diff options
Diffstat (limited to 'sysdeps/ieee754/dbl-64/e_jn.c')
| -rw-r--r-- | sysdeps/ieee754/dbl-64/e_jn.c | 478 |
1 files changed, 258 insertions, 220 deletions
diff --git a/sysdeps/ieee754/dbl-64/e_jn.c b/sysdeps/ieee754/dbl-64/e_jn.c index 0d2a24c93b..f48e43a0d9 100644 --- a/sysdeps/ieee754/dbl-64/e_jn.c +++ b/sysdeps/ieee754/dbl-64/e_jn.c @@ -41,246 +41,284 @@ #include <math_private.h> static const double -invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ -two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ -one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ + invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ + two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ + one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ -static const double zero = 0.00000000000000000000e+00; +static const double zero = 0.00000000000000000000e+00; double -__ieee754_jn(int n, double x) +__ieee754_jn (int n, double x) { - int32_t i,hx,ix,lx, sgn; - double a, b, temp, di; - double z, w; + int32_t i, hx, ix, lx, sgn; + double a, b, temp, di; + double z, w; - /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) - * Thus, J(-n,x) = J(n,-x) - */ - EXTRACT_WORDS(hx,lx,x); - ix = 0x7fffffff&hx; - /* if J(n,NaN) is NaN */ - if(__builtin_expect((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000, 0)) - return x+x; - if(n<0){ - n = -n; - x = -x; - hx ^= 0x80000000; + /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) + * Thus, J(-n,x) = J(n,-x) + */ + EXTRACT_WORDS (hx, lx, x); + ix = 0x7fffffff & hx; + /* if J(n,NaN) is NaN */ + if (__builtin_expect ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000, 0)) + return x + x; + if (n < 0) + { + n = -n; + x = -x; + hx ^= 0x80000000; + } + if (n == 0) + return (__ieee754_j0 (x)); + if (n == 1) + return (__ieee754_j1 (x)); + sgn = (n & 1) & (hx >> 31); /* even n -- 0, odd n -- sign(x) */ + x = fabs (x); + if (__builtin_expect ((ix | lx) == 0 || ix >= 0x7ff00000, 0)) + /* if x is 0 or inf */ + b = zero; + else if ((double) n <= x) + { + /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ + if (ix >= 0x52D00000) /* x > 2**302 */ + { /* (x >> n**2) + * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Let s=sin(x), c=cos(x), + * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then + * + * n sin(xn)*sqt2 cos(xn)*sqt2 + * ---------------------------------- + * 0 s-c c+s + * 1 -s-c -c+s + * 2 -s+c -c-s + * 3 s+c c-s + */ + double s; + double c; + __sincos (x, &s, &c); + switch (n & 3) + { + case 0: temp = c + s; break; + case 1: temp = -c + s; break; + case 2: temp = -c - s; break; + case 3: temp = c - s; break; + } + b = invsqrtpi * temp / __ieee754_sqrt (x); } - if(n==0) return(__ieee754_j0(x)); - if(n==1) return(__ieee754_j1(x)); - sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ - x = fabs(x); - if(__builtin_expect((ix|lx)==0||ix>=0x7ff00000,0)) - /* if x is 0 or inf */ - b = zero; - else if((double)n<=x) { - /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ - if(ix>=0x52D00000) { /* x > 2**302 */ - /* (x >> n**2) - * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Let s=sin(x), c=cos(x), - * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then - * - * n sin(xn)*sqt2 cos(xn)*sqt2 - * ---------------------------------- - * 0 s-c c+s - * 1 -s-c -c+s - * 2 -s+c -c-s - * 3 s+c c-s - */ - double s; - double c; - __sincos (x, &s, &c); - switch(n&3) { - case 0: temp = c + s; break; - case 1: temp = -c + s; break; - case 2: temp = -c - s; break; - case 3: temp = c - s; break; - } - b = invsqrtpi*temp/__ieee754_sqrt(x); - } else { - a = __ieee754_j0(x); - b = __ieee754_j1(x); - for(i=1;i<n;i++){ - temp = b; - b = b*((double)(i+i)/x) - a; /* avoid underflow */ - a = temp; - } + else + { + a = __ieee754_j0 (x); + b = __ieee754_j1 (x); + for (i = 1; i < n; i++) + { + temp = b; + b = b * ((double) (i + i) / x) - a; /* avoid underflow */ + a = temp; } - } else { - if(ix<0x3e100000) { /* x < 2**-29 */ - /* x is tiny, return the first Taylor expansion of J(n,x) - * J(n,x) = 1/n!*(x/2)^n - ... - */ - if(n>33) /* underflow */ - b = zero; - else { - temp = x*0.5; b = temp; - for (a=one,i=2;i<=n;i++) { - a *= (double)i; /* a = n! */ - b *= temp; /* b = (x/2)^n */ - } - b = b/a; + } + } + else + { + if (ix < 0x3e100000) /* x < 2**-29 */ + { /* x is tiny, return the first Taylor expansion of J(n,x) + * J(n,x) = 1/n!*(x/2)^n - ... + */ + if (n > 33) /* underflow */ + b = zero; + else + { + temp = x * 0.5; b = temp; + for (a = one, i = 2; i <= n; i++) + { + a *= (double) i; /* a = n! */ + b *= temp; /* b = (x/2)^n */ } - } else { - /* use backward recurrence */ - /* x x^2 x^2 - * J(n,x)/J(n-1,x) = ---- ------ ------ ..... - * 2n - 2(n+1) - 2(n+2) - * - * 1 1 1 - * (for large x) = ---- ------ ------ ..... - * 2n 2(n+1) 2(n+2) - * -- - ------ - ------ - - * x x x - * - * Let w = 2n/x and h=2/x, then the above quotient - * is equal to the continued fraction: - * 1 - * = ----------------------- - * 1 - * w - ----------------- - * 1 - * w+h - --------- - * w+2h - ... - * - * To determine how many terms needed, let - * Q(0) = w, Q(1) = w(w+h) - 1, - * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), - * When Q(k) > 1e4 good for single - * When Q(k) > 1e9 good for double - * When Q(k) > 1e17 good for quadruple - */ - /* determine k */ - double t,v; - double q0,q1,h,tmp; int32_t k,m; - w = (n+n)/(double)x; h = 2.0/(double)x; - q0 = w; z = w+h; q1 = w*z - 1.0; k=1; - while(q1<1.0e9) { - k += 1; z += h; - tmp = z*q1 - q0; - q0 = q1; - q1 = tmp; + b = b / a; + } + } + else + { + /* use backward recurrence */ + /* x x^2 x^2 + * J(n,x)/J(n-1,x) = ---- ------ ------ ..... + * 2n - 2(n+1) - 2(n+2) + * + * 1 1 1 + * (for large x) = ---- ------ ------ ..... + * 2n 2(n+1) 2(n+2) + * -- - ------ - ------ - + * x x x + * + * Let w = 2n/x and h=2/x, then the above quotient + * is equal to the continued fraction: + * 1 + * = ----------------------- + * 1 + * w - ----------------- + * 1 + * w+h - --------- + * w+2h - ... + * + * To determine how many terms needed, let + * Q(0) = w, Q(1) = w(w+h) - 1, + * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), + * When Q(k) > 1e4 good for single + * When Q(k) > 1e9 good for double + * When Q(k) > 1e17 good for quadruple + */ + /* determine k */ + double t, v; + double q0, q1, h, tmp; int32_t k, m; + w = (n + n) / (double) x; h = 2.0 / (double) x; + q0 = w; z = w + h; q1 = w * z - 1.0; k = 1; + while (q1 < 1.0e9) + { + k += 1; z += h; + tmp = z * q1 - q0; + q0 = q1; + q1 = tmp; + } + m = n + n; + for (t = zero, i = 2 * (n + k); i >= m; i -= 2) + t = one / (i / x - t); + a = t; + b = one; + /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) + * Hence, if n*(log(2n/x)) > ... + * single 8.8722839355e+01 + * double 7.09782712893383973096e+02 + * long double 1.1356523406294143949491931077970765006170e+04 + * then recurrent value may overflow and the result is + * likely underflow to zero + */ + tmp = n; + v = two / x; + tmp = tmp * __ieee754_log (fabs (v * tmp)); + if (tmp < 7.09782712893383973096e+02) + { + for (i = n - 1, di = (double) (i + i); i > 0; i--) + { + temp = b; + b *= di; + b = b / x - a; + a = temp; + di -= two; } - m = n+n; - for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); - a = t; - b = one; - /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) - * Hence, if n*(log(2n/x)) > ... - * single 8.8722839355e+01 - * double 7.09782712893383973096e+02 - * long double 1.1356523406294143949491931077970765006170e+04 - * then recurrent value may overflow and the result is - * likely underflow to zero - */ - tmp = n; - v = two/x; - tmp = tmp*__ieee754_log(fabs(v*tmp)); - if(tmp<7.09782712893383973096e+02) { - for(i=n-1,di=(double)(i+i);i>0;i--){ - temp = b; - b *= di; - b = b/x - a; - a = temp; - di -= two; - } - } else { - for(i=n-1,di=(double)(i+i);i>0;i--){ - temp = b; - b *= di; - b = b/x - a; - a = temp; - di -= two; - /* scale b to avoid spurious overflow */ - if(b>1e100) { - a /= b; - t /= b; - b = one; - } + } + else + { + for (i = n - 1, di = (double) (i + i); i > 0; i--) + { + temp = b; + b *= di; + b = b / x - a; + a = temp; + di -= two; + /* scale b to avoid spurious overflow */ + if (b > 1e100) + { + a /= b; + t /= b; + b = one; } } - /* j0() and j1() suffer enormous loss of precision at and - * near zero; however, we know that their zero points never - * coincide, so just choose the one further away from zero. - */ - z = __ieee754_j0 (x); - w = __ieee754_j1 (x); - if (fabs (z) >= fabs (w)) - b = (t * z / b); - else - b = (t * w / a); } + /* j0() and j1() suffer enormous loss of precision at and + * near zero; however, we know that their zero points never + * coincide, so just choose the one further away from zero. + */ + z = __ieee754_j0 (x); + w = __ieee754_j1 (x); + if (fabs (z) >= fabs (w)) + b = (t * z / b); + else + b = (t * w / a); } - if(sgn==1) return -b; else return b; + } + if (sgn == 1) + return -b; + else + return b; } strong_alias (__ieee754_jn, __jn_finite) double -__ieee754_yn(int n, double x) +__ieee754_yn (int n, double x) { - int32_t i,hx,ix,lx; - int32_t sign; - double a, b, temp; + int32_t i, hx, ix, lx; + int32_t sign; + double a, b, temp; - EXTRACT_WORDS(hx,lx,x); - ix = 0x7fffffff&hx; - /* if Y(n,NaN) is NaN */ - if(__builtin_expect((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000,0)) - return x+x; - if(__builtin_expect((ix|lx)==0, 0)) - return -HUGE_VAL+x; /* -inf and overflow exception. */; - if(__builtin_expect(hx<0, 0)) return zero/(zero*x); - sign = 1; - if(n<0){ - n = -n; - sign = 1 - ((n&1)<<1); + EXTRACT_WORDS (hx, lx, x); + ix = 0x7fffffff & hx; + /* if Y(n,NaN) is NaN */ + if (__builtin_expect ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000, 0)) + return x + x; + if (__builtin_expect ((ix | lx) == 0, 0)) + return -HUGE_VAL + x; + /* -inf and overflow exception. */; + if (__builtin_expect (hx < 0, 0)) + return zero / (zero * x); + sign = 1; + if (n < 0) + { + n = -n; + sign = 1 - ((n & 1) << 1); + } + if (n == 0) + return (__ieee754_y0 (x)); + if (n == 1) + return (sign * __ieee754_y1 (x)); + if (__builtin_expect (ix == 0x7ff00000, 0)) + return zero; + if (ix >= 0x52D00000) /* x > 2**302 */ + { /* (x >> n**2) + * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Let s=sin(x), c=cos(x), + * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then + * + * n sin(xn)*sqt2 cos(xn)*sqt2 + * ---------------------------------- + * 0 s-c c+s + * 1 -s-c -c+s + * 2 -s+c -c-s + * 3 s+c c-s + */ + double c; + double s; + __sincos (x, &s, &c); + switch (n & 3) + { + case 0: temp = s - c; break; + case 1: temp = -s - c; break; + case 2: temp = -s + c; break; + case 3: temp = s + c; break; } - if(n==0) return(__ieee754_y0(x)); - if(n==1) return(sign*__ieee754_y1(x)); - if(__builtin_expect(ix==0x7ff00000, 0)) return zero; - if(ix>=0x52D00000) { /* x > 2**302 */ - /* (x >> n**2) - * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Let s=sin(x), c=cos(x), - * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then - * - * n sin(xn)*sqt2 cos(xn)*sqt2 - * ---------------------------------- - * 0 s-c c+s - * 1 -s-c -c+s - * 2 -s+c -c-s - * 3 s+c c-s - */ - double c; - double s; - __sincos (x, &s, &c); - switch(n&3) { - case 0: temp = s - c; break; - case 1: temp = -s - c; break; - case 2: temp = -s + c; break; - case 3: temp = s + c; break; - } - b = invsqrtpi*temp/__ieee754_sqrt(x); - } else { - u_int32_t high; - a = __ieee754_y0(x); - b = __ieee754_y1(x); - /* quit if b is -inf */ - GET_HIGH_WORD(high,b); - for(i=1;i<n&&high!=0xfff00000;i++){ - temp = b; - b = ((double)(i+i)/x)*b - a; - GET_HIGH_WORD(high,b); - a = temp; - } - /* If B is +-Inf, set up errno accordingly. */ - if (! __finite (b)) - __set_errno (ERANGE); + b = invsqrtpi * temp / __ieee754_sqrt (x); + } + else + { + u_int32_t high; + a = __ieee754_y0 (x); + b = __ieee754_y1 (x); + /* quit if b is -inf */ + GET_HIGH_WORD (high, b); + for (i = 1; i < n && high != 0xfff00000; i++) + { + temp = b; + b = ((double) (i + i) / x) * b - a; + GET_HIGH_WORD (high, b); + a = temp; } - if(sign>0) return b; else return -b; + /* If B is +-Inf, set up errno accordingly. */ + if (!__finite (b)) + __set_errno (ERANGE); + } + if (sign > 0) + return b; + else + return -b; } strong_alias (__ieee754_yn, __yn_finite) |