[PATCH] powerpc: Optimized version of powf()

[PATCH] powerpc: Optimized version of powf()

[Paul Clarke]

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.

2017-06-08  Paul A. Clarke  <pc@us.ibm.com>

	* sysdeps/ieee754/flt-32/e_powf.c: Optimized implementation utilizing
	rearranged code and double float types.
---
Previous submission was architecture-specific with IFUNCs, but discussion
settled on a common implementation, and a much simpler patch.

Tested on powerpc64le, x86_64, and s390x.

 sysdeps/ieee754/flt-32/e_powf.c | 140 ++++++++++++++--------------------------
 1 file changed, 48 insertions(+), 92 deletions(-)

diff --git a/sysdeps/ieee754/flt-32/e_powf.c b/sysdeps/ieee754/flt-32/e_powf.c
index 13b49de..628232a 100644
--- a/sysdeps/ieee754/flt-32/e_powf.c
+++ b/sysdeps/ieee754/flt-32/e_powf.c
@@ -4,7 +4,7 @@
 
 /*
  * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ * Copyright (C) 1993-2017 by Sun Microsystems, Inc. All rights reserved.
  *
  * Developed at SunPro, a Sun Microsystems, Inc. business.
  * Permission to use, copy, modify, and distribute this
@@ -38,11 +38,9 @@ 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 */
@@ -52,14 +50,13 @@ ivln2_l  =  7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
 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 +65,14 @@ __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 +91,21 @@ __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 +131,10 @@ __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 * (ivln2_l + (double)ivln2_h) -
+		 (d2 * d2) * (0.5 - d2 * (0.333333333333 - d2 * 0.25)) * ivln2;
 	} 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 +152,37 @@ __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*((double)cp_h + cp_l);
+	/* log2(ax) = (d2+..)*2/(3*log2) */
+	    d2 = d2+((double)dp_h[k]+dp_l[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 +190,16 @@ __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 * ((double)lg2_h + lg2_l);
+	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 */
-- 
1.8.3.1

Re: [PATCH] powerpc: Optimized version of powf()

[Joseph Myers]

On Thu, 8 Jun 2017, Paul Clarke wrote:

> +	    d2 = d2 * (ivln2_l + (double)ivln2_h) -

> +	    d2 = d1*d2*((double)cp_h + cp_l);

> +	    d2 = d2+((double)dp_h[k]+dp_l[k])+(double)n;

> +	d1 = d1 * ((double)lg2_h + lg2_l);

These expressions all seem like unidiomatic conversions to direct use of 
double.  I'd expect all of ivln2, cp, dp, lg2 to become double constants, 
with rounded-to-nearest double values expressed as hex floats, instead of 
adding high and low parts as float, unless there is some reason that 
wouldn't work.  (dp_h[1] + dp_l[1] doesn't have a comment saying what 
logical value it's meant to be, but it looks like log2(1.5).)

-- 
Joseph S. Myers
joseph@codesourcery.com

Re: [PATCH] powerpc: Optimized version of powf()

[Joseph Myers]

On Thu, 8 Jun 2017, Paul Clarke wrote:

> + * Copyright (C) 1993-2017 by Sun Microsystems, Inc. All rights reserved.

Also, you don't change copyright dates in non-FSF copyright notices.  You 
can add an FSF copyright / LGPL license at the top for the glibc-local 
changes, however, when making substantial changes to such BSD / fdlibm / 
... files.

-- 
Joseph S. Myers
joseph@codesourcery.com

Re: [PATCH] powerpc: Optimized version of powf()

[Joseph Myers]

Also, for performance improvements we generally like to have actual 
benchmark results.  Could you add a powf benchmark, with whatever inputs 
you think are suitably representative, to the benchtests suite, in a 
separate patch, and then include before-and-after results for that 
benchmark, on any one system of your choice, in the submission of the powf 
patch?  That then provides a basis for assessing any future powf 
improvements or reimplementations anyone else comes up with.

-- 
Joseph S. Myers
joseph@codesourcery.com

Re: [PATCH] powerpc: Optimized version of powf()

[Szabolcs Nagy]

* Paul Clarke <pc@us.ibm.com> [2017-06-08 18:37:24 -0500]:
> 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.
> 

note that i plan to post my powf implementation that i think
is better, but i cant promise it for this release cycle.

i have a version that handles error cases inline instead of
via a wrapper (same for exp2f, log2f, expf, logf) and that's
harder to integrate.

this version seems to keep the original float coefficeints,
(and various other constructs from the original code)
i think that's suboptimal.

Re: [PATCH] powerpc: Optimized version of powf()

[Joseph Myers]

On Fri, 9 Jun 2017, Szabolcs Nagy wrote:

> * Paul Clarke <pc@us.ibm.com> [2017-06-08 18:37:24 -0500]:
> > 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.
> 
> note that i plan to post my powf implementation that i think
> is better, but i cant promise it for this release cycle.

I think this illustrates why we need benchmarks so that implementations 
can be compared on the basis of something other than "think is better".

-- 
Joseph S. Myers
joseph@codesourcery.com