-/* 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 one = 1.0, 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 = one - tiny; /* raise inexact flag */
- if (z >= one) {
- z = one + tiny;
- if (q1 == (uint32_t)0xffffffff) {
- q1 = 0;
- q++;
- } else if (z > one) {
- 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;
- ix0 += m << 20;
- INSERT_WORDS(z, ix0, ix1);
- return z;
+ return y;
}
projects / musl / blobdiff