# [v4,04/12] math: Use an improved algorithm for hypotl (ldbl-96)

Message ID 20211203000103.737833-5-adhemerval.zanella@linaro.org Superseded show Improve hypot | expand

## Checks

Context Check Description
dj/TryBot-apply_patch success Patch applied to master at the time it was sent

## Commit Message

Adhemerval Zanella Dec. 3, 2021, midnight UTC
```This implementation is based on 'An Improved Algorithm for hypot(a,b)'
by Carlos F. Borges [1] using the MyHypot3 with the following changes:

- Handle qNaN and sNaN.
- Tune the 'widely varying operands' to avoid spurious underflow
due the multiplication and fix the return value for upwards
rounding mode.
- Handle required underflow exception for subnormal results.

The main advantage of the new algorithm is its precision.  With a
random 1e8 input pairs in the range of [LDBL_MIN, LDBL_MAX], glibc
current implementation shows around 0.02% results with an error of
1 ulp (23158 results) while the new implementation only shows
0.0001% of total (111).

[1] https://arxiv.org/pdf/1904.09481.pdf
---
sysdeps/ieee754/ldbl-96/e_hypotl.c | 231 ++++++++++++-----------------
1 file changed, 98 insertions(+), 133 deletions(-)
```

Wilco Dijkstra Dec. 6, 2021, noon UTC | #1
```Hi Adhemerval,

+  /* If ay is tiny, scale both inputs up.  */
+  if (__glibc_unlikely (ay < TINY_VAL))
+    {
+      if (__glibc_unlikely (ax >= ay / EPS))
+       return ax;

Same here - return ax + ay.

Cheers,
Wilco
```
Adhemerval Zanella Dec. 6, 2021, 12:21 p.m. UTC | #2
```On 06/12/2021 09:00, Wilco Dijkstra wrote:
>
>
> +  /* If ay is tiny, scale both inputs up.  */
> +  if (__glibc_unlikely (ay < TINY_VAL))
> +    {
> +      if (__glibc_unlikely (ax >= ay / EPS))
> +       return ax;
>
> Same here - return ax + ay.

Ack, I have fixed it.
```

## Patch

```diff --git a/sysdeps/ieee754/ldbl-96/e_hypotl.c b/sysdeps/ieee754/ldbl-96/e_hypotl.c
--- a/sysdeps/ieee754/ldbl-96/e_hypotl.c
+++ b/sysdeps/ieee754/ldbl-96/e_hypotl.c
@@ -1,142 +1,107 @@
-/* e_hypotl.c -- long double version of e_hypot.c.
- */
-
-/*
- * ====================================================
- *
- * 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.
- * ====================================================
- */
-
-/* __ieee754_hypotl(x,y)
- *
- * Method :
- *	If (assume round-to-nearest) z=x*x+y*y
- *	has error less than sqrt(2)/2 ulp, than
- *	sqrt(z) has error less than 1 ulp (exercise).
- *
- *	So, compute sqrt(x*x+y*y) with some care as
- *	follows to get the error below 1 ulp:
- *
- *	Assume x>y>0;
- *	(if possible, set rounding to round-to-nearest)
- *	1. if x > 2y  use
- *		x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
- *	where x1 = x with lower 32 bits cleared, x2 = x-x1; else
- *	2. if x <= 2y use
- *		t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
- *	where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1,
- *	y1= y with lower 32 bits chopped, y2 = y-y1.
- *
- *	NOTE: scaling may be necessary if some argument is too
- *	      large or too tiny
- *
- * Special cases:
- *	hypot(x,y) is INF if x or y is +INF or -INF; else
- *	hypot(x,y) is NAN if x or y is NAN.
- *
- * Accuracy:
- *	hypot(x,y) returns sqrt(x^2+y^2) with error less
- *	than 1 ulps (units in the last place)
- */
+/* Euclidean distance function.  Long Double/Binary96 version.
+   Copyright (C) 2021 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
+
+/* This implementation is based on 'An Improved Algorithm for hypot(a,b)' by
+   Carlos F. Borges [1] using the MyHypot3 with the following changes:
+
+   - Handle qNaN and sNaN.
+   - Tune the 'widely varying operands' to avoid spurious underflow
+     due the multiplication and fix the return value for upwards
+     rounding mode.
+   - Handle required underflow exception for subnormal results.
+
+   [1] https://arxiv.org/pdf/1904.09481.pdf  */

#include <math.h>
#include <math_private.h>
#include <math-underflow.h>
#include <libm-alias-finite.h>

-long double __ieee754_hypotl(long double x, long double y)
+#define SCALE      0x8p-8257L
+#define LARGE_VAL  0xb.504f333f9de6484p+8188L
+#define TINY_VAL   0x8p-8194L
+#define EPS        0x8p-68L
+
+/* Hypot kernel. The inputs must be adjusted so that ax >= ay >= 0
+   and squaring ax, ay and (ax - ay) does not overflow or underflow.  */
+static inline long double
+kernel (long double ax, long double ay)
{
-	long double a,b,t1,t2,y1,y2,w;
-	uint32_t j,k,ea,eb;
-
-	GET_LDOUBLE_EXP(ea,x);
-	ea &= 0x7fff;
-	GET_LDOUBLE_EXP(eb,y);
-	eb &= 0x7fff;
-	if(eb > ea) {a=y;b=x;j=ea; ea=eb;eb=j;} else {a=x;b=y;}
-	SET_LDOUBLE_EXP(a,ea);	/* a <- |a| */
-	SET_LDOUBLE_EXP(b,eb);	/* b <- |b| */
-	if((ea-eb)>0x46) {return a+b;} /* x/y > 2**70 */
-	k=0;
-	if(__builtin_expect(ea > 0x5f3f,0)) {	/* a>2**8000 */
-	   if(ea == 0x7fff) {	/* Inf or NaN */
-	       uint32_t exp __attribute__ ((unused));
-	       uint32_t high,low;
-	       w = a+b;			/* for sNaN */
-	       if (issignaling (a) || issignaling (b))
-		 return w;
-	       GET_LDOUBLE_WORDS(exp,high,low,a);
-	       if(((high&0x7fffffff)|low)==0) w = a;
-	       GET_LDOUBLE_WORDS(exp,high,low,b);
-	       if(((eb^0x7fff)|(high&0x7fffffff)|low)==0) w = b;
-	       return w;
-	   }
-	   /* scale a and b by 2**-9600 */
-	   ea -= 0x2580; eb -= 0x2580;	k += 9600;
-	   SET_LDOUBLE_EXP(a,ea);
-	   SET_LDOUBLE_EXP(b,eb);
-	}
-	if(__builtin_expect(eb < 0x20bf, 0)) {	/* b < 2**-8000 */
-	    if(eb == 0) {	/* subnormal b or 0 */
-		uint32_t exp __attribute__ ((unused));
-		uint32_t high,low;
-		GET_LDOUBLE_WORDS(exp,high,low,b);
-		if((high|low)==0) return a;
-		SET_LDOUBLE_WORDS(t1, 0x7ffd, 0x80000000, 0); /* t1=2^16382 */
-		b *= t1;
-		a *= t1;
-		k -= 16382;
-		GET_LDOUBLE_EXP (ea, a);
-		GET_LDOUBLE_EXP (eb, b);
-		if (eb > ea)
-		  {
-		    t1 = a;
-		    a = b;
-		    b = t1;
-		    j = ea;
-		    ea = eb;
-		    eb = j;
-		  }
-	    } else {		/* scale a and b by 2^9600 */
-		ea += 0x2580;	/* a *= 2^9600 */
-		eb += 0x2580;	/* b *= 2^9600 */
-		k -= 9600;
-		SET_LDOUBLE_EXP(a,ea);
-		SET_LDOUBLE_EXP(b,eb);
-	    }
-	}
-    /* medium size a and b */
-	w = a-b;
-	if (w>b) {
-	    uint32_t high;
-	    GET_LDOUBLE_MSW(high,a);
-	    SET_LDOUBLE_WORDS(t1,ea,high,0);
-	    t2 = a-t1;
-	    w  = sqrtl(t1*t1-(b*(-b)-t2*(a+t1)));
-	} else {
-	    uint32_t high;
-	    GET_LDOUBLE_MSW(high,b);
-	    a  = a+a;
-	    SET_LDOUBLE_WORDS(y1,eb,high,0);
-	    y2 = b - y1;
-	    GET_LDOUBLE_MSW(high,a);
-	    SET_LDOUBLE_WORDS(t1,ea+1,high,0);
-	    t2 = a - t1;
-	    w  = sqrtl(t1*y1-(w*(-w)-(t1*y2+t2*b)));
-	}
-	if(k!=0) {
-	    uint32_t exp;
-	    t1 = 1.0;
-	    GET_LDOUBLE_EXP(exp,t1);
-	    SET_LDOUBLE_EXP(t1,exp+k);
-	    w *= t1;
-	    math_check_force_underflow_nonneg (w);
-	    return w;
-	} else return w;
+  long double t1, t2;
+  long double h = sqrtl (ax * ax + ay * ay);
+  if (h <= 2.0L * ay)
+    {
+      long double delta = h - ay;
+      t1 = ax * (2.0L * delta - ax);
+      t2 = (delta - 2.0L * (ax - ay)) * delta;
+    }
+  else
+    {
+      long double delta = h - ax;
+      t1 = 2.0L * delta * (ax - 2.0L * ay);
+      t2 = (4.0L * delta - ay) * ay + delta * delta;
+    }
+
+  h -= (t1 + t2) / (2.0L * h);
+  return h;
+}
+
+long double
+__ieee754_hypotl (long double x, long double y)
+{
+  if (!isfinite(x) || !isfinite(y))
+    {
+      if ((isinf (x) || isinf (y))
+	  && !issignaling (x) && !issignaling (y))
+	return INFINITY;
+      return x + y;
+    }
+
+  x = fabsl (x);
+  y = fabsl (y);
+
+  long double ax = x < y ? y : x;
+  long double ay = x < y ? x : y;
+
+  /* If ax is huge, scale both inputs down.  */
+  if (__glibc_unlikely (ax > LARGE_VAL))
+    {
+      if (__glibc_unlikely (ay <= ax * EPS))
+	return ax + ay;
+
+      return kernel (ax * SCALE, ay * SCALE) / SCALE;
+    }
+
+  /* If ay is tiny, scale both inputs up.  */
+  if (__glibc_unlikely (ay < TINY_VAL))
+    {
+      if (__glibc_unlikely (ax >= ay / EPS))
+	return ax;
+
+      ax = kernel (ax / SCALE, ay / SCALE) * SCALE;
+      math_check_force_underflow_nonneg (ax);
+      return ax;
+    }
+
+  /* Common case: ax is not huge and ay is not tiny.  */
+  if (__glibc_unlikely (ay <= ax * EPS))
+    return ax + ay;
+
+  return kernel (ax, ay);
}
libm_alias_finite (__ieee754_hypotl, __hypotl)

```