/* 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.
+ This file is part of the GNU C Library.
+
+ 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.
+
+ 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.
+
+ 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,
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 = 0xf.64p-4, /* cp high 12 bits. */
-cp_l = -0x7.b11e3p-16, /* 2/(3ln2) - 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*/
+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;
float
__ieee754_powf(float x, float y)
{
- float z,ax,z_h,z_l,p_h,p_l;
- float y1,t1,t2,r,s,t,u,v,w;
+ float z, ax, s;
+ double d1, d2;
int32_t i,j,k,yisint,n;
- int32_t hx,hy,ix,iy,is;
+ int32_t hx,hy,ix,iy;
- GET_FLOAT_WORD(hx,x);
GET_FLOAT_WORD(hy,y);
- ix = hx&0x7fffffff; iy = hy&0x7fffffff;
+ iy = hy&0x7fffffff;
/* y==zero: x**0 = 1 */
if(iy==0 && !issignaling (x)) return one;
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;
- /* 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);
- }
- }
-
/* special value of y */
if (__builtin_expect(iy==0x7f800000, 0)) { /* y is +-inf */
if (ix==0x3f800000)
return __ieee754_sqrtf(x);
}
+ /* 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);
+ }
+ }
+
ax = fabsf(x);
/* special value of x */
if(__builtin_expect(ix==0x7f800000||ix==0||ix==0x3f800000, 0)){
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);
+ d2 = ax-1; /* d2 has 20 trailing zeros. */
+ d2 = d2 * invln2 -
+ (d2 * d2) * (0.5 - d2 * (0.333333333333 - d2 * 0.25)) * invln2;
} 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)
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);
+ /* 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;
}
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;
+ /* 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(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
+ if(ovt>(z-d1)) return s*huge*huge; /* overflow */
}
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 */
+ if(0.0<=(z-d1)) return s*tiny*tiny; /* underflow */
}
/*
- * compute 2**(p_h+p_l)
+ * compute 2**d1
*/
i = j&0x7fffffff;
k = (i>>23)-0x7f;
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));
+ SET_FLOAT_WORD(z,n&~(0x007fffff>>k));
n = ((n&0x007fffff)|0x00800000)>>(23-k);
if(j<0) n = -n;
- p_h -= t;
+ d1 -= z;
}
- 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);
+ 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 */
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