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-rw-r--r--sysdeps/ieee754/flt-32/e_powf.c450
1 files changed, 208 insertions, 242 deletions
diff --git a/sysdeps/ieee754/flt-32/e_powf.c b/sysdeps/ieee754/flt-32/e_powf.c
index c72fe37d3b..ece83f0dd2 100644
--- a/sysdeps/ieee754/flt-32/e_powf.c
+++ b/sysdeps/ieee754/flt-32/e_powf.c
@@ -1,258 +1,224 @@
-/* e_powf.c -- float version of e_pow.c.
- * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
- */
+/* Single-precision pow function.
+ Copyright (C) 2017-2018 Free Software Foundation, Inc.
+ This file is part of the GNU C Library.
-/*
- * ====================================================
- * 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.
- * ====================================================
- */
+ The GNU C Library is free software; you can redistribute it and/or
+ modify it under the terms of the GNU Lesser General Public
+ License as published by the Free Software Foundation; either
+ version 2.1 of the License, or (at your option) any later version.
-#include <math.h>
-#include <math_private.h>
+ The GNU C Library is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ Lesser General Public License for more details.
-static const float huge = 1.0e+30, tiny = 1.0e-30;
+ You should have received a copy of the GNU Lesser General Public
+ License along with the GNU C Library; if not, see
+ <http://www.gnu.org/licenses/>. */
-static const float
-bp[] = {1.0, 1.5,},
-dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
-dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
-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 */
-lg2 = 6.9314718246e-01, /* 0x3f317218 */
-lg2_h = 6.93145752e-01, /* 0x3f317200 */
-lg2_l = 1.42860654e-06, /* 0x35bfbe8c */
-ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
-cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
-cp_h = 9.6179199219e-01, /* 0x3f763800 =head of cp */
-cp_l = 4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
-ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
-ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
-ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
+#include <math.h>
+#include <stdint.h>
+#include <shlib-compat.h>
+#include <libm-alias-float.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,z_h,z_l,p_h,p_l;
- float y1,t1,t2,r,s,t,u,v,w;
- int32_t i,j,k,yisint,n;
- int32_t hx,hy,ix,iy,is;
-
- GET_FLOAT_WORD(hx,x);
- GET_FLOAT_WORD(hy,y);
- ix = hx&0x7fffffff; iy = hy&0x7fffffff;
-
- /* y==zero: x**0 = 1 */
- if(iy==0) return one;
+ /* 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;
+}
- /* x==+-1 */
- if(x == 1.0) return one;
- if(x == -1.0 && isinf(y)) return one;
+#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, uint32_t 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;
+}
- /* +-NaN return x+y */
- if(__builtin_expect(ix > 0x7f800000 ||
- iy > 0x7f800000, 0))
- return x+y;
+/* 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;
+}
- /* 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);
- }
- }
+static inline int
+zeroinfnan (uint32_t ix)
+{
+ return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
+}
- /* 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);
+float
+__powf (float x, float y)
+{
+ uint32_t 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;
}
-
- 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) */
+ if (__glibc_unlikely (zeroinfnan (ix)))
+ {
+ float_t x2 = x * x;
+ if (ix & 0x80000000 && checkint (iy) == 1)
+ {
+ x2 = -x2;
+ sign_bias = 1;
}
- return z;
- }
-
- /* (x<0)**(non-int) is NaN */
- if(__builtin_expect(((((u_int32_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 */
- t = ax-1; /* t has 20 trailing zeros */
- w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
- u = ivln2_h*t; /* ivln2_h has 16 sig. bits */
- v = t*ivln2_l-w*ivln2;
- t1 = u+v;
- GET_FLOAT_WORD(is,t1);
- SET_FLOAT_WORD(t1,is&0xfffff000);
- t2 = v-(t1-u);
- } else {
- float s2,s_h,s_l,t_h,t_l;
- /* 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 s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
- u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
- v = one/(ax+bp[k]);
- s = u*v;
- s_h = s;
- GET_FLOAT_WORD(is,s_h);
- SET_FLOAT_WORD(s_h,is&0xfffff000);
- /* t_h=ax+bp[k] High */
- SET_FLOAT_WORD (t_h,
- ((((ix>>1)|0x20000000)+0x00400000+(k<<21))
- & 0xfffff000));
- t_l = ax - (t_h-bp[k]);
- s_l = v*((u-s_h*t_h)-s_h*t_l);
- /* compute log(ax) */
- s2 = s*s;
- r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
- r += s_l*(s_h+s);
- s2 = s_h*s_h;
- t_h = (float)3.0+s2+r;
- GET_FLOAT_WORD(is,t_h);
- SET_FLOAT_WORD(t_h,is&0xfffff000);
- t_l = r-((t_h-(float)3.0)-s2);
- /* u+v = s*(1+...) */
- u = s_h*t_h;
- v = s_l*t_h+t_l*s;
- /* 2/(3log2)*(s+...) */
- p_h = u+v;
- GET_FLOAT_WORD(is,p_h);
- SET_FLOAT_WORD(p_h,is&0xfffff000);
- p_l = v-(p_h-u);
- z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
- z_l = cp_l*p_h+p_l*cp+dp_l[k];
- /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
- t = (float)n;
- t1 = (((z_h+z_l)+dp_h[k])+t);
- GET_FLOAT_WORD(is,t1);
- SET_FLOAT_WORD(t1,is&0xfffff000);
- t2 = z_l-(((t1-t)-dp_h[k])-z_h);
- }
-
- s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
- if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
- s = -one; /* (-ve)**(odd int) */
-
- /* 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 (__builtin_expect(j>0x43000000, 0)) /* if z > 128 */
- return s*huge*huge; /* overflow */
- else if (__builtin_expect(j==0x43000000, 0)) { /* if z == 128 */
- if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
+#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((u_int32_t) j==0xc3160000, 0)){/* z == -150*/
- if(p_l<=z-p_h) 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**(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;
+ if (ix < 0x00800000)
+ {
+ /* Normalize subnormal x so exponent becomes negative. */
+ ix = asuint (x * 0x1p23f);
+ ix &= 0x7fffffff;
+ ix -= 23 << 23;
}
- t = p_l+p_h;
- GET_FLOAT_WORD(is,t);
- SET_FLOAT_WORD(t,is&0xfffff000);
- 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-two)-(w+z*w);
- z = one-(r-z);
- 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)
+#ifndef __powf
+strong_alias (__powf, __ieee754_powf)
+strong_alias (__powf, __powf_finite)
+versioned_symbol (libm, __powf, powf, GLIBC_2_27);
+libm_alias_float_other (__pow, pow)
+#endif
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