[v2] powerpc: Optimized version of powf()
Commit Message
Most significant changes are code simplification and use of doubles for
intermediate values. Also, some rearrangement to move early
non-dependent code later, out of the faster paths.
2017-06-15 Paul A. Clarke <pc@us.ibm.com>
* sysdeps/ieee754/flt-32/e_powf.c: Optimized implementation utilizing
rearranged code and doubles float types.
---
Changes in v2 (per Joseph's review (Thanks!)):
- added FSF copyright, GPL header (reverted changes to existing copyrights)
- consolidated existing uses of high/low constants to explicit double constants
- based on the very recent "powf" benchtest patch "[PATCH v3] Add powf bench tests",
including performance results:
-- without patch --
{"timing_type": "hp_timing",
"functions": {
"powf": {
"": {
"duration": 5.13314e+09,
"iterations": 6.111e+07,
"max": 434.889,
"min": 5.287,
"mean": 83.9983
}
}
}
}
-- with patch --
{"timing_type": "hp_timing",
"functions": {
"powf": {
"": {
"duration": 5.12187e+09,
"iterations": 1.1466e+08,
"max": 428.786,
"min": 4.738,
"mean": 44.67
}
}
}
}
--
sysdeps/ieee754/flt-32/e_powf.c | 173 +++++++++++++++++-----------------------
1 file changed, 71 insertions(+), 102 deletions(-)
Comments
friendly ping!
On 06/15/2017 03:05 PM, Paul Clarke wrote:
> Most significant changes are code simplification and use of doubles for
> intermediate values. Also, some rearrangement to move early
> non-dependent code later, out of the faster paths.
>
> 2017-06-15 Paul A. Clarke <pc@us.ibm.com>
>
> * sysdeps/ieee754/flt-32/e_powf.c: Optimized implementation utilizing
> rearranged code and doubles float types.
> ---
> Changes in v2 (per Joseph's review (Thanks!)):
> - added FSF copyright, GPL header (reverted changes to existing copyrights)
> - consolidated existing uses of high/low constants to explicit double constants
> - based on the very recent "powf" benchtest patch "[PATCH v3] Add powf bench tests",
> including performance results:
> -- without patch --
> {"timing_type": "hp_timing",
> "functions": {
> "powf": {
> "": {
> "duration": 5.13314e+09,
> "iterations": 6.111e+07,
> "max": 434.889,
> "min": 5.287,
> "mean": 83.9983
> }
> }
> }
> }
> -- with patch --
> {"timing_type": "hp_timing",
> "functions": {
> "powf": {
> "": {
> "duration": 5.12187e+09,
> "iterations": 1.1466e+08,
> "max": 428.786,
> "min": 4.738,
> "mean": 44.67
> }
> }
> }
> }
> --
>
> sysdeps/ieee754/flt-32/e_powf.c | 173 +++++++++++++++++-----------------------
> 1 file changed, 71 insertions(+), 102 deletions(-)
>
> diff --git a/sysdeps/ieee754/flt-32/e_powf.c b/sysdeps/ieee754/flt-32/e_powf.c
> index 13b49de..a456559 100644
> --- a/sysdeps/ieee754/flt-32/e_powf.c
> +++ b/sysdeps/ieee754/flt-32/e_powf.c
> @@ -1,6 +1,22 @@
> /* 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/>. */
>
> /*
> * ====================================================
> @@ -20,8 +36,6 @@ static const float huge = 1.0e+30, tiny = 1.0e-30;
>
> 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,
> @@ -38,28 +52,24 @@ 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 = 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;
> @@ -68,26 +78,14 @@ __ieee754_powf(float x, float y)
> 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)
> @@ -106,6 +104,21 @@ __ieee754_powf(float x, float y)
> 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)){
> @@ -131,16 +144,10 @@ __ieee754_powf(float x, float y)
> 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)
> @@ -158,69 +165,37 @@ __ieee754_powf(float x, float y)
> 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;
> @@ -228,22 +203,16 @@ __ieee754_powf(float x, float y)
> 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 */
>
On Thu, 15 Jun 2017, Paul Clarke wrote:
> Most significant changes are code simplification and use of doubles for
> intermediate values. Also, some rearrangement to move early
> non-dependent code later, out of the faster paths.
>
> 2017-06-15 Paul A. Clarke <pc@us.ibm.com>
>
> * sysdeps/ieee754/flt-32/e_powf.c: Optimized implementation utilizing
> rearranged code and doubles float types.
OK.
Paul Clarke <pc@us.ibm.com> writes:
> Most significant changes are code simplification and use of doubles for
> intermediate values. Also, some rearrangement to move early
> non-dependent code later, out of the faster paths.
>
> 2017-06-15 Paul A. Clarke <pc@us.ibm.com>
>
> * sysdeps/ieee754/flt-32/e_powf.c: Optimized implementation utilizing
> rearranged code and doubles float types.
I removed "powerpc:" from the subject and pushed it as c340290de867.
On 06/23/2017 08:22 AM, Tulio Magno Quites Machado Filho wrote:
> Paul Clarke <pc@us.ibm.com> writes:
>
>> Most significant changes are code simplification and use of doubles for
>> intermediate values. Also, some rearrangement to move early
>> non-dependent code later, out of the faster paths.
>>
>> 2017-06-15 Paul A. Clarke <pc@us.ibm.com>
>>
>> * sysdeps/ieee754/flt-32/e_powf.c: Optimized implementation utilizing
>> rearranged code and doubles float types.
>
> I removed "powerpc:" from the subject and pushed it as c340290de867.
I want to point out that I am immensely pleased to see that on first submission
this kind of patch has a micorbenchmark quote as the reason for making the
change. A job well done to IBM for making objective engineering decisions using
the microbenchmark framework! :-)
@@ -1,6 +1,22 @@
/* 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/>. */
/*
* ====================================================
@@ -20,8 +36,6 @@ static const float huge = 1.0e+30, tiny = 1.0e-30;
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,
@@ -38,28 +52,24 @@ 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 = 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;
@@ -68,26 +78,14 @@ __ieee754_powf(float x, float y)
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)
@@ -106,6 +104,21 @@ __ieee754_powf(float x, float y)
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)){
@@ -131,16 +144,10 @@ __ieee754_powf(float x, float y)
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)
@@ -158,69 +165,37 @@ __ieee754_powf(float x, float y)
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;
@@ -228,22 +203,16 @@ __ieee754_powf(float x, float y)
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 */