Patchwork [v2] powerpc: Optimized version of powf()

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Submitter Paul Clarke
Date June 15, 2017, 8:05 p.m.
Message ID <a0105945-145f-acb0-6e49-a38b4e5c9032@us.ibm.com>
Download mbox | patch
Permalink /patch/21038/
State Committed
Delegated to: Joseph Myers
Headers show

Comments

Paul Clarke - June 15, 2017, 8:05 p.m.
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(-)
Paul Clarke - June 22, 2017, 5:03 p.m.
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 */
>
Joseph Myers - June 22, 2017, 9 p.m.
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.
Tulio Magno Quites Machado Filho - June 23, 2017, 12:22 p.m.
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.
Carlos O'Donell - June 23, 2017, 12:32 p.m.
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! :-)

Patch

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 */