diff options
Diffstat (limited to 'src/math')
| -rw-r--r-- | src/math/exp.c | 121 |
1 files changed, 49 insertions, 72 deletions
diff --git a/src/math/exp.c b/src/math/exp.c index 29bf9609..5c0edee4 100644 --- a/src/math/exp.c +++ b/src/math/exp.c @@ -25,7 +25,7 @@ * 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 + * We use a special Remez 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, @@ -36,15 +36,15 @@ * | 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) + * 2*r + * exp(r) = 1 + ---------- + * R(r) - r + * r*c(r) * = 1 + r + ----------- (for better accuracy) - * 2 - R1(r) + * 2 - c(r) * where - * 2 4 10 - * R1(r) = r - (P1*r + P2*r + ... + P5*r ). + * 2 4 10 + * c(r) = r - (P1*r + P2*r + ... + P5*r ). * * 3. Scale back to obtain exp(x): * From step 1, we have @@ -61,27 +61,16 @@ * * 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. + * if x > 709.782712893383973096 then exp(x) overflows + * if x < -745.133219101941108420 then exp(x) underflows */ #include "libm.h" static const double -halF[2] = {0.5,-0.5,}, -huge = 1.0e+300, -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 */ +half[2] = {0.5,-0.5}, +ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ +ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ @@ -89,68 +78,56 @@ P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ -static const volatile double -twom1000 = 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0 */ - double exp(double x) { - double y,hi=0.0,lo=0.0,c,t,twopk; - int32_t k=0,xsb; + double hi, lo, c, z; + int k, sign; uint32_t hx; GET_HIGH_WORD(hx, x); - xsb = (hx>>31)&1; /* sign bit of x */ + sign = hx>>31; hx &= 0x7fffffff; /* high word of |x| */ - /* filter out non-finite argument */ - if (hx >= 0x40862E42) { /* if |x| >= 709.78... */ - if (hx >= 0x7ff00000) { - uint32_t lx; - - GET_LOW_WORD(lx,x); - if (((hx&0xfffff)|lx) != 0) /* NaN */ - return x+x; - return xsb==0 ? x : 0.0; /* exp(+-inf)={inf,0} */ + /* special cases */ + if (hx >= 0x40862e42) { /* if |x| >= 709.78... */ + if (isnan(x)) + return x; + if (x > 709.782712893383973096) { + /* overflow if x!=inf */ + STRICT_ASSIGN(double, x, 0x1p1023 * x); + return x; + } + if (x < -745.13321910194110842) { + /* underflow if x!=-inf */ + STRICT_ASSIGN(double, x, 0x1p-1000 / -x * 0x1p-1000); + return x; } - 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]; - } + if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ + if (hx < 0x3ff0a2b2) /* if |x| < 1.5 ln2 */ + k = 1 - sign - sign; /* optimization */ + else + k = (int)(invln2*x + half[sign]); + hi = x - k*ln2hi; /* k*ln2hi is exact here */ + lo = k*ln2lo; STRICT_ASSIGN(double, x, hi - lo); - } else if(hx < 0x3e300000) { /* |x| < 2**-28 */ - /* raise inexact */ - if (huge+x > 1.0) - return 1.0+x; - } else + } else if (hx > 0x3e300000) { /* if |x| > 2**-28 */ k = 0; + hi = x; + lo = 0; + } else { + /* inexact if x!=0 */ + FORCE_EVAL(0x1p1023 + x); + return 1 + x; + } /* x is now in primary range */ - t = x*x; - if (k >= -1021) - INSERT_WORDS(twopk, 0x3ff00000+(k<<20), 0); - else - INSERT_WORDS(twopk, 0x3ff00000+((k+1000)<<20), 0); - c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); + z = x*x; + c = x - z*(P1+z*(P2+z*(P3+z*(P4+z*P5)))); + x = 1 + ((x*c/(2-c) - lo) + hi); if (k == 0) - return 1.0 - ((x*c)/(c-2.0) - x); - y = 1.0-((lo-(x*c)/(2.0-c))-hi); - if (k < -1021) - return y*twopk*twom1000; - if (k == 1024) - return y*2.0*0x1p1023; - return y*twopk; + return x; + return scalbn(x, k); } |