mirror of git://git.musl-libc.org/musl
math: new powf
from https://github.com/ARM-software/optimized-routines, commit 04884bd04eac4b251da4026900010ea7d8850edc POWF_SCALE != 1.0 case only matters if TOINT_INTRINSICS is set, which is currently not supported for any target. SNaN is not supported, it would require an issignalingf implementation. code size change: -816 bytes. benchmark on x86_64 before, after, speedup: -Os: powf rthruput: 95.14 ns/call 20.04 ns/call 4.75x powf latency: 137.00 ns/call 34.98 ns/call 3.92x -O3: powf rthruput: 92.48 ns/call 13.67 ns/call 6.77x powf latency: 131.11 ns/call 35.15 ns/call 3.73x
This commit is contained in:
Szabolcs Nagy
Rich Felker
parent
3f94c648ef
commit
d28cd0ad42
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@ -64,6 +64,12 @@ union ldshape {
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/* Support signaling NaNs. */
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#define WANT_SNAN 0
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#if WANT_SNAN
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#error SNaN is unsupported
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#else
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#define issignalingf_inline(x) 0
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#endif
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#ifndef TOINT_INTRINSICS
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#define TOINT_INTRINSICS 0
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#endif
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412
src/math/powf.c
412
src/math/powf.c
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@ -1,259 +1,185 @@
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/* origin: FreeBSD /usr/src/lib/msun/src/e_powf.c */
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/*
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* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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*/
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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* Copyright (c) 2017-2018, Arm Limited.
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* SPDX-License-Identifier: MIT
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*/
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#include <math.h>
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#include <stdint.h>
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#include "libm.h"
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#include "exp2f_data.h"
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#include "powf_data.h"
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static const float
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bp[] = {1.0, 1.5,},
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dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
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dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
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two24 = 16777216.0, /* 0x4b800000 */
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huge = 1.0e30,
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tiny = 1.0e-30,
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/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
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L1 = 6.0000002384e-01, /* 0x3f19999a */
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L2 = 4.2857143283e-01, /* 0x3edb6db7 */
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L3 = 3.3333334327e-01, /* 0x3eaaaaab */
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L4 = 2.7272811532e-01, /* 0x3e8ba305 */
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L5 = 2.3066075146e-01, /* 0x3e6c3255 */
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L6 = 2.0697501302e-01, /* 0x3e53f142 */
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P1 = 1.6666667163e-01, /* 0x3e2aaaab */
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P2 = -2.7777778450e-03, /* 0xbb360b61 */
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P3 = 6.6137559770e-05, /* 0x388ab355 */
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P4 = -1.6533901999e-06, /* 0xb5ddea0e */
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P5 = 4.1381369442e-08, /* 0x3331bb4c */
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lg2 = 6.9314718246e-01, /* 0x3f317218 */
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lg2_h = 6.93145752e-01, /* 0x3f317200 */
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lg2_l = 1.42860654e-06, /* 0x35bfbe8c */
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ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
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cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
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cp_h = 9.6191406250e-01, /* 0x3f764000 =12b cp */
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cp_l = -1.1736857402e-04, /* 0xb8f623c6 =tail of cp_h */
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ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
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ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
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ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
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/*
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POWF_LOG2_POLY_ORDER = 5
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EXP2F_TABLE_BITS = 5
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ULP error: 0.82 (~ 0.5 + relerr*2^24)
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relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
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relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
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relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
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*/
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#define N (1 << POWF_LOG2_TABLE_BITS)
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#define T __powf_log2_data.tab
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#define A __powf_log2_data.poly
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#define OFF 0x3f330000
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/* Subnormal input is normalized so ix has negative biased exponent.
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Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */
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static inline double_t log2_inline(uint32_t ix)
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{
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double_t z, r, r2, r4, p, q, y, y0, invc, logc;
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uint32_t iz, top, tmp;
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int k, i;
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/* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
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The range is split into N subintervals.
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The ith subinterval contains z and c is near its center. */
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tmp = ix - OFF;
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i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
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top = tmp & 0xff800000;
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iz = ix - top;
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k = (int32_t)top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
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invc = T[i].invc;
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logc = T[i].logc;
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z = (double_t)asfloat(iz);
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/* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
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r = z * invc - 1;
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y0 = logc + (double_t)k;
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/* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */
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r2 = r * r;
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y = A[0] * r + A[1];
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p = A[2] * r + A[3];
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r4 = r2 * r2;
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q = A[4] * r + y0;
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q = p * r2 + q;
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y = y * r4 + q;
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return y;
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}
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#undef N
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#undef T
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#define N (1 << EXP2F_TABLE_BITS)
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#define T __exp2f_data.tab
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#define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
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/* The output of log2 and thus the input of exp2 is either scaled by N
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(in case of fast toint intrinsics) or not. The unscaled xd must be
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in [-1021,1023], sign_bias sets the sign of the result. */
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static inline float exp2_inline(double_t xd, uint32_t sign_bias)
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{
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uint64_t ki, ski, t;
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double_t kd, z, r, r2, y, s;
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#if TOINT_INTRINSICS
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#define C __exp2f_data.poly_scaled
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/* N*x = k + r with r in [-1/2, 1/2] */
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kd = roundtoint(xd); /* k */
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ki = converttoint(xd);
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#else
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#define C __exp2f_data.poly
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#define SHIFT __exp2f_data.shift_scaled
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/* x = k/N + r with r in [-1/(2N), 1/(2N)] */
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kd = eval_as_double(xd + SHIFT);
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ki = asuint64(kd);
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kd -= SHIFT; /* k/N */
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#endif
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r = xd - kd;
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/* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
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t = T[ki % N];
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ski = ki + sign_bias;
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t += ski << (52 - EXP2F_TABLE_BITS);
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s = asdouble(t);
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z = C[0] * r + C[1];
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r2 = r * r;
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y = C[2] * r + 1;
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y = z * r2 + y;
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y = y * s;
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return eval_as_float(y);
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}
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/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is
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the bit representation of a non-zero finite floating-point value. */
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static inline int checkint(uint32_t iy)
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{
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int e = iy >> 23 & 0xff;
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if (e < 0x7f)
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return 0;
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if (e > 0x7f + 23)
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return 2;
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if (iy & ((1 << (0x7f + 23 - e)) - 1))
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return 0;
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if (iy & (1 << (0x7f + 23 - e)))
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return 1;
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return 2;
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}
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static inline int zeroinfnan(uint32_t ix)
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{
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return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
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}
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float powf(float x, float y)
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{
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float z,ax,z_h,z_l,p_h,p_l;
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float y1,t1,t2,r,s,sn,t,u,v,w;
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int32_t i,j,k,yisint,n;
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int32_t hx,hy,ix,iy,is;
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uint32_t sign_bias = 0;
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uint32_t ix, iy;
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GET_FLOAT_WORD(hx, x);
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GET_FLOAT_WORD(hy, y);
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ix = hx & 0x7fffffff;
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iy = hy & 0x7fffffff;
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/* x**0 = 1, even if x is NaN */
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if (iy == 0)
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return 1.0f;
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/* 1**y = 1, even if y is NaN */
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if (hx == 0x3f800000)
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return 1.0f;
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/* NaN if either arg is NaN */
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if (ix > 0x7f800000 || iy > 0x7f800000)
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return x + y;
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/* determine if y is an odd int when x < 0
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* yisint = 0 ... y is not an integer
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* yisint = 1 ... y is an odd int
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* yisint = 2 ... y is an even int
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*/
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yisint = 0;
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if (hx < 0) {
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if (iy >= 0x4b800000)
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yisint = 2; /* even integer y */
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else if (iy >= 0x3f800000) {
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k = (iy>>23) - 0x7f; /* exponent */
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j = iy>>(23-k);
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if ((j<<(23-k)) == iy)
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yisint = 2 - (j & 1);
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ix = asuint(x);
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iy = asuint(y);
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if (predict_false(ix - 0x00800000 >= 0x7f800000 - 0x00800000 ||
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zeroinfnan(iy))) {
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/* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */
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if (predict_false(zeroinfnan(iy))) {
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if (2 * iy == 0)
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return issignalingf_inline(x) ? x + y : 1.0f;
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if (ix == 0x3f800000)
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return issignalingf_inline(y) ? x + y : 1.0f;
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if (2 * ix > 2u * 0x7f800000 ||
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2 * iy > 2u * 0x7f800000)
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return x + y;
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if (2 * ix == 2 * 0x3f800000)
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return 1.0f;
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if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
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return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
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return y * y;
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}
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}
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/* special value of y */
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if (iy == 0x7f800000) { /* y is +-inf */
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if (ix == 0x3f800000) /* (-1)**+-inf is 1 */
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return 1.0f;
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else if (ix > 0x3f800000) /* (|x|>1)**+-inf = inf,0 */
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return hy >= 0 ? y : 0.0f;
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else /* (|x|<1)**+-inf = 0,inf */
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return hy >= 0 ? 0.0f: -y;
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}
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if (iy == 0x3f800000) /* y is +-1 */
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return hy >= 0 ? x : 1.0f/x;
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if (hy == 0x40000000) /* y is 2 */
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return x*x;
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if (hy == 0x3f000000) { /* y is 0.5 */
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if (hx >= 0) /* x >= +0 */
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return sqrtf(x);
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}
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ax = fabsf(x);
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/* special value of x */
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if (ix == 0x7f800000 || ix == 0 || ix == 0x3f800000) { /* x is +-0,+-inf,+-1 */
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z = ax;
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if (hy < 0) /* z = (1/|x|) */
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z = 1.0f/z;
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if (hx < 0) {
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if (((ix-0x3f800000)|yisint) == 0) {
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z = (z-z)/(z-z); /* (-1)**non-int is NaN */
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} else if (yisint == 1)
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z = -z; /* (x<0)**odd = -(|x|**odd) */
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if (predict_false(zeroinfnan(ix))) {
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float_t x2 = x * x;
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if (ix & 0x80000000 && checkint(iy) == 1)
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x2 = -x2;
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/* Without the barrier some versions of clang hoist the 1/x2 and
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thus division by zero exception can be signaled spuriously. */
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return iy & 0x80000000 ? fp_barrierf(1 / x2) : x2;
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}
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/* x and y are non-zero finite. */
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if (ix & 0x80000000) {
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/* Finite x < 0. */
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int yint = checkint(iy);
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if (yint == 0)
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return __math_invalidf(x);
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if (yint == 1)
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sign_bias = SIGN_BIAS;
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ix &= 0x7fffffff;
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}
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return z;
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}
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sn = 1.0f; /* sign of result */
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if (hx < 0) {
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if (yisint == 0) /* (x<0)**(non-int) is NaN */
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return (x-x)/(x-x);
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if (yisint == 1) /* (x<0)**(odd int) */
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sn = -1.0f;
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}
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/* |y| is huge */
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if (iy > 0x4d000000) { /* if |y| > 2**27 */
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/* over/underflow if x is not close to one */
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if (ix < 0x3f7ffff8)
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return hy < 0 ? sn*huge*huge : sn*tiny*tiny;
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if (ix > 0x3f800007)
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return hy > 0 ? sn*huge*huge : sn*tiny*tiny;
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/* now |1-x| is tiny <= 2**-20, suffice to compute
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log(x) by x-x^2/2+x^3/3-x^4/4 */
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t = ax - 1; /* t has 20 trailing zeros */
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w = (t*t)*(0.5f - t*(0.333333333333f - t*0.25f));
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u = ivln2_h*t; /* ivln2_h has 16 sig. bits */
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v = t*ivln2_l - w*ivln2;
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t1 = u + v;
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GET_FLOAT_WORD(is, t1);
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SET_FLOAT_WORD(t1, is & 0xfffff000);
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t2 = v - (t1-u);
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} else {
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float s2,s_h,s_l,t_h,t_l;
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n = 0;
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/* take care subnormal number */
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if (ix < 0x00800000) {
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ax *= two24;
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n -= 24;
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GET_FLOAT_WORD(ix, ax);
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/* Normalize subnormal x so exponent becomes negative. */
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ix = asuint(x * 0x1p23f);
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ix &= 0x7fffffff;
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ix -= 23 << 23;
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}
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n += ((ix)>>23) - 0x7f;
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j = ix & 0x007fffff;
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/* determine interval */
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ix = j | 0x3f800000; /* normalize ix */
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if (j <= 0x1cc471) /* |x|<sqrt(3/2) */
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k = 0;
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else if (j < 0x5db3d7) /* |x|<sqrt(3) */
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k = 1;
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else {
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k = 0;
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n += 1;
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ix -= 0x00800000;
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}
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SET_FLOAT_WORD(ax, ix);
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/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
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u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
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v = 1.0f/(ax+bp[k]);
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s = u*v;
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s_h = s;
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GET_FLOAT_WORD(is, s_h);
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SET_FLOAT_WORD(s_h, is & 0xfffff000);
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/* t_h=ax+bp[k] High */
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is = ((ix>>1) & 0xfffff000) | 0x20000000;
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SET_FLOAT_WORD(t_h, is + 0x00400000 + (k<<21));
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t_l = ax - (t_h - bp[k]);
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s_l = v*((u - s_h*t_h) - s_h*t_l);
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/* compute log(ax) */
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s2 = s*s;
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r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
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r += s_l*(s_h+s);
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s2 = s_h*s_h;
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t_h = 3.0f + s2 + r;
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GET_FLOAT_WORD(is, t_h);
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SET_FLOAT_WORD(t_h, is & 0xfffff000);
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t_l = r - ((t_h - 3.0f) - s2);
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/* u+v = s*(1+...) */
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u = s_h*t_h;
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v = s_l*t_h + t_l*s;
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/* 2/(3log2)*(s+...) */
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p_h = u + v;
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GET_FLOAT_WORD(is, p_h);
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SET_FLOAT_WORD(p_h, is & 0xfffff000);
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p_l = v - (p_h - u);
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z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
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z_l = cp_l*p_h + p_l*cp+dp_l[k];
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/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
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t = (float)n;
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t1 = (((z_h + z_l) + dp_h[k]) + t);
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GET_FLOAT_WORD(is, t1);
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SET_FLOAT_WORD(t1, is & 0xfffff000);
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t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
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}
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||||
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||||
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
|
||||
GET_FLOAT_WORD(is, y);
|
||||
SET_FLOAT_WORD(y1, is & 0xfffff000);
|
||||
p_l = (y-y1)*t1 + y*t2;
|
||||
p_h = y1*t1;
|
||||
z = p_l + p_h;
|
||||
GET_FLOAT_WORD(j, z);
|
||||
if (j > 0x43000000) /* if z > 128 */
|
||||
return sn*huge*huge; /* overflow */
|
||||
else if (j == 0x43000000) { /* if z == 128 */
|
||||
if (p_l + ovt > z - p_h)
|
||||
return sn*huge*huge; /* overflow */
|
||||
} else if ((j&0x7fffffff) > 0x43160000) /* z < -150 */ // FIXME: check should be (uint32_t)j > 0xc3160000
|
||||
return sn*tiny*tiny; /* underflow */
|
||||
else if (j == 0xc3160000) { /* z == -150 */
|
||||
if (p_l <= z-p_h)
|
||||
return sn*tiny*tiny; /* underflow */
|
||||
double_t logx = log2_inline(ix);
|
||||
double_t ylogx = y * logx; /* cannot overflow, y is single prec. */
|
||||
if (predict_false((asuint64(ylogx) >> 47 & 0xffff) >=
|
||||
asuint64(126.0 * POWF_SCALE) >> 47)) {
|
||||
/* |y*log(x)| >= 126. */
|
||||
if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
|
||||
return __math_oflowf(sign_bias);
|
||||
if (ylogx <= -150.0 * POWF_SCALE)
|
||||
return __math_uflowf(sign_bias);
|
||||
}
|
||||
/*
|
||||
* compute 2**(p_h+p_l)
|
||||
*/
|
||||
i = j & 0x7fffffff;
|
||||
k = (i>>23) - 0x7f;
|
||||
n = 0;
|
||||
if (i > 0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */
|
||||
n = j + (0x00800000>>(k+1));
|
||||
k = ((n&0x7fffffff)>>23) - 0x7f; /* new k for n */
|
||||
SET_FLOAT_WORD(t, n & ~(0x007fffff>>k));
|
||||
n = ((n&0x007fffff)|0x00800000)>>(23-k);
|
||||
if (j < 0)
|
||||
n = -n;
|
||||
p_h -= t;
|
||||
}
|
||||
t = p_l + p_h;
|
||||
GET_FLOAT_WORD(is, t);
|
||||
SET_FLOAT_WORD(t, is & 0xffff8000);
|
||||
u = t*lg2_h;
|
||||
v = (p_l-(t-p_h))*lg2 + t*lg2_l;
|
||||
z = u + v;
|
||||
w = v - (z - u);
|
||||
t = z*z;
|
||||
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
|
||||
r = (z*t1)/(t1-2.0f) - (w+z*w);
|
||||
z = 1.0f - (r - z);
|
||||
GET_FLOAT_WORD(j, z);
|
||||
j += n<<23;
|
||||
if ((j>>23) <= 0) /* subnormal output */
|
||||
z = scalbnf(z, n);
|
||||
else
|
||||
SET_FLOAT_WORD(z, j);
|
||||
return sn*z;
|
||||
return exp2_inline(ylogx, sign_bias);
|
||||
}
|
||||
|
|
|
|||
|
|
@ -0,0 +1,34 @@
|
|||
/*
|
||||
* Data definition for powf.
|
||||
*
|
||||
* Copyright (c) 2017-2018, Arm Limited.
|
||||
* SPDX-License-Identifier: MIT
|
||||
*/
|
||||
|
||||
#include "powf_data.h"
|
||||
|
||||
const struct powf_log2_data __powf_log2_data = {
|
||||
.tab = {
|
||||
{ 0x1.661ec79f8f3bep+0, -0x1.efec65b963019p-2 * POWF_SCALE },
|
||||
{ 0x1.571ed4aaf883dp+0, -0x1.b0b6832d4fca4p-2 * POWF_SCALE },
|
||||
{ 0x1.49539f0f010bp+0, -0x1.7418b0a1fb77bp-2 * POWF_SCALE },
|
||||
{ 0x1.3c995b0b80385p+0, -0x1.39de91a6dcf7bp-2 * POWF_SCALE },
|
||||
{ 0x1.30d190c8864a5p+0, -0x1.01d9bf3f2b631p-2 * POWF_SCALE },
|
||||
{ 0x1.25e227b0b8eap+0, -0x1.97c1d1b3b7afp-3 * POWF_SCALE },
|
||||
{ 0x1.1bb4a4a1a343fp+0, -0x1.2f9e393af3c9fp-3 * POWF_SCALE },
|
||||
{ 0x1.12358f08ae5bap+0, -0x1.960cbbf788d5cp-4 * POWF_SCALE },
|
||||
{ 0x1.0953f419900a7p+0, -0x1.a6f9db6475fcep-5 * POWF_SCALE },
|
||||
{ 0x1p+0, 0x0p+0 * POWF_SCALE },
|
||||
{ 0x1.e608cfd9a47acp-1, 0x1.338ca9f24f53dp-4 * POWF_SCALE },
|
||||
{ 0x1.ca4b31f026aap-1, 0x1.476a9543891bap-3 * POWF_SCALE },
|
||||
{ 0x1.b2036576afce6p-1, 0x1.e840b4ac4e4d2p-3 * POWF_SCALE },
|
||||
{ 0x1.9c2d163a1aa2dp-1, 0x1.40645f0c6651cp-2 * POWF_SCALE },
|
||||
{ 0x1.886e6037841edp-1, 0x1.88e9c2c1b9ff8p-2 * POWF_SCALE },
|
||||
{ 0x1.767dcf5534862p-1, 0x1.ce0a44eb17bccp-2 * POWF_SCALE },
|
||||
},
|
||||
.poly = {
|
||||
0x1.27616c9496e0bp-2 * POWF_SCALE, -0x1.71969a075c67ap-2 * POWF_SCALE,
|
||||
0x1.ec70a6ca7baddp-2 * POWF_SCALE, -0x1.7154748bef6c8p-1 * POWF_SCALE,
|
||||
0x1.71547652ab82bp0 * POWF_SCALE,
|
||||
}
|
||||
};
|
||||
|
|
@ -0,0 +1,26 @@
|
|||
/*
|
||||
* Copyright (c) 2017-2018, Arm Limited.
|
||||
* SPDX-License-Identifier: MIT
|
||||
*/
|
||||
#ifndef _POWF_DATA_H
|
||||
#define _POWF_DATA_H
|
||||
|
||||
#include "libm.h"
|
||||
#include "exp2f_data.h"
|
||||
|
||||
#define POWF_LOG2_TABLE_BITS 4
|
||||
#define POWF_LOG2_POLY_ORDER 5
|
||||
#if TOINT_INTRINSICS
|
||||
#define POWF_SCALE_BITS EXP2F_TABLE_BITS
|
||||
#else
|
||||
#define POWF_SCALE_BITS 0
|
||||
#endif
|
||||
#define POWF_SCALE ((double)(1 << POWF_SCALE_BITS))
|
||||
extern hidden const struct powf_log2_data {
|
||||
struct {
|
||||
double invc, logc;
|
||||
} tab[1 << POWF_LOG2_TABLE_BITS];
|
||||
double poly[POWF_LOG2_POLY_ORDER];
|
||||
} __powf_log2_data;
|
||||
|
||||
#endif
|
||||
Loading…
Reference in New Issue