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- Log ----------------------------------------------------------------- commit 7adafd921406485dfea0fb8e9290f5ae7f8aa5e5 Author: Raymond Toy toy.raymond@gmail.com Date: Fri Aug 1 23:25:55 2014 -0700
Use the fdlibm routines for exp and log.
diff --git a/src/code/irrat.lisp b/src/code/irrat.lisp index 8f29490..df0bb8c 100644 --- a/src/code/irrat.lisp +++ b/src/code/irrat.lisp @@ -79,8 +79,8 @@ (def-math-rtn "atanh" 1)
;;; Exponential and Logarithmic. -(def-math-rtn "exp" 1) -(def-math-rtn "log" 1) +(def-math-rtn ("__ieee754_exp" %exp) 1) +(def-math-rtn ("__ieee754_log" %log) 1) (def-math-rtn "log10" 1)
(def-math-rtn ("__ieee754_pow" %pow) 2)
commit 1d1ffdf93cb3a67a495b9fe4ea1e3dc679fd401c Author: Raymond Toy toy.raymond@gmail.com Date: Fri Aug 1 23:25:26 2014 -0700
Compile fdlibm routines e_exp.c and e_log.c
diff --git a/src/lisp/Config.x86_darwin b/src/lisp/Config.x86_darwin index ccf021d..8c7c37b 100644 --- a/src/lisp/Config.x86_darwin +++ b/src/lisp/Config.x86_darwin @@ -18,7 +18,7 @@ OS_LIBS =
EXEC_FINAL_OBJ = exec-final.o
-OS_SRC += k_sin.c k_cos.c k_tan.c s_sin.c s_cos.c s_tan.c sincos.c s_log1p.c s_expm1.c e_pow.c +OS_SRC += k_sin.c k_cos.c k_tan.c s_sin.c s_cos.c s_tan.c sincos.c s_log1p.c s_expm1.c e_pow.c e_exp.c e_log.c
k_sin.o : k_sin.c $(CC) -c $(CFLAGS) $(CPPFLAGS) $(CC_REM_PIO2) $< @@ -44,3 +44,7 @@ s_exmp1.o : s_expm1.c
e_pow.o : e_pow.c $(CC) -c $(CFLAGS) $(CPPFLAGS) $(CC_REM_PIO2) $< +e_exp.o : e_exp.c + $(CC) -c $(CFLAGS) $(CPPFLAGS) $(CC_REM_PIO2) $< +e_log.o : e_log.c + $(CC) -c $(CFLAGS) $(CPPFLAGS) $(CC_REM_PIO2) $<
commit 21f9b46373c76e44a72b1f7f73cd292397388962 Author: Raymond Toy toy.raymond@gmail.com Date: Fri Aug 1 23:10:46 2014 -0700
Update to use unions.
diff --git a/src/lisp/e_exp.c b/src/lisp/e_exp.c index e201205..4d94a1e 100644 --- a/src/lisp/e_exp.c +++ b/src/lisp/e_exp.c @@ -108,15 +108,17 @@ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ double y,hi,lo,c,t; int k,xsb; unsigned hx; + union { int i[2]; double d; } ux;
- hx = __HI(x); /* high word of x */ + ux.d = x; + hx = ux.i[HIWORD]; /* high word of x */ xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */
/* filter out non-finite argument */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { - if(((hx&0xfffff)|__LO(x))!=0) + if(((hx&0xfffff)|ux.i[LOWORD])!=0) return x+x; /* NaN */ else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ } @@ -147,10 +149,14 @@ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ if(k==0) return one-((x*c)/(c-2.0)-x); else y = one-((lo-(x*c)/(2.0-c))-hi); if(k >= -1021) { - __HI(y) += (k<<20); /* add k to y's exponent */ + ux.d = y; + ux.i[HIWORD] += (k<<20); /* add k to y's exponent */ + y = ux.d; return y; } else { - __HI(y) += ((k+1000)<<20);/* add k to y's exponent */ + ux.d = y; + ux.i[HIWORD] += ((k+1000)<<20);/* add k to y's exponent */ + y = ux.d; return y*twom1000; } } diff --git a/src/lisp/e_log.c b/src/lisp/e_log.c index 3798bc8..4404ce1 100644 --- a/src/lisp/e_log.c +++ b/src/lisp/e_log.c @@ -92,9 +92,11 @@ static double zero = 0.0; double hfsq,f,s,z,R,w,t1,t2,dk; int k,hx,i,j; unsigned lx; + union { int i[2]; double d; } ux;
- hx = __HI(x); /* high word of x */ - lx = __LO(x); /* low word of x */ + ux.d = x; + hx = ux.i[HIWORD]; /* high word of x */ + lx = ux.i[LOWORD]; /* low word of x */
k=0; if (hx < 0x00100000) { /* x < 2**-1022 */ @@ -102,13 +104,16 @@ static double zero = 0.0; return -two54/zero; /* log(+-0)=-inf */ if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ k -= 54; x *= two54; /* subnormal number, scale up x */ - hx = __HI(x); /* high word of x */ + ux.d = x; + hx = ux.i[HIWORD]; /* high word of x */ } if (hx >= 0x7ff00000) return x+x; k += (hx>>20)-1023; hx &= 0x000fffff; i = (hx+0x95f64)&0x100000; - __HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */ + ux.d = x; + ux.i[HIWORD] = hx|(i^0x3ff00000); /* normalize x or x/2 */ + x = ux.d; k += (i>>20); f = x-1.0; if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
commit 92c7c5d0c4e9904f1a86a6e3b306ca869d710593 Author: Raymond Toy toy.raymond@gmail.com Date: Fri Aug 1 23:05:57 2014 -0700
Add fdlibm routines e_exp and e_log, as is.
diff --git a/src/lisp/e_exp.c b/src/lisp/e_exp.c new file mode 100644 index 0000000..e201205 --- /dev/null +++ b/src/lisp/e_exp.c @@ -0,0 +1,156 @@ + +/* @(#)e_exp.c 1.6 04/04/22 */ +/* + * ==================================================== + * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. + * + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* __ieee754_exp(x) + * Returns the exponential of x. + * + * Method + * 1. Argument reduction: + * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. + * Given x, find r and integer k such that + * + * x = k*ln2 + r, |r| <= 0.5*ln2. + * + * Here r will be represented as r = hi-lo for better + * accuracy. + * + * 2. Approximation of exp(r) by a special rational function on + * the interval [0,0.34658]: + * Write + * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... + * We use a special Remes algorithm on [0,0.34658] to generate + * a polynomial of degree 5 to approximate R. The maximum error + * of this polynomial approximation is bounded by 2**-59. In + * other words, + * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 + * (where z=r*r, and the values of P1 to P5 are listed below) + * and + * | 5 | -59 + * | 2.0+P1*z+...+P5*z - R(z) | <= 2 + * | | + * The computation of exp(r) thus becomes + * 2*r + * exp(r) = 1 + ------- + * R - r + * r*R1(r) + * = 1 + r + ----------- (for better accuracy) + * 2 - R1(r) + * where + * 2 4 10 + * R1(r) = r - (P1*r + P2*r + ... + P5*r ). + * + * 3. Scale back to obtain exp(x): + * From step 1, we have + * exp(x) = 2^k * exp(r) + * + * Special cases: + * exp(INF) is INF, exp(NaN) is NaN; + * exp(-INF) is 0, and + * for finite argument, only exp(0)=1 is exact. + * + * Accuracy: + * according to an error analysis, the error is always less than + * 1 ulp (unit in the last place). + * + * Misc. info. + * For IEEE double + * if x > 7.09782712893383973096e+02 then exp(x) overflow + * if x < -7.45133219101941108420e+02 then exp(x) underflow + * + * 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 + * to produce the hexadecimal values shown. + */ + +#include "fdlibm.h" + +#ifdef __STDC__ +static const double +#else +static double +#endif +one = 1.0, +halF[2] = {0.5,-0.5,}, +huge = 1.0e+300, +twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ +o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ +u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ +ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ + -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ +ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ + -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ +invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ +P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ +P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ +P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ +P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ +P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ + + +#ifdef __STDC__ + double __ieee754_exp(double x) /* default IEEE double exp */ +#else + double __ieee754_exp(x) /* default IEEE double exp */ + double x; +#endif +{ + double y,hi,lo,c,t; + int k,xsb; + unsigned hx; + + hx = __HI(x); /* high word of x */ + xsb = (hx>>31)&1; /* sign bit of x */ + hx &= 0x7fffffff; /* high word of |x| */ + + /* filter out non-finite argument */ + if(hx >= 0x40862E42) { /* if |x|>=709.78... */ + if(hx>=0x7ff00000) { + if(((hx&0xfffff)|__LO(x))!=0) + return x+x; /* NaN */ + else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ + } + if(x > o_threshold) return huge*huge; /* overflow */ + if(x < u_threshold) return twom1000*twom1000; /* underflow */ + } + + /* argument reduction */ + if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ + if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ + hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; + } else { + k = (int)(invln2*x+halF[xsb]); + t = k; + hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ + lo = t*ln2LO[0]; + } + x = hi - lo; + } + else if(hx < 0x3e300000) { /* when |x|<2**-28 */ + if(huge+x>one) return one+x;/* trigger inexact */ + } + else k = 0; + + /* x is now in primary range */ + t = x*x; + c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); + if(k==0) return one-((x*c)/(c-2.0)-x); + else y = one-((lo-(x*c)/(2.0-c))-hi); + if(k >= -1021) { + __HI(y) += (k<<20); /* add k to y's exponent */ + return y; + } else { + __HI(y) += ((k+1000)<<20);/* add k to y's exponent */ + return y*twom1000; + } +} diff --git a/src/lisp/e_log.c b/src/lisp/e_log.c new file mode 100644 index 0000000..3798bc8 --- /dev/null +++ b/src/lisp/e_log.c @@ -0,0 +1,139 @@ + +/* @(#)e_log.c 1.3 95/01/18 */ +/* + * ==================================================== + * 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. + * ==================================================== + */ + +/* __ieee754_log(x) + * Return the logrithm of x + * + * 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 + * of this polynomial approximation is bounded by 2**-58.45. In + * other words, + * 2 4 6 8 10 12 14 + * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s + * (the values of Lg1 to Lg7 are listed in the program) + * and + * | 2 14 | -58.45 + * | 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). + * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) + * 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(+INF) is +INF; log(0) is -INF with signal; + * log(NaN) is that NaN with no signal. + * + * Accuracy: + * according to an error analysis, the error is always less than + * 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 + * to produce the hexadecimal values shown. + */ + +#include "fdlibm.h" + +#ifdef __STDC__ +static const double +#else +static double +#endif +ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ +ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ +two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ +Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ +Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ +Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ +Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ +Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ +Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ +Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ + +static double zero = 0.0; + +#ifdef __STDC__ + double __ieee754_log(double x) +#else + double __ieee754_log(x) + double x; +#endif +{ + double hfsq,f,s,z,R,w,t1,t2,dk; + int k,hx,i,j; + unsigned lx; + + hx = __HI(x); /* high word of x */ + lx = __LO(x); /* low word of x */ + + k=0; + if (hx < 0x00100000) { /* x < 2**-1022 */ + 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 */ + hx = __HI(x); /* high word of x */ + } + if (hx >= 0x7ff00000) return x+x; + k += (hx>>20)-1023; + hx &= 0x000fffff; + i = (hx+0x95f64)&0x100000; + __HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */ + k += (i>>20); + f = x-1.0; + if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ + if(f==zero) if(k==0) return zero; else {dk=(double)k; + return dk*ln2_hi+dk*ln2_lo;} + R = f*f*(0.5-0.33333333333333333*f); + if(k==0) return f-R; else {dk=(double)k; + return dk*ln2_hi-((R-dk*ln2_lo)-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))); + i |= j; + R = t2+t1; + if(i>0) { + hfsq=0.5*f*f; + if(k==0) return f-(hfsq-s*(hfsq+R)); else + return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); + } else { + if(k==0) return f-s*(f-R); else + return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); + } +}
-----------------------------------------------------------------------
Summary of changes: src/code/irrat.lisp | 4 +- src/lisp/Config.x86_darwin | 6 +- src/lisp/e_exp.c | 162 ++++++++++++++++++++++++++++++++++++++++++++ src/lisp/e_log.c | 144 +++++++++++++++++++++++++++++++++++++++ 4 files changed, 313 insertions(+), 3 deletions(-) create mode 100644 src/lisp/e_exp.c create mode 100644 src/lisp/e_log.c
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