diff options
Diffstat (limited to 'sysdeps/ieee754/flt-32/e_powf.c')
| -rw-r--r-- | sysdeps/ieee754/flt-32/e_powf.c | 388 |
1 files changed, 189 insertions, 199 deletions
diff --git a/sysdeps/ieee754/flt-32/e_powf.c b/sysdeps/ieee754/flt-32/e_powf.c index ce8e11f1ea..644a18d05e 100644 --- a/sysdeps/ieee754/flt-32/e_powf.c +++ b/sysdeps/ieee754/flt-32/e_powf.c @@ -1,7 +1,5 @@ -/* e_powf.c -- float version of e_pow.c. - * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. - */ -/* Copyright (C) 2017 Free Software Foundation, Inc. +/* Single-precision pow function. + Copyright (C) 2017 Free Software Foundation, Inc. This file is part of the GNU C Library. The GNU C Library is free software; you can redistribute it and/or @@ -18,210 +16,202 @@ License along with the GNU C Library; if not, see <http://www.gnu.org/licenses/>. */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - #include <math.h> -#include <math_private.h> - -static const float huge = 1.0e+30, tiny = 1.0e-30; - -static const float -bp[] = {1.0, 1.5,}, -zero = 0.0, -one = 1.0, -two = 2.0, -two24 = 16777216.0, /* 0x4b800000 */ - /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ -L1 = 6.0000002384e-01, /* 0x3f19999a */ -L2 = 4.2857143283e-01, /* 0x3edb6db7 */ -L3 = 3.3333334327e-01, /* 0x3eaaaaab */ -L4 = 2.7272811532e-01, /* 0x3e8ba305 */ -L5 = 2.3066075146e-01, /* 0x3e6c3255 */ -L6 = 2.0697501302e-01, /* 0x3e53f142 */ -P1 = 1.6666667163e-01, /* 0x3e2aaaab */ -P2 = -2.7777778450e-03, /* 0xbb360b61 */ -P3 = 6.6137559770e-05, /* 0x388ab355 */ -P4 = -1.6533901999e-06, /* 0xb5ddea0e */ -P5 = 4.1381369442e-08, /* 0x3331bb4c */ -ovt = 4.2995665694e-08; /* -(128-log2(ovfl+.5ulp)) */ - -static const double - dp[] = { 0.0, 0x1.2b803473f7ad1p-1, }, /* log2(1.5) */ - lg2 = M_LN2, - cp = 2.0/3.0/M_LN2, - invln2 = 1.0/M_LN2; +#include <stdint.h> +#include "math_config.h" -float -__ieee754_powf(float x, float y) +/* +POWF_LOG2_POLY_ORDER = 5 +EXP2F_TABLE_BITS = 5 + +ULP error: 0.82 (~ 0.5 + relerr*2^24) +relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2) +relerr_log2: 1.83 * 2^-33 (Relative error of logx.) +relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).) +*/ + +#define N (1 << POWF_LOG2_TABLE_BITS) +#define T __powf_log2_data.tab +#define A __powf_log2_data.poly +#define OFF 0x3f330000 + +/* Subnormal input is normalized so ix has negative biased exponent. + Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */ +static inline double_t +log2_inline (uint32_t ix) { - float z, ax, s; - double d1, d2; - int32_t i,j,k,yisint,n; - int32_t hx,hy,ix,iy; - - GET_FLOAT_WORD(hy,y); - iy = hy&0x7fffffff; - - /* y==zero: x**0 = 1 */ - if(iy==0 && !issignaling (x)) return one; - - /* x==+-1 */ - if(x == 1.0 && !issignaling (y)) return one; - if(x == -1.0 && isinf(y)) return one; - - GET_FLOAT_WORD(hx,x); - ix = hx&0x7fffffff; - - /* +-NaN return x+y */ - if(__builtin_expect(ix > 0x7f800000 || - iy > 0x7f800000, 0)) - return x+y; - - /* special value of y */ - if (__builtin_expect(iy==0x7f800000, 0)) { /* y is +-inf */ - if (ix==0x3f800000) - return y - y; /* inf**+-1 is NaN */ - else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */ - return (hy>=0)? y: zero; - else /* (|x|<1)**-,+inf = inf,0 */ - return (hy<0)?-y: zero; - } - if(iy==0x3f800000) { /* y is +-1 */ - if(hy<0) return one/x; else return x; - } - if(hy==0x40000000) return x*x; /* y is 2 */ - if(hy==0x3f000000) { /* y is 0.5 */ - if(__builtin_expect(hx>=0, 1)) /* x >= +0 */ - return __ieee754_sqrtf(x); - } + /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ + double_t z, r, r2, r4, p, q, y, y0, invc, logc; + uint32_t iz, top, tmp; + int k, i; + + /* x = 2^k z; where z is in range [OFF,2*OFF] and exact. + The range is split into N subintervals. + The ith subinterval contains z and c is near its center. */ + tmp = ix - OFF; + i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N; + top = tmp & 0xff800000; + iz = ix - top; + k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */ + invc = T[i].invc; + logc = T[i].logc; + z = (double_t) asfloat (iz); + + /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */ + r = z * invc - 1; + y0 = logc + (double_t) k; + + /* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */ + r2 = r * r; + y = A[0] * r + A[1]; + p = A[2] * r + A[3]; + r4 = r2 * r2; + q = A[4] * r + y0; + q = p * r2 + q; + y = y * r4 + q; + return y; +} - /* determine if y is an odd int when x < 0 - * yisint = 0 ... y is not an integer - * yisint = 1 ... y is an odd int - * yisint = 2 ... y is an even int - */ - yisint = 0; - if(hx<0) { - if(iy>=0x4b800000) yisint = 2; /* even integer y */ - else if(iy>=0x3f800000) { - k = (iy>>23)-0x7f; /* exponent */ - j = iy>>(23-k); - if((j<<(23-k))==iy) yisint = 2-(j&1); - } - } +#undef N +#undef T +#define N (1 << EXP2F_TABLE_BITS) +#define T __exp2f_data.tab +#define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11)) + +/* The output of log2 and thus the input of exp2 is either scaled by N + (in case of fast toint intrinsics) or not. The unscaled xd must be + in [-1021,1023], sign_bias sets the sign of the result. */ +static inline double_t +exp2_inline (double_t xd, unsigned long sign_bias) +{ + uint64_t ki, ski, t; + /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ + double_t kd, z, r, r2, y, s; + +#if TOINT_INTRINSICS +# define C __exp2f_data.poly_scaled + /* N*x = k + r with r in [-1/2, 1/2] */ + kd = roundtoint (xd); /* k */ + ki = converttoint (xd); +#else +# define C __exp2f_data.poly +# define SHIFT __exp2f_data.shift_scaled + /* x = k/N + r with r in [-1/(2N), 1/(2N)] */ + kd = (double) (xd + SHIFT); /* Rounding to double precision is required. */ + ki = asuint64 (kd); + kd -= SHIFT; /* k/N */ +#endif + r = xd - kd; + + /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */ + t = T[ki % N]; + ski = ki + sign_bias; + t += ski << (52 - EXP2F_TABLE_BITS); + s = asdouble (t); + z = C[0] * r + C[1]; + r2 = r * r; + y = C[2] * r + 1; + y = z * r2 + y; + y = y * s; + return y; +} - ax = fabsf(x); - /* special value of x */ - if(__builtin_expect(ix==0x7f800000||ix==0||ix==0x3f800000, 0)){ - z = ax; /*x is +-0,+-inf,+-1*/ - if(hy<0) z = one/z; /* z = (1/|x|) */ - if(hx<0) { - if(((ix-0x3f800000)|yisint)==0) { - z = (z-z)/(z-z); /* (-1)**non-int is NaN */ - } else if(yisint==1) - z = -z; /* (x<0)**odd = -(|x|**odd) */ - } - return z; - } +/* Returns 0 if not int, 1 if odd int, 2 if even int. */ +static inline int +checkint (uint32_t iy) +{ + int e = iy >> 23 & 0xff; + if (e < 0x7f) + return 0; + if (e > 0x7f + 23) + return 2; + if (iy & ((1 << (0x7f + 23 - e)) - 1)) + return 0; + if (iy & (1 << (0x7f + 23 - e))) + return 1; + return 2; +} - /* (x<0)**(non-int) is NaN */ - if(__builtin_expect(((((uint32_t)hx>>31)-1)|yisint)==0, 0)) - return (x-x)/(x-x); - - /* |y| is huge */ - if(__builtin_expect(iy>0x4d000000, 0)) { /* if |y| > 2**27 */ - /* over/underflow if x is not close to one */ - if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny; - if(ix>0x3f800007) return (hy>0)? huge*huge:tiny*tiny; - /* now |1-x| is tiny <= 2**-20, suffice to compute - log(x) by x-x^2/2+x^3/3-x^4/4 */ - d2 = ax-1; /* d2 has 20 trailing zeros. */ - d2 = d2 * invln2 - - (d2 * d2) * (0.5 - d2 * (0.333333333333 - d2 * 0.25)) * invln2; - } else { - /* Avoid internal underflow for tiny y. The exact value - of y does not matter if |y| <= 2**-32. */ - if (iy < 0x2f800000) - SET_FLOAT_WORD (y, (hy & 0x80000000) | 0x2f800000); - n = 0; - /* take care subnormal number */ - if(ix<0x00800000) - {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); } - n += ((ix)>>23)-0x7f; - j = ix&0x007fffff; - /* determine interval */ - ix = j|0x3f800000; /* normalize ix */ - if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */ - else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */ - else {k=0;n+=1;ix -= 0x00800000;} - SET_FLOAT_WORD(ax,ix); - - /* compute d1 = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ - d1 = (ax-(double)bp[k])/(ax+(double)bp[k]); - /* compute d2 = log(ax) */ - d2 = d1 * d1; - d2 = 3.0 + d2 + d2*d2*(L1+d2*(L2+d2*(L3+d2*(L4+d2*(L5+d2*L6))))); - /* 2/(3log2)*(d2+...) */ - d2 = d1*d2*cp; - /* log2(ax) = (d2+..)*2/(3*log2) */ - d2 = d2+dp[k]+(double)n; - } +static inline int +zeroinfnan (uint32_t ix) +{ + return 2 * ix - 1 >= 2u * 0x7f800000 - 1; +} - s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ - if(((((uint32_t)hx>>31)-1)|(yisint-1))==0) - s = -one; /* (-ve)**(odd int) */ - - /* compute y * d2 */ - d1 = y * d2; - z = d1; - GET_FLOAT_WORD(j,z); - if (__builtin_expect(j>0x43000000, 0)) /* if z > 128 */ - return s*huge*huge; /* overflow */ - else if (__builtin_expect(j==0x43000000, 0)) { /* if z == 128 */ - if(ovt>(z-d1)) return s*huge*huge; /* overflow */ +float +__ieee754_powf (float x, float y) +{ + unsigned long sign_bias = 0; + uint32_t ix, iy; + + ix = asuint (x); + iy = asuint (y); + if (__glibc_unlikely (ix - 0x00800000 >= 0x7f800000 - 0x00800000 + || zeroinfnan (iy))) + { + /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */ + if (__glibc_unlikely (zeroinfnan (iy))) + { + if (2 * iy == 0) + return issignalingf_inline (x) ? x + y : 1.0f; + if (ix == 0x3f800000) + return issignalingf_inline (y) ? x + y : 1.0f; + if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000) + return x + y; + if (2 * ix == 2 * 0x3f800000) + return 1.0f; + if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000)) + return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */ + return y * y; + } + if (__glibc_unlikely (zeroinfnan (ix))) + { + float_t x2 = x * x; + if (ix & 0x80000000 && checkint (iy) == 1) + { + x2 = -x2; + sign_bias = 1; + } +#if WANT_ERRNO + if (2 * ix == 0 && iy & 0x80000000) + return __math_divzerof (sign_bias); +#endif + return iy & 0x80000000 ? 1 / x2 : x2; } - else if (__builtin_expect((j&0x7fffffff)>0x43160000, 0))/* z <= -150 */ - return s*tiny*tiny; /* underflow */ - else if (__builtin_expect((uint32_t) j==0xc3160000, 0)){/* z == -150*/ - if(0.0<=(z-d1)) return s*tiny*tiny; /* underflow */ + /* x and y are non-zero finite. */ + if (ix & 0x80000000) + { + /* Finite x < 0. */ + int yint = checkint (iy); + if (yint == 0) + return __math_invalidf (x); + if (yint == 1) + sign_bias = SIGN_BIAS; + ix &= 0x7fffffff; } - /* - * compute 2**d1 - */ - 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(z,n&~(0x007fffff>>k)); - n = ((n&0x007fffff)|0x00800000)>>(23-k); - if(j<0) n = -n; - d1 -= z; + if (ix < 0x00800000) + { + /* Normalize subnormal x so exponent becomes negative. */ + ix = asuint (x * 0x1p23f); + ix &= 0x7fffffff; + ix -= 23 << 23; } - d1 = d1 * lg2; - d2 = d1*d1; - d2 = d1 - d2*(P1+d2*(P2+d2*(P3+d2*(P4+d2*P5)))); - d2 = (d1*d2)/(d2-two); - z = one - (d2-d1); - GET_FLOAT_WORD(j,z); - j += (n<<23); - if((j>>23)<=0) /* subnormal output */ - { - z = __scalbnf (z, n); - float force_underflow = z * z; - math_force_eval (force_underflow); - } - else SET_FLOAT_WORD(z,j); - return s*z; + } + double_t logx = log2_inline (ix); + double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec. */ + if (__glibc_unlikely ((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); +#if WANT_ERRNO_UFLOW + if (ylogx < -149.0 * POWF_SCALE) + return __math_may_uflowf (sign_bias); +#endif + } + return (float) exp2_inline (ylogx, sign_bias); } strong_alias (__ieee754_powf, __powf_finite) |