author | Jan Vrany <jan.vrany@fit.cvut.cz> |
Tue, 11 Jun 2019 22:03:11 +0100 | |
changeset 3914 | 35aa88fa58c1 |
parent 2380 | 9195eccdcbd9 |
permissions | -rw-r--r-- |
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/* @(#)e_log.c 1.3 95/01/18 */ |
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/* |
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* ==================================================== |
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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* |
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* Developed at SunSoft, a Sun Microsystems, Inc. business. |
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* Permission to use, copy, modify, and distribute this |
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* software is freely granted, provided that this notice |
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* is preserved. |
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* ==================================================== |
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*/ |
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|
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/* __ieee754_log(x) |
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* Return the logrithm of x |
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* |
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* Method : |
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* 1. Argument Reduction: find k and f such that |
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* x = 2^k * (1+f), |
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* where sqrt(2)/2 < 1+f < sqrt(2) . |
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* |
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* 2. Approximation of log(1+f). |
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* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) |
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* = 2s + 2/3 s**3 + 2/5 s**5 + ....., |
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* = 2s + s*R |
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* We use a special Reme algorithm on [0,0.1716] to generate |
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* a polynomial of degree 14 to approximate R The maximum error |
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* of this polynomial approximation is bounded by 2**-58.45. In |
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* other words, |
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* 2 4 6 8 10 12 14 |
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* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s |
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* (the values of Lg1 to Lg7 are listed in the program) |
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* and |
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* | 2 14 | -58.45 |
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* | Lg1*s +...+Lg7*s - R(z) | <= 2 |
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* | | |
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* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. |
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* In order to guarantee error in log below 1ulp, we compute log |
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* by |
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* log(1+f) = f - s*(f - R) (if f is not too large) |
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* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) |
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* |
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* 3. Finally, log(x) = k*ln2 + log(1+f). |
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* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
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* Here ln2 is split into two floating point number: |
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* ln2_hi + ln2_lo, |
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* where n*ln2_hi is always exact for |n| < 2000. |
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* |
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* Special cases: |
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* log(x) is NaN with signal if x < 0 (including -INF) ; |
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* log(+INF) is +INF; log(0) is -INF with signal; |
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* log(NaN) is that NaN with no signal. |
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* |
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* Accuracy: |
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* according to an error analysis, the error is always less than |
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* 1 ulp (unit in the last place). |
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* |
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* Constants: |
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* The hexadecimal values are the intended ones for the following |
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* constants. The decimal values may be used, provided that the |
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* compiler will convert from decimal to binary accurately enough |
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* to produce the hexadecimal values shown. |
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*/ |
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|
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#include "fdlibm.h" |
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|
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#ifdef __STDC__ |
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static const double |
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#else |
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static double |
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#endif |
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ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ |
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ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ |
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two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ |
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Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ |
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Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ |
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Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ |
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Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ |
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Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ |
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Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ |
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Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ |
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|
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static double zero = 0.0; |
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|
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#ifdef __STDC__ |
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86 |
double __ieee754_log(double x) |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
87 |
#else |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
88 |
double __ieee754_log(x) |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
89 |
double x; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
90 |
#endif |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
91 |
{ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
92 |
double hfsq,f,s,z,R,w,t1,t2,dk; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
93 |
int k,hx,i,j; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
94 |
unsigned lx; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
95 |
|
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
96 |
hx = __HI(x); /* high word of x */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
97 |
lx = __LO(x); /* low word of x */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
98 |
|
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
99 |
k=0; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
100 |
if (hx < 0x00100000) { /* x < 2**-1022 */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
101 |
if (((hx&0x7fffffff)|lx)==0) |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
102 |
return -two54/zero; /* log(+-0)=-inf */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
103 |
if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
104 |
k -= 54; x *= two54; /* subnormal number, scale up x */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
105 |
hx = __HI(x); /* high word of x */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
106 |
} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
107 |
if (hx >= 0x7ff00000) return x+x; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
108 |
k += (hx>>20)-1023; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
109 |
hx &= 0x000fffff; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
110 |
i = (hx+0x95f64)&0x100000; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
111 |
__HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
112 |
k += (i>>20); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
113 |
f = x-1.0; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
114 |
if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
115 |
if(f==zero) if(k==0) return zero; else {dk=(double)k; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
116 |
return dk*ln2_hi+dk*ln2_lo;} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
117 |
R = f*f*(0.5-0.33333333333333333*f); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
118 |
if(k==0) return f-R; else {dk=(double)k; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
119 |
return dk*ln2_hi-((R-dk*ln2_lo)-f);} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
120 |
} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
121 |
s = f/(2.0+f); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
122 |
dk = (double)k; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
123 |
z = s*s; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
124 |
i = hx-0x6147a; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
125 |
w = z*z; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
126 |
j = 0x6b851-hx; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
127 |
t1= w*(Lg2+w*(Lg4+w*Lg6)); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
128 |
t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
129 |
i |= j; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
130 |
R = t2+t1; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
131 |
if(i>0) { |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
132 |
hfsq=0.5*f*f; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
133 |
if(k==0) return f-(hfsq-s*(hfsq+R)); else |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
134 |
return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
135 |
} else { |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
136 |
if(k==0) return f-s*(f-R); else |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
137 |
return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
138 |
} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
139 |
} |