math: new software sqrt

approximate 1/sqrt(x) and sqrt(x) with goldschmidt iterations.
this is known to be a fast method for computing sqrt, but it is
tricky to get right, so added detailed comments.

use a lookup table for the initial estimate, this adds 256bytes
rodata but it can be shared between sqrt, sqrtf and sqrtl.
this saves one iteration compared to a linear estimate.

this is for soft float targets, but it supports fenv by using a
floating-point operation to get the final result.  the result
is correctly rounded in all rounding modes.  if fenv support is
turned off then the nearest rounded result is computed and
inexact exception is not signaled.

assumes fast 32bit integer arithmetics and 32 to 64bit mul.

src/math/sqrt.c

@@ -1,184 +1,158 @@
-/* origin: FreeBSD /usr/src/lib/msun/src/e_sqrt.c */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunSoft, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-/* sqrt(x)
- * Return correctly rounded sqrt.
- *           ------------------------------------------
- *           |  Use the hardware sqrt if you have one |
- *           ------------------------------------------
- * Method:
- *   Bit by bit method using integer arithmetic. (Slow, but portable)
- *   1. Normalization
- *      Scale x to y in [1,4) with even powers of 2:
- *      find an integer k such that  1 <= (y=x*2^(2k)) < 4, then
- *              sqrt(x) = 2^k * sqrt(y)
- *   2. Bit by bit computation
- *      Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
- *           i                                                   0
- *                                     i+1         2
- *          s  = 2*q , and      y  =  2   * ( y - q  ).         (1)
- *           i      i            i                 i
- *
- *      To compute q    from q , one checks whether
- *                  i+1       i
- *
- *                            -(i+1) 2
- *                      (q + 2      ) <= y.                     (2)
- *                        i
- *                                                            -(i+1)
- *      If (2) is false, then q   = q ; otherwise q   = q  + 2      .
- *                             i+1   i             i+1   i
- *
- *      With some algebric manipulation, it is not difficult to see
- *      that (2) is equivalent to
- *                             -(i+1)
- *                      s  +  2       <= y                      (3)
- *                       i                i
- *
- *      The advantage of (3) is that s  and y  can be computed by
- *                                    i      i
- *      the following recurrence formula:
- *          if (3) is false
- *
- *          s     =  s  ,       y    = y   ;                    (4)
- *           i+1      i          i+1    i
- *
- *          otherwise,
- *                         -i                     -(i+1)
- *          s     =  s  + 2  ,  y    = y  -  s  - 2             (5)
- *           i+1      i          i+1    i     i
- *
- *      One may easily use induction to prove (4) and (5).
- *      Note. Since the left hand side of (3) contain only i+2 bits,
- *            it does not necessary to do a full (53-bit) comparison
- *            in (3).
- *   3. Final rounding
- *      After generating the 53 bits result, we compute one more bit.
- *      Together with the remainder, we can decide whether the
- *      result is exact, bigger than 1/2ulp, or less than 1/2ulp
- *      (it will never equal to 1/2ulp).
- *      The rounding mode can be detected by checking whether
- *      huge + tiny is equal to huge, and whether huge - tiny is
- *      equal to huge for some floating point number "huge" and "tiny".
- *
- * Special cases:
- *      sqrt(+-0) = +-0         ... exact
- *      sqrt(inf) = inf
- *      sqrt(-ve) = NaN         ... with invalid signal
- *      sqrt(NaN) = NaN         ... with invalid signal for signaling NaN
- */
-
+#include <stdint.h>
+#include <math.h>
 #include "libm.h"
+#include "sqrt_data.h"
 
-static const double tiny = 1.0e-300;
+#define FENV_SUPPORT 1
 
-double sqrt(double x)
+/* returns a*b*2^-32 - e, with error 0 <= e < 1.  */
+static inline uint32_t mul32(uint32_t a, uint32_t b)
 {
-	double z;
-	int32_t sign = (int)0x80000000;
-	int32_t ix0,s0,q,m,t,i;
-	uint32_t r,t1,s1,ix1,q1;
+	return (uint64_t)a*b >> 32;
+}
 
-	EXTRACT_WORDS(ix0, ix1, x);
+/* returns a*b*2^-64 - e, with error 0 <= e < 3.  */
+static inline uint64_t mul64(uint64_t a, uint64_t b)
+{
+	uint64_t ahi = a>>32;
+	uint64_t alo = a&0xffffffff;
+	uint64_t bhi = b>>32;
+	uint64_t blo = b&0xffffffff;
+	return ahi*bhi + (ahi*blo >> 32) + (alo*bhi >> 32);
+}
 
-	/* take care of Inf and NaN */
-	if ((ix0&0x7ff00000) == 0x7ff00000) {
-		return x*x + x;  /* sqrt(NaN)=NaN, sqrt(+inf)=+inf, sqrt(-inf)=sNaN */
-	}
-	/* take care of zero */
-	if (ix0 <= 0) {
-		if (((ix0&~sign)|ix1) == 0)
-			return x;  /* sqrt(+-0) = +-0 */
-		if (ix0 < 0)
-			return (x-x)/(x-x);  /* sqrt(-ve) = sNaN */
-	}
-	/* normalize x */
-	m = ix0>>20;
-	if (m == 0) {  /* subnormal x */
-		while (ix0 == 0) {
-			m -= 21;
-			ix0 |= (ix1>>11);
-			ix1 <<= 21;
-		}
-		for (i=0; (ix0&0x00100000) == 0; i++)
-			ix0<<=1;
-		m -= i - 1;
-		ix0 |= ix1>>(32-i);
-		ix1 <<= i;
-	}
-	m -= 1023;    /* unbias exponent */
-	ix0 = (ix0&0x000fffff)|0x00100000;
-	if (m & 1) {  /* odd m, double x to make it even */
-		ix0 += ix0 + ((ix1&sign)>>31);
-		ix1 += ix1;
-	}
-	m >>= 1;      /* m = [m/2] */
-
-	/* generate sqrt(x) bit by bit */
-	ix0 += ix0 + ((ix1&sign)>>31);
-	ix1 += ix1;
-	q = q1 = s0 = s1 = 0;  /* [q,q1] = sqrt(x) */
-	r = 0x00200000;        /* r = moving bit from right to left */
-
-	while (r != 0) {
-		t = s0 + r;
-		if (t <= ix0) {
-			s0   = t + r;
-			ix0 -= t;
-			q   += r;
-		}
-		ix0 += ix0 + ((ix1&sign)>>31);
-		ix1 += ix1;
-		r >>= 1;
-	}
+double sqrt(double x)
+{
+	uint64_t ix, top, m;
 
-	r = sign;
-	while (r != 0) {
-		t1 = s1 + r;
-		t  = s0;
-		if (t < ix0 || (t == ix0 && t1 <= ix1)) {
-			s1 = t1 + r;
-			if ((t1&sign) == sign && (s1&sign) == 0)
-				s0++;
-			ix0 -= t;
-			if (ix1 < t1)
-				ix0--;
-			ix1 -= t1;
-			q1 += r;
-		}
-		ix0 += ix0 + ((ix1&sign)>>31);
-		ix1 += ix1;
-		r >>= 1;
+	/* special case handling.  */
+	ix = asuint64(x);
+	top = ix >> 52;
+	if (predict_false(top - 0x001 >= 0x7ff - 0x001)) {
+		/* x < 0x1p-1022 or inf or nan.  */
+		if (ix * 2 == 0)
+			return x;
+		if (ix == 0x7ff0000000000000)
+			return x;
+		if (ix > 0x7ff0000000000000)
+			return __math_invalid(x);
+		/* x is subnormal, normalize it.  */
+		ix = asuint64(x * 0x1p52);
+		top = ix >> 52;
+		top -= 52;
 	}
 
-	/* use floating add to find out rounding direction */
-	if ((ix0|ix1) != 0) {
-		z = 1.0 - tiny; /* raise inexact flag */
-		if (z >= 1.0) {
-			z = 1.0 + tiny;
-			if (q1 == (uint32_t)0xffffffff) {
-				q1 = 0;
-				q++;
-			} else if (z > 1.0) {
-				if (q1 == (uint32_t)0xfffffffe)
-					q++;
-				q1 += 2;
-			} else
-				q1 += q1 & 1;
-		}
+	/* argument reduction:
+	   x = 4^e m; with integer e, and m in [1, 4)
+	   m: fixed point representation [2.62]
+	   2^e is the exponent part of the result.  */
+	int even = top & 1;
+	m = (ix << 11) | 0x8000000000000000;
+	if (even) m >>= 1;
+	top = (top + 0x3ff) >> 1;
+
+	/* approximate r ~ 1/sqrt(m) and s ~ sqrt(m) when m in [1,4)
+
+	   initial estimate:
+	   7bit table lookup (1bit exponent and 6bit significand).
+
+	   iterative approximation:
+	   using 2 goldschmidt iterations with 32bit int arithmetics
+	   and a final iteration with 64bit int arithmetics.
+
+	   details:
+
+	   the relative error (e = r0 sqrt(m)-1) of a linear estimate
+	   (r0 = a m + b) is |e| < 0.085955 ~ 0x1.6p-4 at best,
+	   a table lookup is faster and needs one less iteration
+	   6 bit lookup table (128b) gives |e| < 0x1.f9p-8
+	   7 bit lookup table (256b) gives |e| < 0x1.fdp-9
+	   for single and double prec 6bit is enough but for quad
+	   prec 7bit is needed (or modified iterations). to avoid
+	   one more iteration >=13bit table would be needed (16k).
+
+	   a newton-raphson iteration for r is
+	     w = r*r
+	     u = 3 - m*w
+	     r = r*u/2
+	   can use a goldschmidt iteration for s at the end or
+	     s = m*r
+
+	   first goldschmidt iteration is
+	     s = m*r
+	     u = 3 - s*r
+	     r = r*u/2
+	     s = s*u/2
+	   next goldschmidt iteration is
+	     u = 3 - s*r
+	     r = r*u/2
+	     s = s*u/2
+	   and at the end r is not computed only s.
+
+	   they use the same amount of operations and converge at the
+	   same quadratic rate, i.e. if
+	     r1 sqrt(m) - 1 = e, then
+	     r2 sqrt(m) - 1 = -3/2 e^2 - 1/2 e^3
+	   the advantage of goldschmidt is that the mul for s and r
+	   are independent (computed in parallel), however it is not
+	   "self synchronizing": it only uses the input m in the
+	   first iteration so rounding errors accumulate. at the end
+	   or when switching to larger precision arithmetics rounding
+	   errors dominate so the first iteration should be used.
+
+	   the fixed point representations are
+	     m: 2.30 r: 0.32, s: 2.30, d: 2.30, u: 2.30, three: 2.30
+	   and after switching to 64 bit
+	     m: 2.62 r: 0.64, s: 2.62, d: 2.62, u: 2.62, three: 2.62  */
+
+	static const uint64_t three = 0xc0000000;
+	uint64_t r, s, d, u, i;
+
+	i = (ix >> 46) % 128;
+	r = (uint32_t)__rsqrt_tab[i] << 16;
+	/* |r sqrt(m) - 1| < 0x1.fdp-9 */
+	s = mul32(m>>32, r);
+	/* |s/sqrt(m) - 1| < 0x1.fdp-9 */
+	d = mul32(s, r);
+	u = three - d;
+	r = mul32(r, u) << 1;
+	/* |r sqrt(m) - 1| < 0x1.7bp-16 */
+	s = mul32(s, u) << 1;
+	/* |s/sqrt(m) - 1| < 0x1.7bp-16 */
+	d = mul32(s, r);
+	u = three - d;
+	r = mul32(r, u) << 1;
+	/* |r sqrt(m) - 1| < 0x1.3704p-29 (measured worst-case) */
+	r = r << 32;
+	s = mul64(m, r);
+	d = mul64(s, r);
+	u = (three<<32) - d;
+	s = mul64(s, u);  /* repr: 3.61 */
+	/* -0x1p-57 < s - sqrt(m) < 0x1.8001p-61 */
+	s = (s - 2) >> 9; /* repr: 12.52 */
+	/* -0x1.09p-52 < s - sqrt(m) < -0x1.fffcp-63 */
+
+	/* s < sqrt(m) < s + 0x1.09p-52,
+	   compute nearest rounded result:
+	   the nearest result to 52 bits is either s or s+0x1p-52,
+	   we can decide by comparing (2^52 s + 0.5)^2 to 2^104 m.  */
+	uint64_t d0, d1, d2;
+	double y, t;
+	d0 = (m << 42) - s*s;
+	d1 = s - d0;
+	d2 = d1 + s + 1;
+	s += d1 >> 63;
+	s &= 0x000fffffffffffff;
+	s |= top << 52;
+	y = asdouble(s);
+	if (FENV_SUPPORT) {
+		/* handle rounding modes and inexact exception:
+		   only (s+1)^2 == 2^42 m case is exact otherwise
+		   add a tiny value to cause the fenv effects.  */
+		uint64_t tiny = predict_false(d2==0) ? 0 : 0x0010000000000000;
+		tiny |= (d1^d2) & 0x8000000000000000;
+		t = asdouble(tiny);
+		y = eval_as_double(y + t);
 	}
-	ix0 = (q>>1) + 0x3fe00000;
-	ix1 = q1>>1;
-	if (q&1)
-		ix1 |= sign;
-	INSERT_WORDS(z, ix0 + ((uint32_t)m << 20), ix1);
-	return z;
+	return y;
 }

src/math/sqrt_data.c

@@ -0,0 +1,19 @@
+#include "sqrt_data.h"
+const uint16_t __rsqrt_tab[128] = {
+0xb451,0xb2f0,0xb196,0xb044,0xaef9,0xadb6,0xac79,0xab43,
+0xaa14,0xa8eb,0xa7c8,0xa6aa,0xa592,0xa480,0xa373,0xa26b,
+0xa168,0xa06a,0x9f70,0x9e7b,0x9d8a,0x9c9d,0x9bb5,0x9ad1,
+0x99f0,0x9913,0x983a,0x9765,0x9693,0x95c4,0x94f8,0x9430,
+0x936b,0x92a9,0x91ea,0x912e,0x9075,0x8fbe,0x8f0a,0x8e59,
+0x8daa,0x8cfe,0x8c54,0x8bac,0x8b07,0x8a64,0x89c4,0x8925,
+0x8889,0x87ee,0x8756,0x86c0,0x862b,0x8599,0x8508,0x8479,
+0x83ec,0x8361,0x82d8,0x8250,0x81c9,0x8145,0x80c2,0x8040,
+0xff02,0xfd0e,0xfb25,0xf947,0xf773,0xf5aa,0xf3ea,0xf234,
+0xf087,0xeee3,0xed47,0xebb3,0xea27,0xe8a3,0xe727,0xe5b2,
+0xe443,0xe2dc,0xe17a,0xe020,0xdecb,0xdd7d,0xdc34,0xdaf1,
+0xd9b3,0xd87b,0xd748,0xd61a,0xd4f1,0xd3cd,0xd2ad,0xd192,
+0xd07b,0xcf69,0xce5b,0xcd51,0xcc4a,0xcb48,0xca4a,0xc94f,
+0xc858,0xc764,0xc674,0xc587,0xc49d,0xc3b7,0xc2d4,0xc1f4,
+0xc116,0xc03c,0xbf65,0xbe90,0xbdbe,0xbcef,0xbc23,0xbb59,
+0xba91,0xb9cc,0xb90a,0xb84a,0xb78c,0xb6d0,0xb617,0xb560,
+};

src/math/sqrt_data.h

@@ -0,0 +1,13 @@
+#ifndef _SQRT_DATA_H
+#define _SQRT_DATA_H
+
+#include <features.h>
+#include <stdint.h>
+
+/* if x in [1,2): i = (int)(64*x);
+   if x in [2,4): i = (int)(32*x-64);
+   __rsqrt_tab[i]*2^-16 is estimating 1/sqrt(x) with small relative error:
+   |__rsqrt_tab[i]*0x1p-16*sqrt(x) - 1| < -0x1.fdp-9 < 2^-8 */
+extern hidden const uint16_t __rsqrt_tab[128];
+
+#endif