diff options
Diffstat (limited to 'sysdeps/libm-ieee754')
| -rw-r--r-- | sysdeps/libm-ieee754/e_log.c | 40 | ||||
| -rw-r--r-- | sysdeps/libm-ieee754/e_sqrt.c | 95 | ||||
| -rw-r--r-- | sysdeps/libm-ieee754/k_rem_pio2.c | 62 |
3 files changed, 98 insertions, 99 deletions
diff --git a/sysdeps/libm-ieee754/e_log.c b/sysdeps/libm-ieee754/e_log.c index c27e0a9d64..f584694686 100644 --- a/sysdeps/libm-ieee754/e_log.c +++ b/sysdeps/libm-ieee754/e_log.c @@ -5,7 +5,7 @@ * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ @@ -15,19 +15,19 @@ static char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $"; #endif /* __ieee754_log(x) - * Return the logrithm of x + * Return the logarithm of x * - * Method : - * 1. Argument Reduction: find k and f such that - * x = 2^k * (1+f), + * Method : + * 1. Argument Reduction: find k and f such that + * x = 2^k * (1+f), * where sqrt(2)/2 < 1+f < sqrt(2) . * * 2. Approximation of log(1+f). * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) * = 2s + 2/3 s**3 + 2/5 s**5 + ....., * = 2s + s*R - * We use a special Reme algorithm on [0,0.1716] to generate - * a polynomial of degree 14 to approximate R The maximum error + * We use a special Reme algorithm on [0,0.1716] to generate + * a polynomial of degree 14 to approximate R The maximum error * of this polynomial approximation is bounded by 2**-58.45. In * other words, * 2 4 6 8 10 12 14 @@ -35,22 +35,22 @@ static char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $"; * (the values of Lg1 to Lg7 are listed in the program) * and * | 2 14 | -58.45 - * | Lg1*s +...+Lg7*s - R(z) | <= 2 + * | Lg1*s +...+Lg7*s - R(z) | <= 2 * | | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. * In order to guarantee error in log below 1ulp, we compute log * by * log(1+f) = f - s*(f - R) (if f is not too large) * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) - * - * 3. Finally, log(x) = k*ln2 + log(1+f). + * + * 3. Finally, log(x) = k*ln2 + log(1+f). * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) - * Here ln2 is split into two floating point number: + * Here ln2 is split into two floating point number: * ln2_hi + ln2_lo, * where n*ln2_hi is always exact for |n| < 2000. * * Special cases: - * log(x) is NaN with signal if x < 0 (including -INF) ; + * log(x) is NaN with signal if x < 0 (including -INF) ; * log(+INF) is +INF; log(0) is -INF with signal; * log(NaN) is that NaN with no signal. * @@ -59,9 +59,9 @@ static char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $"; * 1 ulp (unit in the last place). * * Constants: - * The hexadecimal values are the intended ones for the following - * constants. The decimal values may be used, provided that the - * compiler will convert from decimal to binary accurately enough + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ @@ -105,12 +105,12 @@ static double zero = 0.0; k=0; if (hx < 0x00100000) { /* x < 2**-1022 */ - if (((hx&0x7fffffff)|lx)==0) + if (((hx&0x7fffffff)|lx)==0) return -two54/zero; /* log(+-0)=-inf */ if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ k -= 54; x *= two54; /* subnormal number, scale up x */ GET_HIGH_WORD(hx,x); - } + } if (hx >= 0x7ff00000) return x+x; k += (hx>>20)-1023; hx &= 0x000fffff; @@ -125,14 +125,14 @@ static double zero = 0.0; if(k==0) return f-R; else {dk=(double)k; return dk*ln2_hi-((R-dk*ln2_lo)-f);} } - s = f/(2.0+f); + s = f/(2.0+f); dk = (double)k; z = s*s; i = hx-0x6147a; w = z*z; j = 0x6b851-hx; - t1= w*(Lg2+w*(Lg4+w*Lg6)); - t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); + t1= w*(Lg2+w*(Lg4+w*Lg6)); + t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); i |= j; R = t2+t1; if(i>0) { diff --git a/sysdeps/libm-ieee754/e_sqrt.c b/sysdeps/libm-ieee754/e_sqrt.c index 15fba001d3..67da5455f9 100644 --- a/sysdeps/libm-ieee754/e_sqrt.c +++ b/sysdeps/libm-ieee754/e_sqrt.c @@ -5,7 +5,7 @@ * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ @@ -19,10 +19,10 @@ static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $"; * ------------------------------------------ * | Use the hardware sqrt if you have one | * ------------------------------------------ - * Method: - * Bit by bit method using integer arithmetic. (Slow, but portable) + * Method: + * Bit by bit method using integer arithmetic. (Slow, but portable) * 1. Normalization - * Scale x to y in [1,4) with even powers of 2: + * Scale x to y in [1,4) with even powers of 2: * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then * sqrt(x) = 2^k * sqrt(y) * 2. Bit by bit computation @@ -31,9 +31,9 @@ static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $"; * i+1 2 * s = 2*q , and y = 2 * ( y - q ). (1) * i i i i - * - * To compute q from q , one checks whether - * i+1 i + * + * To compute q from q , one checks whether + * i+1 i * * -(i+1) 2 * (q + 2 ) <= y. (2) @@ -42,13 +42,13 @@ static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $"; * If (2) is false, then q = q ; otherwise q = q + 2 . * i+1 i i+1 i * - * With some algebric manipulation, it is not difficult to see - * that (2) is equivalent to + * With some algebraic manipulation, it is not difficult to see + * that (2) is equivalent to * -(i+1) * s + 2 <= y (3) * i i * - * The advantage of (3) is that s and y can be computed by + * The advantage of (3) is that s and y can be computed by * i i * the following recurrence formula: * if (3) is false @@ -60,10 +60,10 @@ static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $"; * -i -(i+1) * s = s + 2 , y = y - s - 2 (5) * i+1 i i+1 i i - * - * One may easily use induction to prove (4) and (5). + * + * One may easily use induction to prove (4) and (5). * Note. Since the left hand side of (3) contain only i+2 bits, - * it does not necessary to do a full (53-bit) comparison + * it does not necessary to do a full (53-bit) comparison * in (3). * 3. Final rounding * After generating the 53 bits result, we compute one more bit. @@ -73,7 +73,7 @@ static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $"; * 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 @@ -101,17 +101,17 @@ static double one = 1.0, tiny=1.0e-300; #endif { double z; - int32_t sign = (int)0x80000000; + int32_t sign = (int)0x80000000; int32_t ix0,s0,q,m,t,i; u_int32_t r,t1,s1,ix1,q1; EXTRACT_WORDS(ix0,ix1,x); /* take care of Inf and NaN */ - if((ix0&0x7ff00000)==0x7ff00000) { + 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 */ @@ -145,12 +145,12 @@ static double one = 1.0, tiny=1.0e-300; 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; - } + t = s0+r; + if(t<=ix0) { + s0 = t+r; + ix0 -= t; + q += r; + } ix0 += ix0 + ((ix1&sign)>>31); ix1 += ix1; r>>=1; @@ -158,9 +158,9 @@ static double one = 1.0, tiny=1.0e-300; r = sign; while(r!=0) { - t1 = s1+r; + t1 = s1+r; t = s0; - if((t<ix0)||((t==ix0)&&(t1<=ix1))) { + if((t<ix0)||((t==ix0)&&(t1<=ix1))) { s1 = t1+r; if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1; ix0 -= t; @@ -181,7 +181,7 @@ static double one = 1.0, tiny=1.0e-300; if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;} else if (z>one) { if (q1==(u_int32_t)0xfffffffe) q+=1; - q1+=2; + q1+=2; } else q1 += (q1&1); } @@ -197,18 +197,18 @@ static double one = 1.0, tiny=1.0e-300; /* Other methods (use floating-point arithmetic) ------------- -(This is a copy of a drafted paper by Prof W. Kahan +(This is a copy of a drafted paper by Prof W. Kahan and K.C. Ng, written in May, 1986) - Two algorithms are given here to implement sqrt(x) + Two algorithms are given here to implement sqrt(x) (IEEE double precision arithmetic) in software. Both supply sqrt(x) correctly rounded. The first algorithm (in Section A) uses newton iterations and involves four divisions. The second one uses reciproot iterations to avoid division, but requires more multiplications. Both algorithms need the ability - to chop results of arithmetic operations instead of round them, + to chop results of arithmetic operations instead of round them, and the INEXACT flag to indicate when an arithmetic operation - is executed exactly with no roundoff error, all part of the + is executed exactly with no roundoff error, all part of the standard (IEEE 754-1985). The ability to perform shift, add, subtract and logical AND operations upon 32-bit words is needed too, though not part of the standard. @@ -218,7 +218,7 @@ A. sqrt(x) by Newton Iteration (1) Initial approximation Let x0 and x1 be the leading and the trailing 32-bit words of - a floating point number x (in IEEE double format) respectively + a floating point number x (in IEEE double format) respectively 1 11 52 ...widths ------------------------------------------------------ @@ -226,7 +226,7 @@ A. sqrt(x) by Newton Iteration ------------------------------------------------------ msb lsb msb lsb ...order - + ------------------------ ------------------------ x0: |s| e | f1 | x1: | f2 | ------------------------ ------------------------ @@ -251,7 +251,7 @@ A. sqrt(x) by Newton Iteration (2) Iterative refinement - Apply Heron's rule three times to y, we have y approximates + Apply Heron's rule three times to y, we have y approximates sqrt(x) to within 1 ulp (Unit in the Last Place): y := (y+x/y)/2 ... almost 17 sig. bits @@ -276,12 +276,12 @@ A. sqrt(x) by Newton Iteration it requires more multiplications and additions. Also x must be scaled in advance to avoid spurious overflow in evaluating the expression 3y*y+x. Hence it is not recommended uless division - is slow. If division is very slow, then one should use the + is slow. If division is very slow, then one should use the reciproot algorithm given in section B. (3) Final adjustment - By twiddling y's last bit it is possible to force y to be + By twiddling y's last bit it is possible to force y to be correctly rounded according to the prevailing rounding mode as follows. Let r and i be copies of the rounding mode and inexact flag before entering the square root program. Also we @@ -312,7 +312,7 @@ A. sqrt(x) by Newton Iteration I := i; ... restore inexact flag R := r; ... restore rounded mode return sqrt(x):=y. - + (4) Special cases Square root of +inf, +-0, or NaN is itself; @@ -331,7 +331,7 @@ B. sqrt(x) by Reciproot Iteration k := 0x5fe80000 - (x0>>1); y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits - Here k is a 32-bit integer and T2[] is an integer array + Here k is a 32-bit integer and T2[] is an integer array containing correction terms. Now magically the floating value of y (y's leading 32-bit word is y0, the value of its trailing word y1 is set to zero) approximates 1/sqrt(x) @@ -352,9 +352,9 @@ B. sqrt(x) by Reciproot Iteration Apply Reciproot iteration three times to y and multiply the result by x to get an approximation z that matches sqrt(x) - to about 1 ulp. To be exact, we will have + to about 1 ulp. To be exact, we will have -1ulp < sqrt(x)-z<1.0625ulp. - + ... set rounding mode to Round-to-nearest y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x) y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x) @@ -363,14 +363,14 @@ B. sqrt(x) by Reciproot Iteration z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x) Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that - (a) the term z*y in the final iteration is always less than 1; + (a) the term z*y in the final iteration is always less than 1; (b) the error in the final result is biased upward so that -1 ulp < sqrt(x) - z < 1.0625 ulp instead of |sqrt(x)-z|<1.03125ulp. (3) Final adjustment - By twiddling y's last bit it is possible to force y to be + By twiddling y's last bit it is possible to force y to be correctly rounded according to the prevailing rounding mode as follows. Let r and i be copies of the rounding mode and inexact flag before entering the square root program. Also we @@ -410,27 +410,27 @@ B. sqrt(x) by Reciproot Iteration I := 1; ... Raise Inexact flag: z is not exact else { j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2 - k := z1 >> 26; ... get z's 25-th and 26-th + k := z1 >> 26; ... get z's 25-th and 26-th fraction bits I := i or (k&j) or ((k&(j+j+1))!=(x1&3)); } R:= r ... restore rounded mode return sqrt(x):=z. - If multiplication is cheaper then the foregoing red tape, the + If multiplication is cheaper then the foregoing red tape, the Inexact flag can be evaluated by I := i; I := (z*z!=x) or I. - Note that z*z can overwrite I; this value must be sensed if it is + Note that z*z can overwrite I; this value must be sensed if it is True. Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be zero. -------------------- - z1: | f2 | + z1: | f2 | -------------------- bit 31 bit 0 @@ -447,7 +447,6 @@ B. sqrt(x) by Reciproot Iteration 11 01 even ------------------------------------------------- - (4) Special cases (see (4) of Section A). - + (4) Special cases (see (4) of Section A). + */ - diff --git a/sysdeps/libm-ieee754/k_rem_pio2.c b/sysdeps/libm-ieee754/k_rem_pio2.c index 5e29b05312..ccf1633bd4 100644 --- a/sysdeps/libm-ieee754/k_rem_pio2.c +++ b/sysdeps/libm-ieee754/k_rem_pio2.c @@ -5,7 +5,7 @@ * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ @@ -17,12 +17,12 @@ static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $ /* * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) * double x[],y[]; int e0,nx,prec; int ipio2[]; - * - * __kernel_rem_pio2 return the last three digits of N with + * + * __kernel_rem_pio2 return the last three digits of N with * y = x - N*pi/2 * so that |y| < pi/2. * - * The method is to compute the integer (mod 8) and fraction parts of + * The method is to compute the integer (mod 8) and fraction parts of * (2/pi)*x without doing the full multiplication. In general we * skip the part of the product that are known to be a huge integer ( * more accurately, = 0 mod 8 ). Thus the number of operations are @@ -31,10 +31,10 @@ static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $ * (2/pi) is represented by an array of 24-bit integers in ipio2[]. * * Input parameters: - * x[] The input value (must be positive) is broken into nx + * x[] The input value (must be positive) is broken into nx * pieces of 24-bit integers in double precision format. - * x[i] will be the i-th 24 bit of x. The scaled exponent - * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 + * x[i] will be the i-th 24 bit of x. The scaled exponent + * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 * match x's up to 24 bits. * * Example of breaking a double positive z into x[0]+x[1]+x[2]: @@ -52,7 +52,7 @@ static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $ * 64-bit precision 2 * 113-bit precision 3 * The actual value is the sum of them. Thus for 113-bit - * precison, one may have to do something like: + * precision, one may have to do something like: * * long double t,w,r_head, r_tail; * t = (long double)y[2] + (long double)y[1]; @@ -71,8 +71,8 @@ static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $ * 3 113 bits (quad) * * ipio2[] - * integer array, contains the (24*i)-th to (24*i+23)-th - * bit of 2/pi after binary point. The corresponding + * integer array, contains the (24*i)-th to (24*i+23)-th + * bit of 2/pi after binary point. The corresponding * floating value is * * ipio2[i] * 2^(-24(i+1)). @@ -87,8 +87,8 @@ static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $ * in the computation. The recommended value is 2,3,4, * 6 for single, double, extended,and quad. * - * jz local integer variable indicating the number of - * terms of ipio2[] used. + * jz local integer variable indicating the number of + * terms of ipio2[] used. * * jx nx - 1 * @@ -108,9 +108,9 @@ static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $ * exponent for q[i] would be q0-24*i. * * PIo2[] double precision array, obtained by cutting pi/2 - * into 24 bits chunks. + * into 24 bits chunks. * - * f[] ipio2[] in floating point + * f[] ipio2[] in floating point * * iq[] integer array by breaking up q[] in 24-bits chunk. * @@ -124,9 +124,9 @@ static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $ /* * Constants: - * The hexadecimal values are the intended ones for the following - * constants. The decimal values may be used, provided that the - * compiler will convert from decimal to binary accurately enough + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ @@ -136,7 +136,7 @@ static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $ #ifdef __STDC__ static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ #else -static int init_jk[] = {2,3,4,6}; +static int init_jk[] = {2,3,4,6}; #endif #ifdef __STDC__ @@ -155,9 +155,9 @@ static double PIo2[] = { }; #ifdef __STDC__ -static const double +static const double #else -static double +static double #endif zero = 0.0, one = 1.0, @@ -165,9 +165,9 @@ two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ #ifdef __STDC__ - int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2) + int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2) #else - int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) + int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) double x[], y[]; int e0,nx,prec; int32_t ipio2[]; #endif { @@ -211,7 +211,7 @@ recompute: i = (iq[jz-1]>>(24-q0)); n += i; iq[jz-1] -= i<<(24-q0); ih = iq[jz-1]>>(23-q0); - } + } else if(q0==0) ih = iq[jz-1]>>23; else if(z>=0.5) ih=2; @@ -262,7 +262,7 @@ recompute: while(iq[jz]==0) { jz--; q0-=24;} } else { /* break z into 24-bit if necessary */ z = __scalbn(z,-q0); - if(z>=two24) { + if(z>=two24) { fw = (double)((int32_t)(twon24*z)); iq[jz] = (int32_t)(z-two24*fw); jz += 1; q0 += 24; @@ -287,29 +287,29 @@ recompute: case 0: fw = 0.0; for (i=jz;i>=0;i--) fw += fq[i]; - y[0] = (ih==0)? fw: -fw; + y[0] = (ih==0)? fw: -fw; break; case 1: case 2: fw = 0.0; - for (i=jz;i>=0;i--) fw += fq[i]; - y[0] = (ih==0)? fw: -fw; + for (i=jz;i>=0;i--) fw += fq[i]; + y[0] = (ih==0)? fw: -fw; fw = fq[0]-fw; for (i=1;i<=jz;i++) fw += fq[i]; - y[1] = (ih==0)? fw: -fw; + y[1] = (ih==0)? fw: -fw; break; case 3: /* painful */ for (i=jz;i>0;i--) { - fw = fq[i-1]+fq[i]; + fw = fq[i-1]+fq[i]; fq[i] += fq[i-1]-fw; fq[i-1] = fw; } for (i=jz;i>1;i--) { - fw = fq[i-1]+fq[i]; + fw = fq[i-1]+fq[i]; fq[i] += fq[i-1]-fw; fq[i-1] = fw; } - for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; + for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; if(ih==0) { y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; } else { |