi) + x + gl; // // Now we pass b presented as sum b_hi + b_lo to special version // of logl function which accept a pair of arguments as // 'mutiprecision' value. // // Special log algorithm overview // ================================ // Here we use a table lookup method. The basic idea is that in // order to compute logl(Arg) = logl (Arg-1) for an argument Arg in [1,2), // we construct a value G such that G*Arg is close to 1 and that // logl(1/G) is obtainable easily from a table of values calculated // beforehand. Thus // // logl(Arg) = logl(1/G) + logl((G*Arg - 1)) // // Because |G*Arg - 1| is small, the second term on the right hand // side can be approximated by a short polynomial. We elaborate // this method in four steps. // // Step 0: Initialization // // We need to calculate logl( X ). Obtain N, S_hi such that // // X = 2^N * ( S_hi + S_lo ) exactly // // where S_hi in [1,2) and S_lo is a correction to S_hi in the sense // that |S_lo| <= ulp(S_hi). // // For the special version of logl: S_lo = b_lo // !-----------------------------------------------! // // Step 1: Argument Reduction // // Based on S_hi, obtain G_1, G_2, G_3 from a table and calculate // // G := G_1 * G_2 * G_3 // r := (G * S_hi - 1) + G * S_lo // // These G_j's have the property that the product is exactly // representable and that |r| < 2^(-12) as a result. // // Step 2: Approximation // // logl(1 + r) is approximated by a short polynomial poly(r). // // Step 3: Reconstruction // // Finally, // // logl( X ) = logl( 2^N * (S_hi + S_lo) ) // ~=~ N*logl(2) + logl(1/G) + logl(1 + r) // ~=~ N*logl(2) + logl(1/G) + poly(r). // // For detailed description see logl or log1pl function, regular path. // // Registers used //============================================================== // Floating Point registers used: // f8, input // f32 -> f101 (70 registers) // General registers used: // r32 -> r57 (26 registers) // Predicate registers used: // p6 -> p11 // p6 for '0, NaNs, Inf' path // p7 for '+ denormals' path // p8 for 'near 0' path // p9 for 'huges' path // p10 for '- denormals' path // p11 for negative values // // Data tables //============================================================== RODATA .align 64 // C7, C9 'near 0' polynomial coefficients LOCAL_OBJECT_START(Poly_C_near_0_79) data8 0xF8DC939BBEDD5A54, 0x00003FF9 data8 0xB6DB6DAB21565AC5, 0x0000BFFA LOCAL_OBJECT_END(Poly_C_near_0_79) // C3, C5 'near 0' polynomial coefficients LOCAL_OBJECT_START(Poly_C_near_0_35) data8 0x999999999991D582, 0x00003FFB data8 0xAAAAAAAAAAAAAAA9, 0x0000BFFC LOCAL_OBJECT_END(Poly_C_near_0_35) // Q coeffs LOCAL_OBJECT_START(Constants_Q) data4 0x00000000,0xB1721800,0x00003FFE,0x00000000 data4 0x4361C4C6,0x82E30865,0x0000BFE2,0x00000000 data4 0x328833CB,0xCCCCCAF2,0x00003FFC,0x00000000 data4 0xA9D4BAFB,0x80000077,0x0000BFFD,0x00000000 data4 0xAAABE3D2,0xAAAAAAAA,0x00003FFD,0x00000000 data4 0xFFFFDAB7,0xFFFFFFFF,0x0000BFFD,0x00000000 LOCAL_OBJECT_END(Constants_Q) // Z1 - 16 bit fixed LOCAL_OBJECT_START(Constants_Z_1) data4 0x00008000 data4 0x00007879 data4 0x000071C8 data4 0x00006BCB data4 0x00006667 data4 0x00006187 data4 0x00005D18 data4 0x0000590C data4 0x00005556 data4 0x000051EC data4 0x00004EC5 data4 0x00004BDB data4 0x00004925 data4 0x0000469F data4 0x00004445 data4 0x00004211 LOCAL_OBJECT_END(Constants_Z_1) // G1 and H1 - IEEE single and h1 - IEEE double LOCAL_OBJECT_START(Constants_G_H_h1) data4 0x3F800000,0x00000000 data8 0x0000000000000000 data4 0x3F70F0F0,0x3D785196 data8 0x3DA163A6617D741C data4 0x3F638E38,0x3DF13843 data8 0x3E2C55E6CBD3D5BB data4 0x3F579430,0x3E2FF9A0 data8 0xBE3EB0BFD86EA5E7 data4 0x3F4CCCC8,0x3E647FD6 data8 0x3E2E6A8C86B12760 data4 0x3F430C30,0x3E8B3AE7 data8 0x3E47574C5C0739BA data4 0x3F3A2E88,0x3EA30C68 data8 0x3E20E30F13E8AF2F data4 0x3F321640,0x3EB9CEC8 data8 0xBE42885BF2C630BD data4 0x3F2AAAA8,0x3ECF9927 data8 0x3E497F3497E577C6 data4 0x3F23D708,0x3EE47FC5 data8 0x3E3E6A6EA6B0A5AB data4 0x3F1D89D8,0x3EF8947D data8 0xBDF43E3CD328D9BE data4 0x3F17B420,0x3F05F3A1 data8 0x3E4094C30ADB090A data4 0x3F124920,0x3F0F4303 data8 0xBE28FBB2FC1FE510 data4 0x3F0D3DC8,0x3F183EBF data8 0x3E3A789510FDE3FA data4 0x3F088888,0x3F20EC80 data8 0x3E508CE57CC8C98F data4 0x3F042108,0x3F29516A data8 0xBE534874A223106C LOCAL_OBJECT_END(Constants_G_H_h1) // Z2 - 16 bit fixed LOCAL_OBJECT_START(Constants_Z_2) data4 0x00008000 data4 0x00007F81 data4 0x00007F02 data4 0x00007E85 data4 0x00007E08 data4 0x00007D8D data4 0x00007D12 data4 0x00007C98 data4 0x00007C20 data4 0x00007BA8 data4 0x00007B31 data4 0x00007ABB data4 0x00007A45 data4 0x000079D1 data4 0x0000795D data4 0x000078EB LOCAL_OBJECT_END(Constants_Z_2) // G2 and H2 - IEEE single and h2 - IEEE double LOCAL_OBJECT_START(Constants_G_H_h2) data4 0x3F800000,0x00000000 data8 0x0000000000000000 data4 0x3F7F00F8,0x3B7F875D data8 0x3DB5A11622C42273 data4 0x3F7E03F8,0x3BFF015B data8 0x3DE620CF21F86ED3 data4 0x3F7D08E0,0x3C3EE393 data8 0xBDAFA07E484F34ED data4 0x3F7C0FC0,0x3C7E0586 data8 0xBDFE07F03860BCF6 data4 0x3F7B1880,0x3C9E75D2 data8 0x3DEA370FA78093D6 data4 0x3F7A2328,0x3CBDC97A data8 0x3DFF579172A753D0 data4 0x3F792FB0,0x3CDCFE47 data8 0x3DFEBE6CA7EF896B data4 0x3F783E08,0x3CFC15D0 data8 0x3E0CF156409ECB43 data4 0x3F774E38,0x3D0D874D data8 0xBE0B6F97FFEF71DF data4 0x3F766038,0x3D1CF49B data8 0xBE0804835D59EEE8 data4 0x3F757400,0x3D2C531D data8 0x3E1F91E9A9192A74 data4 0x3F748988,0x3D3BA322 data8 0xBE139A06BF72A8CD data4 0x3F73A0D0,0x3D4AE46F data8 0x3E1D9202F8FBA6CF data4 0x3F72B9D0,0x3D5A1756 data8 0xBE1DCCC4BA796223 data4 0x3F71D488,0x3D693B9D data8 0xBE049391B6B7C239 LOCAL_OBJECT_END(Constants_G_H_h2) // G3 and H3 - IEEE single and h3 - IEEE double LOCAL_OBJECT_START(Constants_G_H_h3) data4 0x3F7FFC00,0x38800100 data8 0x3D355595562224CD data4 0x3F7FF400,0x39400480 data8 0x3D8200A206136FF6 data4 0x3F7FEC00,0x39A00640 data8 0x3DA4D68DE8DE9AF0 data4 0x3F7FE400,0x39E00C41 data8 0xBD8B4291B10238DC data4 0x3F7FDC00,0x3A100A21 data8 0xBD89CCB83B1952CA data4 0x3F7FD400,0x3A300F22 data8 0xBDB107071DC46826 data4 0x3F7FCC08,0x3A4FF51C data8 0x3DB6FCB9F43307DB data4 0x3F7FC408,0x3A6FFC1D data8 0xBD9B7C4762DC7872 data4 0x3F7FBC10,0x3A87F20B data8 0xBDC3725E3F89154A data4 0x3F7FB410,0x3A97F68B data8 0xBD93519D62B9D392 data4 0x3F7FAC18,0x3AA7EB86 data8 0x3DC184410F21BD9D data4 0x3F7FA420,0x3AB7E101 data8 0xBDA64B952245E0A6 data4 0x3F7F9C20,0x3AC7E701 data8 0x3DB4B0ECAABB34B8 data4 0x3F7F9428,0x3AD7DD7B data8 0x3D9923376DC40A7E data4 0x3F7F8C30,0x3AE7D474 data8 0x3DC6E17B4F2083D3 data4 0x3F7F8438,0x3AF7CBED data8 0x3DAE314B811D4394 data4 0x3F7F7C40,0x3B03E1F3 data8 0xBDD46F21B08F2DB1 data4 0x3F7F7448,0x3B0BDE2F data8 0xBDDC30A46D34522B data4 0x3F7F6C50,0x3B13DAAA data8 0x3DCB0070B1F473DB data4 0x3F7F6458,0x3B1BD766 data8 0xBDD65DDC6AD282FD data4 0x3F7F5C68,0x3B23CC5C data8 0xBDCDAB83F153761A data4 0x3F7F5470,0x3B2BC997 data8 0xBDDADA40341D0F8F data4 0x3F7F4C78,0x3B33C711 data8 0x3DCD1BD7EBC394E8 data4 0x3F7F4488,0x3B3BBCC6 data8 0xBDC3532B52E3E695 data4 0x3F7F3C90,0x3B43BAC0 data8 0xBDA3961EE846B3DE data4 0x3F7F34A0,0x3B4BB0F4 data8 0xBDDADF06785778D4 data4 0x3F7F2CA8,0x3B53AF6D data8 0x3DCC3ED1E55CE212 data4 0x3F7F24B8,0x3B5BA620 data8 0xBDBA31039E382C15 data4 0x3F7F1CC8,0x3B639D12 data8 0x3D635A0B5C5AF197 data4 0x3F7F14D8,0x3B6B9444 data8 0xBDDCCB1971D34EFC data4 0x3F7F0CE0,0x3B7393BC data8 0x3DC7450252CD7ADA data4 0x3F7F04F0,0x3B7B8B6D data8 0xBDB68F177D7F2A42 LOCAL_OBJECT_END(Constants_G_H_h3) // Assembly macros //============================================================== // Floating Point Registers FR_Arg = f8 FR_Res = f8 FR_AX = f32 FR_XLog_Hi = f33 FR_XLog_Lo = f34 // Special logl registers FR_Y_hi = f35 FR_Y_lo = f36 FR_Scale = f37 FR_X_Prime = f38 FR_S_hi = f39 FR_W = f40 FR_G = f41 FR_H = f42 FR_wsq = f43 FR_w4 = f44 FR_h = f45 FR_w6 = f46 FR_G2 = f47 FR_H2 = f48 FR_poly_lo = f49 FR_P8 = f50 FR_poly_hi = f51 FR_P7 = f52 FR_h2 = f53 FR_rsq = f54 FR_P6 = f55 FR_r = f56 FR_log2_hi = f57 FR_log2_lo = f58 FR_float_N = f59 FR_Q4 = f60 FR_G3 = f61 FR_H3 = f62 FR_h3 = f63 FR_Q3 = f64 FR_Q2 = f65 FR_1LN10_hi = f66 FR_Q1 = f67 FR_1LN10_lo = f68 FR_P5 = f69 FR_rcub = f70 FR_Neg_One = f71 FR_Z = f72 FR_AA = f73 FR_BB = f74 FR_S_lo = f75 FR_2_to_minus_N = f76 // Huge & Main path prolog registers FR_Half = f77 FR_Two = f78 FR_X2 = f79 FR_P2 = f80 FR_P2L = f81 FR_Rcp = f82 FR_GG = f83 FR_HH = f84 FR_EE = f85 FR_DD = f86 FR_GL = f87 FR_A = f88 FR_AL = f89 FR_B = f90 FR_BL = f91 FR_Tmp = f92 // Near 0 & Huges path prolog registers FR_C3 = f93 FR_C5 = f94 FR_C7 = f95 FR_C9 = f96 FR_X3 = f97 FR_X4 = f98 FR_P9 = f99 FR_P5 = f100 FR_P3 = f101 // General Purpose Registers // General prolog registers GR_PFS = r32 GR_TwoN7 = r40 GR_TwoP63 = r41 GR_ExpMask = r42 GR_ArgExp = r43 GR_Half = r44 // Near 0 path prolog registers GR_Poly_C_35 = r45 GR_Poly_C_79 = r46 // Special logl registers GR_Index1 = r34 GR_Index2 = r35 GR_signif = r36 GR_X_0 = r37 GR_X_1 = r38 GR_X_2 = r39 GR_Z_1 = r40 GR_Z_2 = r41 GR_N = r42 GR_Bias = r43 GR_M = r44 GR_Index3 = r45 GR_exp_2tom80 = r45 GR_exp_mask = r47 GR_exp_2tom7 = r48 GR_ad_ln10 = r49 GR_ad_tbl_1 = r50 GR_ad_tbl_2 = r51 GR_ad_tbl_3 = r52 GR_ad_q = r53 GR_ad_z_1 = r54 GR_ad_z_2 = r55 GR_ad_z_3 = r56 GR_minus_N = r57 .section .text GLOBAL_LIBM_ENTRY(asinhl) { .mfi alloc GR_PFS = ar.pfs,0,27,0,0 fma.s1 FR_P2 = FR_Arg, FR_Arg, f1 // p2 = x^2 + 1 mov GR_Half = 0xfffe // 0.5's exp } { .mfi addl GR_Poly_C_79 = @ltoff(Poly_C_near_0_79), gp // C7, C9 coeffs fma.s1 FR_X2 = FR_Arg, FR_Arg, f0 // Obtain x^2 addl GR_Poly_C_35 = @ltoff(Poly_C_near_0_35), gp // C3, C5 coeffs };; { .mfi getf.exp GR_ArgExp = FR_Arg // get arument's exponent fabs FR_AX = FR_Arg // absolute value of argument mov GR_TwoN7 = 0xfff8 // 2^-7 exp } { .mfi ld8 GR_Poly_C_79 = [GR_Poly_C_79] // get actual coeff table address fma.s0 FR_Two = f1, f1, f1 // construct 2.0 mov GR_ExpMask = 0x1ffff // mask for exp };; { .mfi ld8 GR_Poly_C_35 = [GR_Poly_C_35] // get actual coeff table address fclass.m p6,p0 = FR_Arg, 0xe7 // if arg NaN inf zero mov GR_TwoP63 = 0x1003e // 2^63 exp } { .mfi addl GR_ad_z_1 = @ltoff(Constants_Z_1#),gp nop.f 0 nop.i 0 };; { .mfi setf.exp FR_Half = GR_Half // construct 0.5 fclass.m p7,p0 = FR_Arg, 0x09 // if arg + denorm and GR_ArgExp = GR_ExpMask, GR_ArgExp // select exp } { .mfb ld8 GR_ad_z_1 = [GR_ad_z_1] // Get pointer to Constants_Z_1 nop.f 0 nop.b 0 };; { .mfi ldfe FR_C9 = [GR_Poly_C_79],16 // load C9 fclass.m p10,p0 = FR_Arg, 0x0a // if arg - denorm cmp.gt p8, p0 = GR_TwoN7, GR_ArgExp // if arg < 2^-7 ('near 0') } { .mfb cmp.le p9, p0 = GR_TwoP63, GR_ArgExp // if arg > 2^63 ('huges') (p6) fma.s0 FR_Res = FR_Arg,f1,FR_Arg // r = a + a (p6) br.ret.spnt b0 // return };; // (X^2 + 1) computation { .mfi (p8) ldfe FR_C5 = [GR_Poly_C_35],16 // load C5 fms.s1 FR_Tmp = f1, f1, FR_P2 // Tmp = 1 - p2 add GR_ad_tbl_1 = 0x040, GR_ad_z_1 // Point to Constants_G_H_h1 } { .mfb (p8) ldfe FR_C7 = [GR_Poly_C_79],16 // load C7 (p7) fnma.s0 FR_Res = FR_Arg,FR_Arg,FR_Arg // r = a - a*a (p7) br.ret.spnt b0 // return };; { .mfi (p8) ldfe FR_C3 = [GR_Poly_C_35],16 // load C3 fcmp.lt.s1 p11, p12 = FR_Arg, f0 // if arg is negative add GR_ad_q = -0x60, GR_ad_z_1 // Point to Constants_P } { .mfb add GR_ad_z_2 = 0x140, GR_ad_z_1 // Point to Constants_Z_2 (p10) fma.s0 FR_Res = FR_Arg,FR_Arg,FR_Arg // r = a + a*a (p10) br.ret.spnt b0 // return };; { .mfi add GR_ad_tbl_2 = 0x180, GR_ad_z_1 // Point to Constants_G_H_h2 frsqrta.s1 FR_Rcp, p0 = FR_P2 // Rcp = 1/p2 reciprocal appr. add GR_ad_tbl_3 = 0x280, GR_ad_z_1 // Point to Constants_G_H_h3 } { .mfi nop.m 0 fms.s1 FR_P2L = FR_AX, FR_AX, FR_X2 //low part of p2=fma(X*X-p2) mov GR_Bias = 0x0FFFF // Create exponent bias };; { .mfb nop.m 0 (p9) fms.s1 FR_XLog_Hi = FR_Two, FR_AX, f0 // Hi of log1p arg = 2*X - 1 (p9) br.cond.spnt huges_logl // special version of log1p };; { .mfb ldfe FR_log2_hi = [GR_ad_q],16 // Load log2_hi (p8) fma.s1 FR_X3 = FR_X2, FR_Arg, f0 // x^3 = x^2 * x (p8) br.cond.spnt near_0 // Go to near 0 branch };; { .mfi ldfe FR_log2_lo = [GR_ad_q],16 // Load log2_lo nop.f 0 nop.i 0 };; { .mfi ldfe FR_Q4 = [GR_ad_q],16 // Load Q4 fma.s1 FR_Tmp = FR_Tmp, f1, FR_X2 // Tmp = Tmp + x^2 mov GR_exp_mask = 0x1FFFF // Create exponent mask };; { .mfi ldfe FR_Q3 = [GR_ad_q],16 // Load Q3 fma.s1 FR_GG = FR_Rcp, FR_P2, f0 // g = Rcp * p2 // 8 bit Newton Raphson iteration nop.i 0 } { .mfi nop.m 0 fma.s1 FR_HH = FR_Half, FR_Rcp, f0 // h = 0.5 * Rcp nop.i 0 };; { .mfi ldfe FR_Q2 = [GR_ad_q],16 // Load Q2 fnma.s1 FR_EE = FR_GG, FR_HH, FR_Half // e = 0.5 - g * h nop.i 0 } { .mfi nop.m 0 fma.s1 FR_P2L = FR_Tmp, f1, FR_P2L // low part of p2 = Tmp + p2l nop.i 0 };; { .mfi ldfe FR_Q1 = [GR_ad_q] // Load Q1 fma.s1 FR_GG = FR_GG, FR_EE, FR_GG // g = g * e + g // 16 bit Newton Raphson iteration nop.i 0 } { .mfi nop.m 0 fma.s1 FR_HH = FR_HH, FR_EE, FR_HH // h = h * e + h nop.i 0 };; { .mfi nop.m 0 fnma.s1 FR_EE = FR_GG, FR_HH, FR_Half // e = 0.5 - g * h nop.i 0 };; { .mfi nop.m 0 fma.s1 FR_GG = FR_GG, FR_EE, FR_GG // g = g * e + g // 32 bit Newton Raphson iteration nop.i 0 } { .mfi nop.m 0 fma.s1 FR_HH = FR_HH, FR_EE, FR_HH // h = h * e + h nop.i 0 };; { .mfi nop.m 0 fnma.s1 FR_EE = FR_GG, FR_HH, FR_Half // e = 0.5 - g * h nop.i 0 };; { .mfi nop.m 0 fma.s1 FR_GG = FR_GG, FR_EE, FR_GG // g = g * e + g // 64 bit Newton Raphson iteration nop.i 0 } { .mfi nop.m 0 fma.s1 FR_HH = FR_HH, FR_EE, FR_HH // h = h * e + h nop.i 0 };; { .mfi nop.m 0 fnma.s1 FR_DD = FR_GG, FR_GG, FR_P2 // Remainder d = g * g - p2 nop.i 0 } { .mfi nop.m 0 fma.s1 FR_XLog_Hi = FR_AX, f1, FR_GG // bh = z + gh nop.i 0 };; { .mfi nop.m 0 fma.s1 FR_DD = FR_DD, f1, FR_P2L // add p2l: d = d + p2l nop.i 0 };; { .mfi getf.sig GR_signif = FR_XLog_Hi // Get significand of x+1 fmerge.ns FR_Neg_One = f1, f1 // Form -1.0 mov GR_exp_2tom7 = 0x0fff8 // Exponent of 2^-7 };; { .mfi nop.m 0 fma.s1 FR_GL = FR_DD, FR_HH, f0 // gl = d * h extr.u GR_Index1 = GR_signif, 59, 4 // Get high 4 bits of signif } { .mfi nop.m 0 fma.s1 FR_XLog_Hi = FR_DD, FR_HH, FR_XLog_Hi // bh = bh + gl nop.i 0 };; { .mmi shladd GR_ad_z_1 = GR_Index1, 2, GR_ad_z_1 // Point to Z_1 shladd GR_ad_tbl_1 = GR_Index1, 4, GR_ad_tbl_1 // Point to G_1 extr.u GR_X_0 = GR_signif, 49, 15 // Get high 15 bits of signif. };; { .mmi ld4 GR_Z_1 = [GR_ad_z_1] // Load Z_1 nop.m 0 nop.i 0 };; { .mmi ldfps FR_G, FR_H = [GR_ad_tbl_1],8 // Load G_1, H_1 nop.m 0 nop.i 0 };; { .mfi nop.m 0 fms.s1 FR_XLog_Lo = FR_GG, f1, FR_XLog_Hi // bl = gh - bh pmpyshr2.u GR_X_1 = GR_X_0,GR_Z_1,15 // Get bits 30-15 of X_0 * Z_1 };; // WE CANNOT USE GR_X_1 IN NEXT 3 CYCLES BECAUSE OF POSSIBLE 10 CLOCKS STALL! // "DEAD" ZONE! { .mfi nop.m 0 nop.f 0 nop.i 0 };; { .mfi nop.m 0 fmerge.se FR_S_hi = f1,FR_XLog_Hi // Form |x+1| nop.i 0 };; { .mmi getf.exp GR_N = FR_XLog_Hi // Get N = exponent of x+1 ldfd FR_h = [GR_ad_tbl_1] // Load h_1 nop.i 0 };; { .mfi nop.m 0 nop.f 0 extr.u GR_Index2 = GR_X_1, 6, 4 // Extract bits 6-9 of X_1 };; { .mfi shladd GR_ad_tbl_2 = GR_Index2, 4, GR_ad_tbl_2 // Point to G_2 fma.s1 FR_XLog_Lo = FR_XLog_Lo, f1, FR_AX // bl = bl + x mov GR_exp_2tom80 = 0x0ffaf // Exponent of 2^-80 } { .mfi shladd GR_ad_z_2 = GR_Index2, 2, GR_ad_z_2 // Point to Z_2 nop.f 0 sub GR_N = GR_N, GR_Bias // sub bias from exp };; { .mmi ldfps FR_G2, FR_H2 = [GR_ad_tbl_2],8 // Load G_2, H_2 ld4 GR_Z_2 = [GR_ad_z_2] // Load Z_2 sub GR_minus_N = GR_Bias, GR_N // Form exponent of 2^(-N) };; { .mmi ldfd FR_h2 = [GR_ad_tbl_2] // Load h_2 nop.m 0 nop.i 0 };; { .mmi setf.sig FR_float_N = GR_N // Put integer N into rightmost sign setf.exp FR_2_to_minus_N = GR_minus_N // Form 2^(-N) pmpyshr2.u GR_X_2 = GR_X_1,GR_Z_2,15 // Get bits 30-15 of X_1 * Z_2 };; // WE CANNOT USE GR_X_2 IN NEXT 3 CYCLES ("DEAD" ZONE!) // BECAUSE OF POSSIBLE 10 CLOCKS STALL! // So we can negate Q coefficients there for negative values { .mfi nop.m 0 (p11) fma.s1 FR_Q1 = FR_Q1, FR_Neg_One, f0 // Negate Q1 nop.i 0 } { .mfi nop.m 0 fma.s1 FR_XLog_Lo = FR_XLog_Lo, f1, FR_GL // bl = bl + gl nop.i 0 };; { .mfi nop.m 0 (p11) fma.s1 FR_Q2 = FR_Q2, FR_Neg_One, f0 // Negate Q2 nop.i 0 };; { .mfi nop.m 0 (p11) fma.s1 FR_Q3 = FR_Q3, FR_Neg_One, f0 // Negate Q3 nop.i 0 };; { .mfi nop.m 0 (p11) fma.s1 FR_Q4 = FR_Q4, FR_Neg_One, f0 // Negate Q4 extr.u GR_Index3 = GR_X_2, 1, 5 // Extract bits 1-5 of X_2 };; { .mfi shladd GR_ad_tbl_3 = GR_Index3, 4, GR_ad_tbl_3 // Point to G_3 nop.f 0 nop.i 0 };; { .mfi ldfps FR_G3, FR_H3 = [GR_ad_tbl_3],8 // Load G_3, H_3 nop.f 0 nop.i 0 };; { .mfi ldfd FR_h3 = [GR_ad_tbl_3] // Load h_3 fcvt.xf FR_float_N = FR_float_N nop.i 0 };; { .mfi nop.m 0 fmpy.s1 FR_G = FR_G, FR_G2 // G = G_1 * G_2 nop.i 0 } { .mfi nop.m 0 fadd.s1 FR_H = FR_H, FR_H2 // H = H_1 + H_2 nop.i 0 };; { .mfi nop.m 0 fadd.s1 FR_h = FR_h, FR_h2 // h = h_1 + h_2 nop.i 0 } { .mfi nop.m 0 fma.s1 FR_S_lo = FR_XLog_Lo, FR_2_to_minus_N, f0 //S_lo=S_lo*2^-N nop.i 0 };; { .mfi nop.m 0 fmpy.s1 FR_G = FR_G, FR_G3 // G = (G_1 * G_2) * G_3 nop.i 0 } { .mfi nop.m 0 fadd.s1 FR_H = FR_H, FR_H3 // H = (H_1 + H_2) + H_3 nop.i 0 };; { .mfi nop.m 0 fadd.s1 FR_h = FR_h, FR_h3 // h = (h_1 + h_2) + h_3 nop.i 0 };; { .mfi nop.m 0 fms.s1 FR_r = FR_G, FR_S_hi, f1 // r = G * S_hi - 1 nop.i 0 } { .mfi nop.m 0 fma.s1 FR_Y_hi = FR_float_N, FR_log2_hi, FR_H // Y_hi=N*log2_hi+H nop.i 0 };; { .mfi nop.m 0 fma.s1 FR_h = FR_float_N, FR_log2_lo, FR_h // h=N*log2_lo+h nop.i 0 } { .mfi nop.m 0 fma.s1 FR_r = FR_G, FR_S_lo, FR_r // r=G*S_lo+(G*S_hi-1) nop.i 0 };; { .mfi nop.m 0 fma.s1 FR_poly_lo = FR_r, FR_Q4, FR_Q3 // poly_lo = r * Q4 + Q3 nop.i 0 } { .mfi nop.m 0 fmpy.s1 FR_rsq = FR_r, FR_r // rsq = r * r nop.i 0 };; { .mfi nop.m 0 fma.s1 FR_poly_lo = FR_poly_lo, FR_r, FR_Q2 // poly_lo=poly_lo*r+Q2 nop.i 0 } { .mfi nop.m 0 fma.s1 FR_rcub = FR_rsq, FR_r, f0 // rcub = r^3 nop.i 0 };; .pred.rel "mutex",p12,p11 { .mfi nop.m 0 (p12) fma.s1 FR_poly_hi = FR_Q1, FR_rsq, FR_r // poly_hi = Q1*rsq + r nop.i 0 } { .mfi nop.m 0 (p11) fms.s1 FR_poly_hi = FR_Q1, FR_rsq, FR_r // poly_hi = Q1*rsq + r nop.i 0 };; .pred.rel "mutex",p12,p11 { .mfi nop.m 0 (p12) fma.s1 FR_poly_lo = FR_poly_lo, FR_rcub, FR_h//poly_lo=poly_lo*r^3+h nop.i 0 } { .mfi nop.m 0 (p11) fms.s1 FR_poly_lo = FR_poly_lo, FR_rcub, FR_h//poly_lo=poly_lo*r^3+h nop.i 0 } ;; { .mfi nop.m 0 fadd.s0 FR_Y_lo = FR_poly_hi, FR_poly_lo // Y_lo=poly_hi+poly_lo nop.i 0 } { .mfi nop.m 0 (p11) fma.s0 FR_Y_hi = FR_Y_hi, FR_Neg_One, f0 // FR_Y_hi sign for neg nop.i 0 };; { .mfb nop.m 0 fadd.s0 FR_Res = FR_Y_lo,FR_Y_hi // Result=Y_lo+Y_hi br.ret.sptk b0 // Common exit for 2^-7 < x < inf };; // * SPECIAL VERSION OF LOGL FOR HUGE ARGUMENTS * huges_logl: { .mfi getf.sig GR_signif = FR_XLog_Hi // Get significand of x+1 fmerge.ns FR_Neg_One = f1, f1 // Form -1.0 mov GR_exp_2tom7 = 0x0fff8 // Exponent of 2^-7 };; { .mfi add GR_ad_tbl_1 = 0x040, GR_ad_z_1 // Point to Constants_G_H_h1 nop.f 0 add GR_ad_q = -0x60, GR_ad_z_1 // Point to Constants_P } { .mfi add GR_ad_z_2 = 0x140, GR_ad_z_1 // Point to Constants_Z_2 nop.f 0 add GR_ad_tbl_2 = 0x180, GR_ad_z_1 // Point to Constants_G_H_h2 };; { .mfi nop.m 0 nop.f 0 extr.u GR_Index1 = GR_signif, 59, 4 // Get high 4 bits of signif } { .mfi add GR_ad_tbl_3 = 0x280, GR_ad_z_1 // Point to Constants_G_H_h3 nop.f 0 nop.i 0 };; { .mfi shladd GR_ad_z_1 = GR_Index1, 2, GR_ad_z_1 // Point to Z_1 nop.f 0 extr.u GR_X_0 = GR_signif, 49, 15 // Get high 15 bits of signif. };; { .mfi ld4 GR_Z_1 = [GR_ad_z_1] // Load Z_1 nop.f 0 mov GR_exp_mask = 0x1FFFF // Create exponent mask } { .mfi shladd GR_ad_tbl_1 = GR_Index1, 4, GR_ad_tbl_1 // Point to G_1 nop.f 0 mov GR_Bias = 0x0FFFF // Create exponent bias };; { .mfi ldfps FR_G, FR_H = [GR_ad_tbl_1],8 // Load G_1, H_1 fmerge.se FR_S_hi = f1,FR_XLog_Hi // Form |x+1| nop.i 0 };; { .mmi getf.exp GR_N = FR_XLog_Hi // Get N = exponent of x+1 ldfd FR_h = [GR_ad_tbl_1] // Load h_1 nop.i 0 };; { .mfi ldfe FR_log2_hi = [GR_ad_q],16 // Load log2_hi nop.f 0 pmpyshr2.u GR_X_1 = GR_X_0,GR_Z_1,15 // Get bits 30-15 of X_0 * Z_1 };; // WE CANNOT USE GR_X_1 IN NEXT 3 CYCLES BECAUSE OF POSSIBLE 10 CLOCKS STALL! // "DEAD" ZONE! { .mmi ldfe FR_log2_lo = [GR_ad_q],16 // Load log2_lo sub GR_N = GR_N, GR_Bias mov GR_exp_2tom80 = 0x0ffaf // Exponent of 2^-80 };; { .mfi ldfe FR_Q4 = [GR_ad_q],16 // Load Q4 nop.f 0 sub GR_minus_N = GR_Bias, GR_N // Form exponent of 2^(-N) };; { .mmf ldfe FR_Q3 = [GR_ad_q],16 // Load Q3 setf.sig FR_float_N = GR_N // Put integer N into rightmost sign nop.f 0 };; { .mmi nop.m 0 ldfe FR_Q2 = [GR_ad_q],16 // Load Q2 extr.u GR_Index2 = GR_X_1, 6, 4 // Extract bits 6-9 of X_1 };; { .mmi ldfe FR_Q1 = [GR_ad_q] // Load Q1 shladd GR_ad_z_2 = GR_Index2, 2, GR_ad_z_2 // Point to Z_2 nop.i 0 };; { .mmi ld4 GR_Z_2 = [GR_ad_z_2] // Load Z_2 shladd GR_ad_tbl_2 = GR_Index2, 4, GR_ad_tbl_2 // Point to G_2 nop.i 0 };; { .mmi ldfps FR_G2, FR_H2 = [GR_ad_tbl_2],8 // Load G_2, H_2 nop.m 0 nop.i 0 };; { .mfi ldfd FR_h2 = [GR_ad_tbl_2] // Load h_2 nop.f 0 nop.i 0 } { .mfi setf.exp FR_2_to_minus_N = GR_minus_N // Form 2^(-N) nop.f 0 nop.i 0 };; { .mfi nop.m 0 nop.f 0 pmpyshr2.u GR_X_2 = GR_X_1,GR_Z_2,15 // Get bits 30-15 of X_1 * Z_2 };; // WE CANNOT USE GR_X_2 IN NEXT 3 CYCLES BECAUSE OF POSSIBLE 10 CLOCKS STALL! // "DEAD" ZONE! // JUST HAVE TO INSERT 3 NOP CYCLES (nothing to do here) { .mfi nop.m 0 nop.f 0 nop.i 0 };; { .mfi nop.m 0 nop.f 0 nop.i 0 };; { .mfi nop.m 0 nop.f 0 nop.i 0 };; { .mfi nop.m 0 (p11) fma.s1 FR_Q4 = FR_Q4, FR_Neg_One, f0 // Negate Q4 extr.u GR_Index3 = GR_X_2, 1, 5 // Extract bits 1-5 of X_2 };; { .mfi shladd GR_ad_tbl_3 = GR_Index3, 4, GR_ad_tbl_3 // Point to G_3 fcvt.xf FR_float_N = FR_float_N nop.i 0 } { .mfi nop.m 0 (p11) fma.s1 FR_Q3 = FR_Q3, FR_Neg_One, f0 // Negate Q3 nop.i 0 };; { .mfi ldfps FR_G3, FR_H3 = [GR_ad_tbl_3],8 // Load G_3, H_3 (p11) fma.s1 FR_Q2 = FR_Q2, FR_Neg_One, f0 // Negate Q2 nop.i 0 } { .mfi nop.m 0 (p11) fma.s1 FR_Q1 = FR_Q1, FR_Neg_One, f0 // Negate Q1 nop.i 0 };; { .mfi ldfd FR_h3 = [GR_ad_tbl_3] // Load h_3 fmpy.s1 FR_G = FR_G, FR_G2 // G = G_1 * G_2 nop.i 0 } { .mfi nop.m 0 fadd.s1 FR_H = FR_H, FR_H2 // H = H_1 + H_2 nop.i 0 };; { .mmf nop.m 0 nop.m 0 fadd.s1 FR_h = FR_h, FR_h2 // h = h_1 + h_2 };; { .mfi nop.m 0 fmpy.s1 FR_G = FR_G, FR_G3 // G = (G_1 * G_2) * G_3 nop.i 0 } { .mfi nop.m 0 fadd.s1 FR_H = FR_H, FR_H3 // H = (H_1 + H_2) + H_3 nop.i 0 };; { .mfi nop.m 0 fadd.s1 FR_h = FR_h, FR_h3 // h = (h_1 + h_2) + h_3 nop.i 0 };; { .mfi nop.m 0 fms.s1 FR_r = FR_G, FR_S_hi, f1 // r = G * S_hi - 1 nop.i 0 } { .mfi nop.m 0 fma.s1 FR_Y_hi = FR_float_N, FR_log2_hi, FR_H // Y_hi=N*log2_hi+H nop.i 0 };; { .mfi nop.m 0 fma.s1 FR_h = FR_float_N, FR_log2_lo, FR_h // h=N*log2_lo+h nop.i 0 };; { .mfi nop.m 0 fma.s1 FR_poly_lo = FR_r, FR_Q4, FR_Q3 // poly_lo = r * Q4 + Q3 nop.i 0 } { .mfi nop.m 0 fmpy.s1 FR_rsq = FR_r, FR_r // rsq = r * r nop.i 0 };; { .mfi nop.m 0 fma.s1 FR_poly_lo = FR_poly_lo, FR_r, FR_Q2 // poly_lo=poly_lo*r+Q2 nop.i 0 } { .mfi nop.m 0 fma.s1 FR_rcub = FR_rsq, FR_r, f0 // rcub = r^3 nop.i 0 };; .pred.rel "mutex",p12,p11 { .mfi nop.m 0 (p12) fma.s1 FR_poly_hi = FR_Q1, FR_rsq, FR_r // poly_hi = Q1*rsq + r nop.i 0 } { .mfi nop.m 0 (p11) fms.s1 FR_poly_hi = FR_Q1, FR_rsq, FR_r // poly_hi = Q1*rsq + r nop.i 0 };; .pred.rel "mutex",p12,p11 { .mfi nop.m 0 (p12) fma.s1 FR_poly_lo = FR_poly_lo, FR_rcub, FR_h//poly_lo=poly_lo*r^3+h nop.i 0 } { .mfi nop.m 0 (p11) fms.s1 FR_poly_lo = FR_poly_lo, FR_rcub, FR_h//poly_lo=poly_lo*r^3+h nop.i 0 };; { .mfi nop.m 0 fadd.s0 FR_Y_lo = FR_poly_hi, FR_poly_lo // Y_lo=poly_hi+poly_lo nop.i 0 } { .mfi nop.m 0 (p11) fma.s0 FR_Y_hi = FR_Y_hi, FR_Neg_One, f0 // FR_Y_hi sign for neg nop.i 0 };; { .mfb nop.m 0 fadd.s0 FR_Res = FR_Y_lo,FR_Y_hi // Result=Y_lo+Y_hi br.ret.sptk b0 // Common exit for 2^-7 < x < inf };; // NEAR ZERO POLYNOMIAL INTERVAL near_0: { .mfi nop.m 0 fma.s1 FR_X4 = FR_X2, FR_X2, f0 // x^4 = x^2 * x^2 nop.i 0 };; { .mfi nop.m 0 fma.s1 FR_P9 = FR_C9,FR_X2,FR_C7 // p9 = C9*x^2 + C7 nop.i 0 } { .mfi nop.m 0 fma.s1 FR_P5 = FR_C5,FR_X2,FR_C3 // p5 = C5*x^2 + C3 nop.i 0 };; { .mfi nop.m 0 fma.s1 FR_P3 = FR_P9,FR_X4,FR_P5 // p3 = p9*x^4 + p5 nop.i 0 };; { .mfb nop.m 0 fma.s0 FR_Res = FR_P3,FR_X3,FR_Arg // res = p3*C3 + x br.ret.sptk b0 // Near 0 path return };; GLOBAL_LIBM_END(asinhl)