/////////////////////////////////////// // Implements: // double sqrt(double x); /////////////////////////////////////// // Notes: // This is taken from newlib. // Only included because pow() needs it. // @nolint /* * ==================================================== * 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. * ==================================================== */ #include #include typedef union { double value; struct { uint32_t lsw; uint32_t msw; } parts; } ieee_double_shape_type; #define EXTRACT_WORDS(ix0,ix1,d) \ do { \ ieee_double_shape_type ew_u; \ ew_u.value = (d); \ (ix0) = ew_u.parts.msw; \ (ix1) = ew_u.parts.lsw; \ } while (0) /* Get the more significant 32 bit int from a double. */ #define GET_HIGH_WORD(i,d) \ do { \ ieee_double_shape_type gh_u; \ gh_u.value = (d); \ (i) = gh_u.parts.msw; \ } while (0) /* Get the less significant 32 bit int from a double. */ #define GET_LOW_WORD(i,d) \ do { \ ieee_double_shape_type gl_u; \ gl_u.value = (d); \ (i) = gl_u.parts.lsw; \ } while (0) #define INSERT_WORDS(d,ix0,ix1) \ do { \ ieee_double_shape_type iw_u; \ iw_u.parts.msw = (ix0); \ iw_u.parts.lsw = (ix1); \ (d) = iw_u.value; \ } while (0) /* Set the more significant 32 bits of a double from an int. */ #define SET_HIGH_WORD(d,v) \ do { \ ieee_double_shape_type sh_u; \ sh_u.value = (d); \ sh_u.parts.msw = (v); \ (d) = sh_u.value; \ } while (0) /* Set the less significant 32 bits of a double from an int. */ #define SET_LOW_WORD(d,v) \ do { \ ieee_double_shape_type sl_u; \ sl_u.value = (d); \ sl_u.parts.lsw = (v); \ (d) = sl_u.value; \ } while (0) /* __ieee754_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 * * Other methods : see the appended file at the end of the program below. *--------------- */ static const double one = 1.0, tiny=1.0e-300; double sqrt(double x) { double z; int32_t sign = 0x80000000; uint32_t r,t1,s1,ix1,q1; int32_t ix0,s0,q,m,t,i; EXTRACT_WORDS(ix0,ix1,x); /* 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 */ else 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; } r = sign; while(r!=0) { t1 = s1+r; t = s0; if((t>31); ix1 += ix1; r>>=1; } /* use floating add to find out rounding direction */ if((ix0|ix1)!=0) { z = one-tiny; /* trigger inexact flag */ if (z>=one) { z = one+tiny; if (q1==(uint32_t)0xffffffff) { q1=0; q += 1;} else if (z>one) { if (q1==(uint32_t)0xfffffffe) q+=1; q1+=2; } else q1 += (q1&1); } } ix0 = (q>>1)+0x3fe00000; ix1 = q1>>1; if ((q&1)==1) ix1 |= sign; ix0 += (m <<20); INSERT_WORDS(z,ix0,ix1); return z; }