diff options
| -rw-r--r-- | src/math/jn.c | 171 | ||||
| -rw-r--r-- | src/math/jnf.c | 154 |
2 files changed, 161 insertions, 164 deletions
diff --git a/src/math/jn.c b/src/math/jn.c index d95af15d..4878a54f 100644 --- a/src/math/jn.c +++ b/src/math/jn.c @@ -20,7 +20,7 @@ * Note 2. About jn(n,x), yn(n,x) * For n=0, j0(x) is called, * for n=1, j1(x) is called, - * for n<x, forward recursion us used starting + * for n<=x, forward recursion is used starting * from values of j0(x) and j1(x). * for n>x, a continued fraction approximation to * j(n,x)/j(n-1,x) is evaluated and then backward @@ -32,7 +32,6 @@ * yn(n,x) is similar in all respects, except * that forward recursion is used for all * values of n>1. - * */ #include "libm.h" @@ -41,33 +40,39 @@ static const double invsqrtpi = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x504 double jn(int n, double x) { - int32_t i,hx,ix,lx,sgn; - double a, b, temp, di; - double z, w; + uint32_t ix, lx; + int nm1, i, sign; + double a, b, temp; + + EXTRACT_WORDS(ix, lx, x); + sign = ix>>31; + ix &= 0x7fffffff; + + if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */ + return x; /* 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 ((ix|((uint32_t)(lx|-lx))>>31) > 0x7ff00000) - return x+x; + /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */ + if (n == 0) + return j0(x); if (n < 0) { - n = -n; + nm1 = -(n+1); x = -x; - hx ^= 0x80000000; - } - if (n == 0) return j0(x); - if (n == 1) return j1(x); + sign ^= 1; + } else + nm1 = n-1; + if (nm1 == 0) + return j1(x); - sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ + sign &= n; /* even n: 0, odd n: signbit(x) */ x = fabs(x); - if ((ix|lx) == 0 || ix >= 0x7ff00000) /* if x is 0 or inf */ + if ((ix|lx) == 0 || ix == 0x7ff00000) /* if x is 0 or inf */ b = 0.0; - else if ((double)n <= x) { + else if (nm1 < x) { /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ - if (ix >= 0x52D00000) { /* x > 2**302 */ + 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) @@ -81,19 +86,21 @@ double jn(int n, double x) * 2 -s+c -c-s * 3 s+c c-s */ - switch(n&3) { - case 0: temp = cos(x)+sin(x); break; - case 1: temp = -cos(x)+sin(x); break; - case 2: temp = -cos(x)-sin(x); break; - case 3: temp = cos(x)-sin(x); break; + switch(nm1&3) { + case 0: temp = -cos(x)+sin(x); break; + case 1: temp = -cos(x)-sin(x); break; + case 2: temp = cos(x)-sin(x); break; + default: + case 3: temp = cos(x)+sin(x); break; } b = invsqrtpi*temp/sqrt(x); } else { a = j0(x); b = j1(x); - for (i=1; i<n; i++){ + for (i=0; i<nm1; ) { + i++; temp = b; - b = b*((double)(i+i)/x) - a; /* avoid underflow */ + b = b*(2.0*i/x) - a; /* avoid underflow */ a = temp; } } @@ -102,12 +109,13 @@ double jn(int n, double x) /* x is tiny, return the first Taylor expansion of J(n,x) * J(n,x) = 1/n!*(x/2)^n - ... */ - if (n > 33) /* underflow */ + if (nm1 > 32) /* underflow */ b = 0.0; else { temp = x*0.5; b = temp; - for (a=1.0,i=2; i<=n; i++) { + a = 1.0; + for (i=2; i<=nm1+1; i++) { a *= (double)i; /* a = n! */ b *= temp; /* b = (x/2)^n */ } @@ -143,13 +151,14 @@ double jn(int n, double x) * When Q(k) > 1e17 good for quadruple */ /* determine k */ - double t,v; - double q0,q1,h,tmp; - int32_t k,m; + double t,q0,q1,w,h,z,tmp,nf; + int k; - w = (n+n)/(double)x; h = 2.0/(double)x; - q0 = w; + nf = nm1 + 1.0; + w = 2*nf/x; + h = 2/x; z = w+h; + q0 = w; q1 = w*z - 1.0; k = 1; while (q1 < 1.0e9) { @@ -159,9 +168,8 @@ double jn(int n, double x) q0 = q1; q1 = tmp; } - m = n+n; - for (t=0.0, i = 2*(n+k); i>=m; i -= 2) - t = 1.0/(i/x-t); + for (t=0.0, i=k; i>=0; i--) + t = 1/(2*(i+nf)/x - t); a = t; b = 1.0; /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) @@ -172,26 +180,20 @@ double jn(int n, double x) * then recurrent value may overflow and the result is * likely underflow to zero */ - tmp = n; - v = 2.0/x; - tmp = tmp*log(fabs(v*tmp)); + tmp = nf*log(fabs(w)); if (tmp < 7.09782712893383973096e+02) { - for (i=n-1,di=(double)(i+i); i>0; i--) { + for (i=nm1; i>0; i--) { temp = b; - b *= di; - b = b/x - a; + b = b*(2.0*i)/x - a; a = temp; - di -= 2.0; } } else { - for (i=n-1,di=(double)(i+i); i>0; i--) { + for (i=nm1; i>0; i--) { temp = b; - b *= di; - b = b/x - a; + b = b*(2.0*i)/x - a; a = temp; - di -= 2.0; /* scale b to avoid spurious overflow */ - if (b > 1e100) { + if (b > 0x1p500) { a /= b; t /= b; b = 1.0; @@ -206,39 +208,40 @@ double jn(int n, double x) b = t*w/a; } } - if (sgn==1) return -b; - return b; + return sign ? -b : b; } - double yn(int n, double x) { - int32_t i,hx,ix,lx; - int32_t sign; + uint32_t ix, lx, ib; + int nm1, sign, i; double a, b, temp; - EXTRACT_WORDS(hx, lx, x); - ix = 0x7fffffff & hx; - /* if Y(n,NaN) is NaN */ - if ((ix|((uint32_t)(lx|-lx))>>31) > 0x7ff00000) - return x+x; - if ((ix|lx) == 0) - return -1.0/0.0; - if (hx < 0) - return 0.0/0.0; - sign = 1; - if (n < 0) { - n = -n; - sign = 1 - ((n&1)<<1); - } - if (n == 0) - return y0(x); - if (n == 1) - return sign*y1(x); + EXTRACT_WORDS(ix, lx, x); + sign = ix>>31; + ix &= 0x7fffffff; + + if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */ + return x; + if (sign && (ix|lx)!=0) /* x < 0 */ + return 0/0.0; if (ix == 0x7ff00000) return 0.0; - if (ix >= 0x52D00000) { /* x > 2**302 */ + + if (n == 0) + return y0(x); + if (n < 0) { + nm1 = -(n+1); + sign = n&1; + } else { + nm1 = n-1; + sign = 0; + } + if (nm1 == 0) + return sign ? -y1(x) : y1(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) @@ -252,26 +255,26 @@ double yn(int n, double x) * 2 -s+c -c-s * 3 s+c c-s */ - switch(n&3) { - case 0: temp = sin(x)-cos(x); break; - case 1: temp = -sin(x)-cos(x); break; - case 2: temp = -sin(x)+cos(x); break; - case 3: temp = sin(x)+cos(x); break; + switch(nm1&3) { + case 0: temp = -sin(x)-cos(x); break; + case 1: temp = -sin(x)+cos(x); break; + case 2: temp = sin(x)+cos(x); break; + default: + case 3: temp = sin(x)-cos(x); break; } b = invsqrtpi*temp/sqrt(x); } else { - uint32_t high; a = y0(x); b = y1(x); /* quit if b is -inf */ - GET_HIGH_WORD(high, b); - for (i=1; i<n && high!=0xfff00000; i++){ + GET_HIGH_WORD(ib, b); + for (i=0; i<nm1 && ib!=0xfff00000; ){ + i++; temp = b; - b = ((double)(i+i)/x)*b - a; - GET_HIGH_WORD(high, b); + b = (2.0*i/x)*b - a; + GET_HIGH_WORD(ib, b); a = temp; } } - if (sign > 0) return b; - return -b; + return sign ? -b : b; } diff --git a/src/math/jnf.c b/src/math/jnf.c index fd291103..f63c062f 100644 --- a/src/math/jnf.c +++ b/src/math/jnf.c @@ -18,55 +18,57 @@ float jnf(int n, float x) { - int32_t i,hx,ix, sgn; - float a, b, temp, di; - float z, w; + uint32_t ix; + int nm1, sign, i; + float a, b, temp; + + GET_FLOAT_WORD(ix, x); + sign = ix>>31; + ix &= 0x7fffffff; + if (ix > 0x7f800000) /* nan */ + return x; - /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) - * Thus, J(-n,x) = J(n,-x) - */ - GET_FLOAT_WORD(hx, x); - ix = 0x7fffffff & hx; - /* if J(n,NaN) is NaN */ - if (ix > 0x7f800000) - return x+x; + /* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */ + if (n == 0) + return j0f(x); if (n < 0) { - n = -n; + nm1 = -(n+1); x = -x; - hx ^= 0x80000000; - } - if (n == 0) return j0f(x); - if (n == 1) return j1f(x); + sign ^= 1; + } else + nm1 = n-1; + if (nm1 == 0) + return j1f(x); - sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ + sign &= n; /* even n: 0, odd n: signbit(x) */ x = fabsf(x); - if (ix == 0 || ix >= 0x7f800000) /* if x is 0 or inf */ + if (ix == 0 || ix == 0x7f800000) /* if x is 0 or inf */ b = 0.0f; - else if((float)n <= x) { + else if (nm1 < x) { /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ a = j0f(x); b = j1f(x); - for (i=1; i<n; i++){ + for (i=0; i<nm1; ){ + i++; temp = b; - b = b*((float)(i+i)/x) - a; /* avoid underflow */ + b = b*(2.0f*i/x) - a; a = temp; } } else { - if (ix < 0x30800000) { /* x < 2**-29 */ + if (ix < 0x35800000) { /* x < 2**-20 */ /* 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 = 0.0f; - else { - temp = 0.5f * x; - b = temp; - for (a=1.0f,i=2; i<=n; i++) { - a *= (float)i; /* a = n! */ - b *= temp; /* b = (x/2)^n */ - } - b = b/a; + if (nm1 > 8) /* underflow */ + nm1 = 8; + temp = 0.5f * x; + b = temp; + a = 1.0f; + for (i=2; i<=nm1+1; i++) { + a *= (float)i; /* a = n! */ + b *= temp; /* b = (x/2)^n */ } + b = b/a; } else { /* use backward recurrence */ /* x x^2 x^2 @@ -97,26 +99,25 @@ float jnf(int n, float x) * When Q(k) > 1e17 good for quadruple */ /* determine k */ - float t,v; - float q0,q1,h,tmp; - int32_t k,m; + float t,q0,q1,w,h,z,tmp,nf; + int k; - w = (n+n)/x; - h = 2.0f/x; + nf = nm1+1.0f; + w = 2*nf/x; + h = 2/x; z = w+h; q0 = w; q1 = w*z - 1.0f; k = 1; - while (q1 < 1.0e9f) { + while (q1 < 1.0e4f) { k += 1; z += h; tmp = z*q1 - q0; q0 = q1; q1 = tmp; } - m = n+n; - for (t=0.0f, i = 2*(n+k); i>=m; i -= 2) - t = 1.0f/(i/x-t); + for (t=0.0f, i=k; i>=0; i--) + t = 1.0f/(2*(i+nf)/x-t); a = t; b = 1.0f; /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) @@ -127,26 +128,20 @@ float jnf(int n, float x) * then recurrent value may overflow and the result is * likely underflow to zero */ - tmp = n; - v = 2.0f/x; - tmp = tmp*logf(fabsf(v*tmp)); + tmp = nf*logf(fabsf(w)); if (tmp < 88.721679688f) { - for (i=n-1,di=(float)(i+i); i>0; i--) { + for (i=nm1; i>0; i--) { temp = b; - b *= di; - b = b/x - a; + b = 2.0f*i*b/x - a; a = temp; - di -= 2.0f; } } else { - for (i=n-1,di=(float)(i+i); i>0; i--){ + for (i=nm1; i>0; i--){ temp = b; - b *= di; - b = b/x - a; + b = 2.0f*i*b/x - a; a = temp; - di -= 2.0f; /* scale b to avoid spurious overflow */ - if (b > 1e10f) { + if (b > 0x1p60f) { a /= b; t /= b; b = 1.0f; @@ -161,48 +156,47 @@ float jnf(int n, float x) b = t*w/a; } } - if (sgn == 1) return -b; - return b; + return sign ? -b : b; } float ynf(int n, float x) { - int32_t i,hx,ix,ib; - int32_t sign; + uint32_t ix, ib; + int nm1, sign, i; float a, b, temp; - GET_FLOAT_WORD(hx, x); - ix = 0x7fffffff & hx; - /* if Y(n,NaN) is NaN */ - if (ix > 0x7f800000) - return x+x; - if (ix == 0) - return -1.0f/0.0f; - if (hx < 0) - return 0.0f/0.0f; - sign = 1; - if (n < 0) { - n = -n; - sign = 1 - ((n&1)<<1); - } - if (n == 0) - return y0f(x); - if (n == 1) - return sign*y1f(x); + GET_FLOAT_WORD(ix, x); + sign = ix>>31; + ix &= 0x7fffffff; + if (ix > 0x7f800000) /* nan */ + return x; + if (sign && ix != 0) /* x < 0 */ + return 0/0.0f; if (ix == 0x7f800000) return 0.0f; + if (n == 0) + return y0f(x); + if (n < 0) { + nm1 = -(n+1); + sign = n&1; + } else { + nm1 = n-1; + sign = 0; + } + if (nm1 == 0) + return sign ? -y1f(x) : y1f(x); + a = y0f(x); b = y1f(x); /* quit if b is -inf */ GET_FLOAT_WORD(ib,b); - for (i = 1; i < n && ib != 0xff800000; i++){ + for (i = 0; i < nm1 && ib != 0xff800000; ) { + i++; temp = b; - b = ((float)(i+i)/x)*b - a; + b = (2.0f*i/x)*b - a; GET_FLOAT_WORD(ib, b); a = temp; } - if (sign > 0) - return b; - return -b; + return sign ? -b : b; } |