forked from len0rd/rockbox
PDBox: Undoing incorrect changes of math functions.
git-svn-id: svn://svn.rockbox.org/rockbox/trunk@22159 a1c6a512-1295-4272-9138-f99709370657
This commit is contained in:
Wincent Balin
parent
010bb8e6ba
commit
1f9675227a
1 changed files with 819 additions and 45 deletions
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@ -582,55 +582,284 @@ float rb_log10(float x)
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}
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/* Power function, taken from glibc-2.8 and dietlibc-0.32 */
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/* Power function,
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Taken from glibc-2.8 */
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int rb_isinf(float x)
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{
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int32_t ix, t;
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GET_FLOAT_WORD(ix,x);
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t = ix & 0x7fffffff;
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t ^= 0x7f800000;
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t |= -t;
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return ~(t >> 31) & (ix >> 30);
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}
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float rb_copysignf(float x, float y)
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{
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uint32_t ix, iy;
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GET_FLOAT_WORD(ix,x);
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GET_FLOAT_WORD(iy,y);
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SET_FLOAT_WORD(x,(ix&0x7fffffff)|(iy&0x80000000));
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return x;
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}
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static const float
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huge = 1.0e+30,
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tiny = 1.0e-30,
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one = 1.0f;
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twom25 = 2.9802322388e-08; /* 0x33000000 */
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float rb_scalbnf(float x, int n)
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{
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int32_t k, ix;
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GET_FLOAT_WORD(ix,x);
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k = (ix&0x7f800000)>>23; /* extract exponent */
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if (k==0) { /* 0 or subnormal x */
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if ((ix&0x7fffffff)==0) return x; /* +-0 */
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x *= two25;
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GET_FLOAT_WORD(ix,x);
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k = ((ix&0x7f800000)>>23) - 25;
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}
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if (k==0xff) return x+x; /* NaN or Inf */
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k = k+n;
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if (n> 50000 || k > 0xfe)
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return huge*rb_copysignf(huge,x); /* overflow */
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if (n< -50000)
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return tiny*rb_copysignf(tiny,x); /*underflow*/
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if (k > 0) /* normal result */
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{SET_FLOAT_WORD(x,(ix&0x807fffff)|(k<<23)); return x;}
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if (k <= -25)
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return tiny*rb_copysignf(tiny,x); /*underflow*/
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k += 25; /* subnormal result */
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SET_FLOAT_WORD(x,(ix&0x807fffff)|(k<<23));
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return x*twom25;
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}
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static const float
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bp[] = {1.0, 1.5,},
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dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
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dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
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one = 1.0,
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two = 2.0,
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two24 = 16777216.0, /* 0x4b800000 */
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/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
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L1 = 6.0000002384e-01, /* 0x3f19999a */
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L2 = 4.2857143283e-01, /* 0x3edb6db7 */
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L3 = 3.3333334327e-01, /* 0x3eaaaaab */
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L4 = 2.7272811532e-01, /* 0x3e8ba305 */
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L5 = 2.3066075146e-01, /* 0x3e6c3255 */
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L6 = 2.0697501302e-01, /* 0x3e53f142 */
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P1 = 1.6666667163e-01, /* 0x3e2aaaab */
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P2 = -2.7777778450e-03, /* 0xbb360b61 */
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P3 = 6.6137559770e-05, /* 0x388ab355 */
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P4 = -1.6533901999e-06, /* 0xb5ddea0e */
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P5 = 4.1381369442e-08; /* 0x3331bb4c */
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static const float
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lg2 = 6.9314718246e-01, /* 0x3f317218 */
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lg2_h = 6.93145752e-01, /* 0x3f317200 */
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lg2_l = 1.42860654e-06, /* 0x35bfbe8c */
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ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
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cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
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cp_h = 9.6179199219e-01, /* 0x3f763800 =head of cp */
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cp_l = 4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
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ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
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ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
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ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
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float rb_pow(float x, float y)
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{
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unsigned int e;
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float result;
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float z, ax, z_h, z_l, p_h, p_l;
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float y1, t1, t2, r, s, t, u, v, w;
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int32_t i, j, k, yisint, n;
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int32_t hx, hy, ix, iy, is;
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/* Special cases 0^x */
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if(x == 0.0f)
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{
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if(y > 0.0f)
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return 0.0f;
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else if(y == 0.0f)
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return 1.0f;
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else
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return 1.0f / x;
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GET_FLOAT_WORD(hx,x);
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GET_FLOAT_WORD(hy,y);
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ix = hx&0x7fffffff;
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iy = hy&0x7fffffff;
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/* y==zero: x**0 = 1 */
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if(iy==0) return one;
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/* x==+-1 */
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if(x == 1.0) return one;
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if(x == -1.0 && rb_isinf(y)) return one;
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/* +-NaN return x+y */
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if(ix > 0x7f800000 || iy > 0x7f800000)
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return x+y;
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/* determine if y is an odd int when x < 0
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* yisint = 0 ... y is not an integer
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* yisint = 1 ... y is an odd int
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* yisint = 2 ... y is an even int
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*/
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yisint = 0;
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if(hx<0) {
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if(iy>=0x4b800000) yisint = 2; /* even integer y */
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else if(iy>=0x3f800000) {
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k = (iy>>23)-0x7f; /* exponent */
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j = iy>>(23-k);
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if((j<<(23-k))==iy) yisint = 2-(j&1);
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}
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}
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/* Special case x^n where n is integer */
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if(y == (int) (e = (int) y))
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{
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if((int) e < 0)
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{
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e = -e;
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x = 1.0f / x;
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}
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result = 1.0f;
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while(1)
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{
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if(e & 1)
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result *= x;
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if((e >>= 1) == 0)
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break;
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x *= x;
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}
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return result;
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/* special value of y */
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if (iy==0x7f800000) { /* y is +-inf */
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if (ix==0x3f800000)
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return y - y; /* inf**+-1 is NaN */
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else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
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return (hy>=0)? y: zero;
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else /* (|x|<1)**-,+inf = inf,0 */
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return (hy<0)?-y: zero;
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}
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if(iy==0x3f800000) { /* y is +-1 */
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if(hy<0) return one/x; else return x;
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}
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if(hy==0x40000000) return x*x; /* y is 2 */
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if(hy==0x3f000000) { /* y is 0.5 */
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if(hx>=0) /* x >= +0 */
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return rb_sqrt(x);
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}
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/* Normal case */
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return rb_exp(rb_log(x) * y);
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ax = rb_fabs(x);
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/* special value of x */
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if(ix==0x7f800000||ix==0||ix==0x3f800000){
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z = ax; /*x is +-0,+-inf,+-1*/
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if(hy<0) z = one/z; /* z = (1/|x|) */
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if(hx<0) {
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if(((ix-0x3f800000)|yisint)==0) {
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z = (z-z)/(z-z); /* (-1)**non-int is NaN */
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} else if(yisint==1)
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z = -z; /* (x<0)**odd = -(|x|**odd) */
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}
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return z;
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}
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/* (x<0)**(non-int) is NaN */
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if(((((uint32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
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/* |y| is huge */
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if(iy>0x4d000000) { /* if |y| > 2**27 */
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/* over/underflow if x is not close to one */
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if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny;
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if(ix>0x3f800007) return (hy>0)? huge*huge:tiny*tiny;
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/* now |1-x| is tiny <= 2**-20, suffice to compute
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log(x) by x-x^2/2+x^3/3-x^4/4 */
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t = x-1; /* t has 20 trailing zeros */
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w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
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u = ivln2_h*t; /* ivln2_h has 16 sig. bits */
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v = t*ivln2_l-w*ivln2;
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t1 = u+v;
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GET_FLOAT_WORD(is,t1);
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SET_FLOAT_WORD(t1,is&0xfffff000);
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t2 = v-(t1-u);
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} else {
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float s2, s_h, s_l, t_h, t_l;
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n = 0;
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/* take care subnormal number */
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if(ix<0x00800000)
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{ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
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n += ((ix)>>23)-0x7f;
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j = ix&0x007fffff;
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/* determine interval */
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ix = j|0x3f800000; /* normalize ix */
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if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */
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else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */
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else {k=0;n+=1;ix -= 0x00800000;}
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SET_FLOAT_WORD(ax,ix);
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/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
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u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
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v = one/(ax+bp[k]);
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s = u*v;
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s_h = s;
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GET_FLOAT_WORD(is,s_h);
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SET_FLOAT_WORD(s_h,is&0xfffff000);
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/* t_h=ax+bp[k] High */
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SET_FLOAT_WORD(t_h,((ix>>1)|0x20000000)+0x0040000+(k<<21));
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t_l = ax - (t_h-bp[k]);
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s_l = v*((u-s_h*t_h)-s_h*t_l);
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/* compute log(ax) */
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s2 = s*s;
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r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
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r += s_l*(s_h+s);
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s2 = s_h*s_h;
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t_h = (float)3.0+s2+r;
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GET_FLOAT_WORD(is,t_h);
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SET_FLOAT_WORD(t_h,is&0xfffff000);
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t_l = r-((t_h-(float)3.0)-s2);
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/* u+v = s*(1+...) */
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u = s_h*t_h;
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v = s_l*t_h+t_l*s;
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/* 2/(3log2)*(s+...) */
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p_h = u+v;
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GET_FLOAT_WORD(is,p_h);
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SET_FLOAT_WORD(p_h,is&0xfffff000);
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p_l = v-(p_h-u);
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z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
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z_l = cp_l*p_h+p_l*cp+dp_l[k];
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/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
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t = (float)n;
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t1 = (((z_h+z_l)+dp_h[k])+t);
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GET_FLOAT_WORD(is,t1);
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SET_FLOAT_WORD(t1,is&0xfffff000);
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t2 = z_l-(((t1-t)-dp_h[k])-z_h);
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}
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s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
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if(((((uint32_t)hx>>31)-1)|(yisint-1))==0)
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s = -one; /* (-ve)**(odd int) */
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/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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GET_FLOAT_WORD(is,y);
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SET_FLOAT_WORD(y1,is&0xfffff000);
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p_l = (y-y1)*t1+y*t2;
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p_h = y1*t1;
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z = p_l+p_h;
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GET_FLOAT_WORD(j,z);
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if (j>0x43000000) /* if z > 128 */
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return s*huge*huge; /* overflow */
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else if (j==0x43000000) { /* if z == 128 */
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if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
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}
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else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */
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return s*tiny*tiny; /* underflow */
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else if ((uint32_t) j==0xc3160000){ /* z == -150 */
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if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
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}
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/*
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* compute 2**(p_h+p_l)
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*/
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i = j&0x7fffffff;
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k = (i>>23)-0x7f;
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n = 0;
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if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */
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n = j+(0x00800000>>(k+1));
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k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */
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SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
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n = ((n&0x007fffff)|0x00800000)>>(23-k);
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if(j<0) n = -n;
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p_h -= t;
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}
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t = p_l+p_h;
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GET_FLOAT_WORD(is,t);
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SET_FLOAT_WORD(t,is&0xfffff000);
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u = t*lg2_h;
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v = (p_l-(t-p_h))*lg2+t*lg2_l;
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z = u+v;
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w = v-(z-u);
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t = z*z;
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t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
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r = (z*t1)/(t1-two)-(w+z*w);
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z = one-(r-z);
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GET_FLOAT_WORD(j,z);
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j += (n<<23);
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if((j>>23)<=0) z = rb_scalbnf(z,n); /* subnormal output */
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else SET_FLOAT_WORD(z,j);
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return s*z;
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}
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@ -701,10 +930,14 @@ float rb_sqrt(float x)
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return z;
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}
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/* Absolute value, simple calculus */
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/* Absolute value,
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taken from glibc-2.8 */
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float rb_fabs(float x)
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{
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return (x < 0.0f) ? -x : x;
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uint32_t ix;
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GET_FLOAT_WORD(ix,x);
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SET_FLOAT_WORD(x,ix&0x7fffffff);
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return x;
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}
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/* Arc tangent,
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@ -860,18 +1093,559 @@ float rb_atan2(float x, float y)
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}
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/* Sine hyperbolic, taken from dietlibc-0.32 */
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/* Sine hyperbolic,
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taken from glibc-2.8 */
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static const float
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o_threshold = 8.8721679688e+01,/* 0x42b17180 */
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invln2 = 1.4426950216e+00,/* 0x3fb8aa3b */
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/* scaled coefficients related to expm1 */
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Q1 = -3.3333335072e-02, /* 0xbd088889 */
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Q2 = 1.5873016091e-03, /* 0x3ad00d01 */
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Q3 = -7.9365076090e-05, /* 0xb8a670cd */
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Q4 = 4.0082177293e-06, /* 0x36867e54 */
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Q5 = -2.0109921195e-07; /* 0xb457edbb */
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float rb_expm1(float x)
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{
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float y,hi,lo,c=0,t,e,hxs,hfx,r1;
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int32_t k,xsb;
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uint32_t hx;
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GET_FLOAT_WORD(hx,x);
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xsb = hx&0x80000000; /* sign bit of x */
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if(xsb==0) y=x; else y= -x; /* y = |x| */
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hx &= 0x7fffffff; /* high word of |x| */
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/* filter out huge and non-finite argument */
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if(hx >= 0x4195b844) { /* if |x|>=27*ln2 */
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if(hx >= 0x42b17218) { /* if |x|>=88.721... */
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if(hx>0x7f800000)
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return x+x; /* NaN */
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if(hx==0x7f800000)
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return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
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if(x > o_threshold) return huge*huge; /* overflow */
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}
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if(xsb!=0) { /* x < -27*ln2, return -1.0 with inexact */
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if(x+tiny<(float)0.0) /* raise inexact */
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return tiny-one; /* return -1 */
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}
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}
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/* argument reduction */
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if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */
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if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */
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if(xsb==0)
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{hi = x - ln2_hi; lo = ln2_lo; k = 1;}
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else
|
||||
{hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
|
||||
} else {
|
||||
k = invln2*x+((xsb==0)?(float)0.5:(float)-0.5);
|
||||
t = k;
|
||||
hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
|
||||
lo = t*ln2_lo;
|
||||
}
|
||||
x = hi - lo;
|
||||
c = (hi-x)-lo;
|
||||
}
|
||||
else if(hx < 0x33000000) { /* when |x|<2**-25, return x */
|
||||
t = huge+x; /* return x with inexact flags when x!=0 */
|
||||
return x - (t-(huge+x));
|
||||
}
|
||||
else k = 0;
|
||||
|
||||
/* x is now in primary range */
|
||||
hfx = (float)0.5*x;
|
||||
hxs = x*hfx;
|
||||
r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
|
||||
t = (float)3.0-r1*hfx;
|
||||
e = hxs*((r1-t)/((float)6.0 - x*t));
|
||||
if(k==0) return x - (x*e-hxs); /* c is 0 */
|
||||
else {
|
||||
e = (x*(e-c)-c);
|
||||
e -= hxs;
|
||||
if(k== -1) return (float)0.5*(x-e)-(float)0.5;
|
||||
if(k==1) {
|
||||
if(x < (float)-0.25) return -(float)2.0*(e-(x+(float)0.5));
|
||||
else return one+(float)2.0*(x-e);
|
||||
}
|
||||
if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
|
||||
int32_t i;
|
||||
y = one-(e-x);
|
||||
GET_FLOAT_WORD(i,y);
|
||||
SET_FLOAT_WORD(y,i+(k<<23)); /* add k to y's exponent */
|
||||
return y-one;
|
||||
}
|
||||
t = one;
|
||||
if(k<23) {
|
||||
int32_t i;
|
||||
SET_FLOAT_WORD(t,0x3f800000 - (0x1000000>>k)); /* t=1-2^-k */
|
||||
y = t-(e-x);
|
||||
GET_FLOAT_WORD(i,y);
|
||||
SET_FLOAT_WORD(y,i+(k<<23)); /* add k to y's exponent */
|
||||
} else {
|
||||
int32_t i;
|
||||
SET_FLOAT_WORD(t,((0x7f-k)<<23)); /* 2^-k */
|
||||
y = x-(e+t);
|
||||
y += one;
|
||||
GET_FLOAT_WORD(i,y);
|
||||
SET_FLOAT_WORD(y,i+(k<<23)); /* add k to y's exponent */
|
||||
}
|
||||
}
|
||||
return y;
|
||||
}
|
||||
|
||||
static const float shuge = 1.0e37;
|
||||
|
||||
float rb_sinh(float x)
|
||||
{
|
||||
float y = rb_exp(x);
|
||||
return (y - 1.0f / y) * 0.5f;
|
||||
float t,w,h;
|
||||
int32_t ix,jx;
|
||||
|
||||
GET_FLOAT_WORD(jx,x);
|
||||
ix = jx&0x7fffffff;
|
||||
|
||||
/* x is INF or NaN */
|
||||
if(ix>=0x7f800000) return x+x;
|
||||
|
||||
h = 0.5;
|
||||
if (jx<0) h = -h;
|
||||
/* |x| in [0,22], return sign(x)*0.5*(E+E/(E+1))) */
|
||||
if (ix < 0x41b00000) { /* |x|<22 */
|
||||
if (ix<0x31800000) /* |x|<2**-28 */
|
||||
if(shuge+x>one) return x;/* sinh(tiny) = tiny with inexact */
|
||||
t = rb_expm1(rb_fabs(x));
|
||||
if(ix<0x3f800000) return h*((float)2.0*t-t*t/(t+one));
|
||||
return h*(t+t/(t+one));
|
||||
}
|
||||
|
||||
/* |x| in [22, log(maxdouble)] return 0.5*exp(|x|) */
|
||||
if (ix < 0x42b17180) return h*rb_exp(rb_fabs(x));
|
||||
|
||||
/* |x| in [log(maxdouble), overflowthresold] */
|
||||
if (ix<=0x42b2d4fc) {
|
||||
w = rb_exp((float)0.5*rb_fabs(x));
|
||||
t = h*w;
|
||||
return t*w;
|
||||
}
|
||||
|
||||
/* |x| > overflowthresold, sinh(x) overflow */
|
||||
return x*shuge;
|
||||
}
|
||||
|
||||
|
||||
/* Tangent, simple calculus solution. */
|
||||
/* Tangent,
|
||||
taken from glibc-2.8 */
|
||||
|
||||
static const float
|
||||
pio4 = 7.8539812565e-01, /* 0x3f490fda */
|
||||
pio4lo= 3.7748947079e-08, /* 0x33222168 */
|
||||
T[] = {
|
||||
3.3333334327e-01, /* 0x3eaaaaab */
|
||||
1.3333334029e-01, /* 0x3e088889 */
|
||||
5.3968254477e-02, /* 0x3d5d0dd1 */
|
||||
2.1869488060e-02, /* 0x3cb327a4 */
|
||||
8.8632395491e-03, /* 0x3c11371f */
|
||||
3.5920790397e-03, /* 0x3b6b6916 */
|
||||
1.4562094584e-03, /* 0x3abede48 */
|
||||
5.8804126456e-04, /* 0x3a1a26c8 */
|
||||
2.4646313977e-04, /* 0x398137b9 */
|
||||
7.8179444245e-05, /* 0x38a3f445 */
|
||||
7.1407252108e-05, /* 0x3895c07a */
|
||||
-1.8558637748e-05, /* 0xb79bae5f */
|
||||
2.5907305826e-05, /* 0x37d95384 */
|
||||
};
|
||||
|
||||
float kernel_tan(float x, float y, int iy)
|
||||
{
|
||||
float z,r,v,w,s;
|
||||
int32_t ix,hx;
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff; /* high word of |x| */
|
||||
if(ix<0x31800000) /* x < 2**-28 */
|
||||
{if((int)x==0) { /* generate inexact */
|
||||
if((ix|(iy+1))==0) return one/rb_fabs(x);
|
||||
else return (iy==1)? x: -one/x;
|
||||
}
|
||||
}
|
||||
if(ix>=0x3f2ca140) { /* |x|>=0.6744 */
|
||||
if(hx<0) {x = -x; y = -y;}
|
||||
z = pio4-x;
|
||||
w = pio4lo-y;
|
||||
x = z+w; y = 0.0;
|
||||
}
|
||||
z = x*x;
|
||||
w = z*z;
|
||||
/* Break x^5*(T[1]+x^2*T[2]+...) into
|
||||
* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
|
||||
* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
|
||||
*/
|
||||
r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
|
||||
v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
|
||||
s = z*x;
|
||||
r = y + z*(s*(r+v)+y);
|
||||
r += T[0]*s;
|
||||
w = x+r;
|
||||
if(ix>=0x3f2ca140) {
|
||||
v = (float)iy;
|
||||
return (float)(1-((hx>>30)&2))*(v-(float)2.0*(x-(w*w/(w+v)-r)));
|
||||
}
|
||||
if(iy==1) return w;
|
||||
else { /* if allow error up to 2 ulp,
|
||||
simply return -1.0/(x+r) here */
|
||||
/* compute -1.0/(x+r) accurately */
|
||||
float a,t;
|
||||
int32_t i;
|
||||
z = w;
|
||||
GET_FLOAT_WORD(i,z);
|
||||
SET_FLOAT_WORD(z,i&0xfffff000);
|
||||
v = r-(z - x); /* z+v = r+x */
|
||||
t = a = -(float)1.0/w; /* a = -1.0/w */
|
||||
GET_FLOAT_WORD(i,t);
|
||||
SET_FLOAT_WORD(t,i&0xfffff000);
|
||||
s = (float)1.0+t*z;
|
||||
return t+a*(s+t*v);
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
|
||||
static const int init_jk[] = {4,7,9}; /* initial value for jk */
|
||||
|
||||
static const float PIo2[] = {
|
||||
1.5703125000e+00, /* 0x3fc90000 */
|
||||
4.5776367188e-04, /* 0x39f00000 */
|
||||
2.5987625122e-05, /* 0x37da0000 */
|
||||
7.5437128544e-08, /* 0x33a20000 */
|
||||
6.0026650317e-11, /* 0x2e840000 */
|
||||
7.3896444519e-13, /* 0x2b500000 */
|
||||
5.3845816694e-15, /* 0x27c20000 */
|
||||
5.6378512969e-18, /* 0x22d00000 */
|
||||
8.3009228831e-20, /* 0x1fc40000 */
|
||||
3.2756352257e-22, /* 0x1bc60000 */
|
||||
6.3331015649e-25, /* 0x17440000 */
|
||||
};
|
||||
|
||||
static const float
|
||||
two8 = 2.5600000000e+02, /* 0x43800000 */
|
||||
twon8 = 3.9062500000e-03; /* 0x3b800000 */
|
||||
|
||||
int kernel_rem_pio2(float *x, float *y, int e0, int nx, int prec, const int32_t *ipio2)
|
||||
{
|
||||
int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
|
||||
float z,fw,f[20],fq[20],q[20];
|
||||
|
||||
/* initialize jk*/
|
||||
jk = init_jk[prec];
|
||||
jp = jk;
|
||||
|
||||
/* determine jx,jv,q0, note that 3>q0 */
|
||||
jx = nx-1;
|
||||
jv = (e0-3)/8; if(jv<0) jv=0;
|
||||
q0 = e0-8*(jv+1);
|
||||
|
||||
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
|
||||
j = jv-jx; m = jx+jk;
|
||||
for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (float) ipio2[j];
|
||||
|
||||
/* compute q[0],q[1],...q[jk] */
|
||||
for (i=0;i<=jk;i++) {
|
||||
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
|
||||
}
|
||||
|
||||
jz = jk;
|
||||
recompute:
|
||||
/* distill q[] into iq[] reversingly */
|
||||
for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
|
||||
fw = (float)((int32_t)(twon8* z));
|
||||
iq[i] = (int32_t)(z-two8*fw);
|
||||
z = q[j-1]+fw;
|
||||
}
|
||||
|
||||
/* compute n */
|
||||
z = rb_scalbnf(z,q0); /* actual value of z */
|
||||
z -= (float)8.0*rb_floor(z*(float)0.125); /* trim off integer >= 8 */
|
||||
n = (int32_t) z;
|
||||
z -= (float)n;
|
||||
ih = 0;
|
||||
if(q0>0) { /* need iq[jz-1] to determine n */
|
||||
i = (iq[jz-1]>>(8-q0)); n += i;
|
||||
iq[jz-1] -= i<<(8-q0);
|
||||
ih = iq[jz-1]>>(7-q0);
|
||||
}
|
||||
else if(q0==0) ih = iq[jz-1]>>8;
|
||||
else if(z>=(float)0.5) ih=2;
|
||||
|
||||
if(ih>0) { /* q > 0.5 */
|
||||
n += 1; carry = 0;
|
||||
for(i=0;i<jz ;i++) { /* compute 1-q */
|
||||
j = iq[i];
|
||||
if(carry==0) {
|
||||
if(j!=0) {
|
||||
carry = 1; iq[i] = 0x100- j;
|
||||
}
|
||||
} else iq[i] = 0xff - j;
|
||||
}
|
||||
if(q0>0) { /* rare case: chance is 1 in 12 */
|
||||
switch(q0) {
|
||||
case 1:
|
||||
iq[jz-1] &= 0x7f; break;
|
||||
case 2:
|
||||
iq[jz-1] &= 0x3f; break;
|
||||
}
|
||||
}
|
||||
if(ih==2) {
|
||||
z = one - z;
|
||||
if(carry!=0) z -= rb_scalbnf(one,q0);
|
||||
}
|
||||
}
|
||||
|
||||
/* check if recomputation is needed */
|
||||
if(z==zero) {
|
||||
j = 0;
|
||||
for (i=jz-1;i>=jk;i--) j |= iq[i];
|
||||
if(j==0) { /* need recomputation */
|
||||
for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
|
||||
|
||||
for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
|
||||
f[jx+i] = (float) ipio2[jv+i];
|
||||
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
|
||||
q[i] = fw;
|
||||
}
|
||||
jz += k;
|
||||
goto recompute;
|
||||
}
|
||||
}
|
||||
|
||||
/* chop off zero terms */
|
||||
if(z==(float)0.0) {
|
||||
jz -= 1; q0 -= 8;
|
||||
while(iq[jz]==0) { jz--; q0-=8;}
|
||||
} else { /* break z into 8-bit if necessary */
|
||||
z = rb_scalbnf(z,-q0);
|
||||
if(z>=two8) {
|
||||
fw = (float)((int32_t)(twon8*z));
|
||||
iq[jz] = (int32_t)(z-two8*fw);
|
||||
jz += 1; q0 += 8;
|
||||
iq[jz] = (int32_t) fw;
|
||||
} else iq[jz] = (int32_t) z ;
|
||||
}
|
||||
|
||||
/* convert integer "bit" chunk to floating-point value */
|
||||
fw = rb_scalbnf(one,q0);
|
||||
for(i=jz;i>=0;i--) {
|
||||
q[i] = fw*(float)iq[i]; fw*=twon8;
|
||||
}
|
||||
|
||||
/* compute PIo2[0,...,jp]*q[jz,...,0] */
|
||||
for(i=jz;i>=0;i--) {
|
||||
for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
|
||||
fq[jz-i] = fw;
|
||||
}
|
||||
|
||||
/* compress fq[] into y[] */
|
||||
switch(prec) {
|
||||
case 0:
|
||||
fw = 0.0;
|
||||
for (i=jz;i>=0;i--) fw += fq[i];
|
||||
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;
|
||||
fw = fq[0]-fw;
|
||||
for (i=1;i<=jz;i++) fw += fq[i];
|
||||
y[1] = (ih==0)? fw: -fw;
|
||||
break;
|
||||
case 3: /* painful */
|
||||
for (i=jz;i>0;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];
|
||||
fq[i] += fq[i-1]-fw;
|
||||
fq[i-1] = fw;
|
||||
}
|
||||
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 {
|
||||
y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
|
||||
}
|
||||
}
|
||||
return n&7;
|
||||
}
|
||||
|
||||
|
||||
static const int32_t two_over_pi[] = {
|
||||
0xA2, 0xF9, 0x83, 0x6E, 0x4E, 0x44, 0x15, 0x29, 0xFC,
|
||||
0x27, 0x57, 0xD1, 0xF5, 0x34, 0xDD, 0xC0, 0xDB, 0x62,
|
||||
0x95, 0x99, 0x3C, 0x43, 0x90, 0x41, 0xFE, 0x51, 0x63,
|
||||
0xAB, 0xDE, 0xBB, 0xC5, 0x61, 0xB7, 0x24, 0x6E, 0x3A,
|
||||
0x42, 0x4D, 0xD2, 0xE0, 0x06, 0x49, 0x2E, 0xEA, 0x09,
|
||||
0xD1, 0x92, 0x1C, 0xFE, 0x1D, 0xEB, 0x1C, 0xB1, 0x29,
|
||||
0xA7, 0x3E, 0xE8, 0x82, 0x35, 0xF5, 0x2E, 0xBB, 0x44,
|
||||
0x84, 0xE9, 0x9C, 0x70, 0x26, 0xB4, 0x5F, 0x7E, 0x41,
|
||||
0x39, 0x91, 0xD6, 0x39, 0x83, 0x53, 0x39, 0xF4, 0x9C,
|
||||
0x84, 0x5F, 0x8B, 0xBD, 0xF9, 0x28, 0x3B, 0x1F, 0xF8,
|
||||
0x97, 0xFF, 0xDE, 0x05, 0x98, 0x0F, 0xEF, 0x2F, 0x11,
|
||||
0x8B, 0x5A, 0x0A, 0x6D, 0x1F, 0x6D, 0x36, 0x7E, 0xCF,
|
||||
0x27, 0xCB, 0x09, 0xB7, 0x4F, 0x46, 0x3F, 0x66, 0x9E,
|
||||
0x5F, 0xEA, 0x2D, 0x75, 0x27, 0xBA, 0xC7, 0xEB, 0xE5,
|
||||
0xF1, 0x7B, 0x3D, 0x07, 0x39, 0xF7, 0x8A, 0x52, 0x92,
|
||||
0xEA, 0x6B, 0xFB, 0x5F, 0xB1, 0x1F, 0x8D, 0x5D, 0x08,
|
||||
0x56, 0x03, 0x30, 0x46, 0xFC, 0x7B, 0x6B, 0xAB, 0xF0,
|
||||
0xCF, 0xBC, 0x20, 0x9A, 0xF4, 0x36, 0x1D, 0xA9, 0xE3,
|
||||
0x91, 0x61, 0x5E, 0xE6, 0x1B, 0x08, 0x65, 0x99, 0x85,
|
||||
0x5F, 0x14, 0xA0, 0x68, 0x40, 0x8D, 0xFF, 0xD8, 0x80,
|
||||
0x4D, 0x73, 0x27, 0x31, 0x06, 0x06, 0x15, 0x56, 0xCA,
|
||||
0x73, 0xA8, 0xC9, 0x60, 0xE2, 0x7B, 0xC0, 0x8C, 0x6B,
|
||||
};
|
||||
|
||||
static const int32_t npio2_hw[] = {
|
||||
0x3fc90f00, 0x40490f00, 0x4096cb00, 0x40c90f00, 0x40fb5300, 0x4116cb00,
|
||||
0x412fed00, 0x41490f00, 0x41623100, 0x417b5300, 0x418a3a00, 0x4196cb00,
|
||||
0x41a35c00, 0x41afed00, 0x41bc7e00, 0x41c90f00, 0x41d5a000, 0x41e23100,
|
||||
0x41eec200, 0x41fb5300, 0x4203f200, 0x420a3a00, 0x42108300, 0x4216cb00,
|
||||
0x421d1400, 0x42235c00, 0x4229a500, 0x422fed00, 0x42363600, 0x423c7e00,
|
||||
0x4242c700, 0x42490f00
|
||||
};
|
||||
|
||||
/*
|
||||
* invpio2: 24 bits of 2/pi
|
||||
* pio2_1: first 17 bit of pi/2
|
||||
* pio2_1t: pi/2 - pio2_1
|
||||
* pio2_2: second 17 bit of pi/2
|
||||
* pio2_2t: pi/2 - (pio2_1+pio2_2)
|
||||
* pio2_3: third 17 bit of pi/2
|
||||
* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
|
||||
*/
|
||||
|
||||
static const float
|
||||
half = 5.0000000000e-01, /* 0x3f000000 */
|
||||
invpio2 = 6.3661980629e-01, /* 0x3f22f984 */
|
||||
pio2_1 = 1.5707855225e+00, /* 0x3fc90f80 */
|
||||
pio2_1t = 1.0804334124e-05, /* 0x37354443 */
|
||||
pio2_2 = 1.0804273188e-05, /* 0x37354400 */
|
||||
pio2_2t = 6.0770999344e-11, /* 0x2e85a308 */
|
||||
pio2_3 = 6.0770943833e-11, /* 0x2e85a300 */
|
||||
pio2_3t = 6.1232342629e-17; /* 0x248d3132 */
|
||||
|
||||
int32_t rem_pio2(float x, float *y)
|
||||
{
|
||||
float z,w,t,r,fn;
|
||||
float tx[3];
|
||||
int32_t e0,i,j,nx,n,ix,hx;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix<=0x3f490fd8) /* |x| ~<= pi/4 , no need for reduction */
|
||||
{y[0] = x; y[1] = 0; return 0;}
|
||||
if(ix<0x4016cbe4) { /* |x| < 3pi/4, special case with n=+-1 */
|
||||
if(hx>0) {
|
||||
z = x - pio2_1;
|
||||
if((ix&0xfffffff0)!=0x3fc90fd0) { /* 24+24 bit pi OK */
|
||||
y[0] = z - pio2_1t;
|
||||
y[1] = (z-y[0])-pio2_1t;
|
||||
} else { /* near pi/2, use 24+24+24 bit pi */
|
||||
z -= pio2_2;
|
||||
y[0] = z - pio2_2t;
|
||||
y[1] = (z-y[0])-pio2_2t;
|
||||
}
|
||||
return 1;
|
||||
} else { /* negative x */
|
||||
z = x + pio2_1;
|
||||
if((ix&0xfffffff0)!=0x3fc90fd0) { /* 24+24 bit pi OK */
|
||||
y[0] = z + pio2_1t;
|
||||
y[1] = (z-y[0])+pio2_1t;
|
||||
} else { /* near pi/2, use 24+24+24 bit pi */
|
||||
z += pio2_2;
|
||||
y[0] = z + pio2_2t;
|
||||
y[1] = (z-y[0])+pio2_2t;
|
||||
}
|
||||
return -1;
|
||||
}
|
||||
}
|
||||
if(ix<=0x43490f80) { /* |x| ~<= 2^7*(pi/2), medium size */
|
||||
t = rb_fabs(x);
|
||||
n = (int32_t) (t*invpio2+half);
|
||||
fn = (float)n;
|
||||
r = t-fn*pio2_1;
|
||||
w = fn*pio2_1t; /* 1st round good to 40 bit */
|
||||
if(n<32&&(int32_t)(ix&0xffffff00)!=npio2_hw[n-1]) {
|
||||
y[0] = r-w; /* quick check no cancellation */
|
||||
} else {
|
||||
uint32_t high;
|
||||
j = ix>>23;
|
||||
y[0] = r-w;
|
||||
GET_FLOAT_WORD(high,y[0]);
|
||||
i = j-((high>>23)&0xff);
|
||||
if(i>8) { /* 2nd iteration needed, good to 57 */
|
||||
t = r;
|
||||
w = fn*pio2_2;
|
||||
r = t-w;
|
||||
w = fn*pio2_2t-((t-r)-w);
|
||||
y[0] = r-w;
|
||||
GET_FLOAT_WORD(high,y[0]);
|
||||
i = j-((high>>23)&0xff);
|
||||
if(i>25) { /* 3rd iteration need, 74 bits acc */
|
||||
t = r; /* will cover all possible cases */
|
||||
w = fn*pio2_3;
|
||||
r = t-w;
|
||||
w = fn*pio2_3t-((t-r)-w);
|
||||
y[0] = r-w;
|
||||
}
|
||||
}
|
||||
}
|
||||
y[1] = (r-y[0])-w;
|
||||
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
|
||||
else return n;
|
||||
}
|
||||
/*
|
||||
* all other (large) arguments
|
||||
*/
|
||||
if(ix>=0x7f800000) { /* x is inf or NaN */
|
||||
y[0]=y[1]=x-x; return 0;
|
||||
}
|
||||
/* set z = scalbn(|x|,ilogb(x)-7) */
|
||||
e0 = (ix>>23)-134; /* e0 = ilogb(z)-7; */
|
||||
SET_FLOAT_WORD(z, ix - ((int32_t)(e0<<23)));
|
||||
for(i=0;i<2;i++) {
|
||||
tx[i] = (float)((int32_t)(z));
|
||||
z = (z-tx[i])*two8;
|
||||
}
|
||||
tx[2] = z;
|
||||
nx = 3;
|
||||
while(tx[nx-1]==zero) nx--; /* skip zero term */
|
||||
n = kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
|
||||
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
|
||||
return n;
|
||||
}
|
||||
|
||||
float rb_tan(float x)
|
||||
{
|
||||
return rb_sin(x) / rb_cos(x);
|
||||
float y[2],z=0.0;
|
||||
int32_t n, ix;
|
||||
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
|
||||
/* |x| ~< pi/4 */
|
||||
ix &= 0x7fffffff;
|
||||
if(ix <= 0x3f490fda) return kernel_tan(x,z,1);
|
||||
|
||||
/* tan(Inf or NaN) is NaN */
|
||||
else if (ix>=0x7f800000) return x-x; /* NaN */
|
||||
|
||||
/* argument reduction needed */
|
||||
else {
|
||||
n = rem_pio2(x,y);
|
||||
return kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even
|
||||
-1 -- n odd */
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue