float y0f(float x)
{
float z,s,c,ss,cc,u,v;
int32_t hx,ix;
GET_FLOAT_WORD(hx, x);
ix = 0x7fffffff & hx;
/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
if (ix >= 0x7f800000)
return 1.0f/(x+x*x);
if (ix == 0)
return -1.0f/0.0f;
if (hx < 0)
return 0.0f/0.0f;
if (ix >= 0x40000000) { /* |x| >= 2.0 */
/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
* where x0 = x-pi/4
* Better formula:
* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
* = 1/sqrt(2) * (sin(x) + cos(x))
* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
* = 1/sqrt(2) * (sin(x) - cos(x))
* To avoid cancellation, use
* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
* to compute the worse one.
*/
s = sinf(x);
c = cosf(x);
ss = s-c;
cc = s+c;
/*
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
*/
if (ix < 0x7f000000) { /* make sure x+x not overflow */
z = -cosf(x+x);
if (s*c < 0.0f)
cc = z/ss;
else
ss = z/cc;
}
if (ix > 0x80000000)
z = (invsqrtpi*ss)/sqrtf(x);
else {
u = pzerof(x);
v = qzerof(x);
z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
}
return z;
}
if (ix <= 0x32000000) { /* x < 2**-27 */
return u00 + tpi*logf(x);
}
z = x*x;
u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
v = 1.0f+z*(v01+z*(v02+z*(v03+z*v04)));
return u/v + tpi*(j0f(x)*logf(x));
}
int main()
{
float y;
int i;
for (i = 0; i< 20; i++)
{
y = j0f(z[i]);
printf("%.9e\n", y);
}
exit(0);
}
static void RDFtoRP(const Curve &rdf, int npoints, Curve *rp) {
const float wstep = 1.f / sqrtf(npoints);
Curve tmp(rdf);
for (int i = 0; i < rp->size(); ++i) {
const float u0 = rp->ToX(i);
const float u = TWOPI * u0;
const float wndsize = rdf.x1 * std::min(0.5f, std::max(0.2f, 4.f * u0 * wstep));
for (int j = 0; j < tmp.size(); ++j) {
float x = rdf.ToX(j);
float wnd = BlackmanWindow(x, wndsize);
tmp[j] = (rdf[j] - 1) * j0f(u*x) * x * wnd;
}
(*rp)[i] = fabsf(1.f + TWOPI * Integrate(tmp) * npoints);
}
}
int main(void)
{
#pragma STDC FENV_ACCESS ON
float y;
float d;
int e, i, err = 0;
struct f_f *p;
for (i = 0; i < sizeof t/sizeof *t; i++) {
p = t + i;
if (p->r < 0)
continue;
fesetround(p->r);
feclearexcept(FE_ALL_EXCEPT);
y = j0f(p->x);
e = fetestexcept(INEXACT|INVALID|DIVBYZERO|UNDERFLOW|OVERFLOW);
if (!checkexcept(e, p->e, p->r)) {
printf("%s:%d: bad fp exception: %s j0f(%a)=%a, want %s",
p->file, p->line, rstr(p->r), p->x, p->y, estr(p->e));
printf(" got %s\n", estr(e));
err++;
}
d = ulperrf(y, p->y, p->dy);
if (!checkulp(d, p->r)) {
// printf("%s:%d: %s j0f(%a) want %a got %a ulperr %.3f = %a + %a\n",
// p->file, p->line, rstr(p->r), p->x, p->y, y, d, d-p->dy, p->dy);
err++;
// TODO: avoid spamming the output
printf(__FILE__ ": known to be broken near zeros\n");
break;
}
}
return !!err;
}
void
domathf (void)
{
#ifndef NO_FLOAT
float f1;
float f2;
int i1;
f1 = acosf(0.0);
fprintf( stdout, "acosf : %f\n", f1);
f1 = acoshf(0.0);
fprintf( stdout, "acoshf : %f\n", f1);
f1 = asinf(1.0);
fprintf( stdout, "asinf : %f\n", f1);
f1 = asinhf(1.0);
fprintf( stdout, "asinhf : %f\n", f1);
f1 = atanf(M_PI_4);
fprintf( stdout, "atanf : %f\n", f1);
f1 = atan2f(2.3, 2.3);
fprintf( stdout, "atan2f : %f\n", f1);
f1 = atanhf(1.0);
fprintf( stdout, "atanhf : %f\n", f1);
f1 = cbrtf(27.0);
fprintf( stdout, "cbrtf : %f\n", f1);
f1 = ceilf(3.5);
fprintf( stdout, "ceilf : %f\n", f1);
f1 = copysignf(3.5, -2.5);
fprintf( stdout, "copysignf : %f\n", f1);
f1 = cosf(M_PI_2);
fprintf( stdout, "cosf : %f\n", f1);
f1 = coshf(M_PI_2);
fprintf( stdout, "coshf : %f\n", f1);
f1 = erff(42.0);
fprintf( stdout, "erff : %f\n", f1);
f1 = erfcf(42.0);
fprintf( stdout, "erfcf : %f\n", f1);
f1 = expf(0.42);
fprintf( stdout, "expf : %f\n", f1);
f1 = exp2f(0.42);
fprintf( stdout, "exp2f : %f\n", f1);
f1 = expm1f(0.00042);
fprintf( stdout, "expm1f : %f\n", f1);
f1 = fabsf(-1.123);
fprintf( stdout, "fabsf : %f\n", f1);
f1 = fdimf(1.123, 2.123);
fprintf( stdout, "fdimf : %f\n", f1);
f1 = floorf(0.5);
fprintf( stdout, "floorf : %f\n", f1);
f1 = floorf(-0.5);
fprintf( stdout, "floorf : %f\n", f1);
f1 = fmaf(2.1, 2.2, 3.01);
fprintf( stdout, "fmaf : %f\n", f1);
f1 = fmaxf(-0.42, 0.42);
fprintf( stdout, "fmaxf : %f\n", f1);
f1 = fminf(-0.42, 0.42);
fprintf( stdout, "fminf : %f\n", f1);
f1 = fmodf(42.0, 3.0);
fprintf( stdout, "fmodf : %f\n", f1);
/* no type-specific variant */
i1 = fpclassify(1.0);
fprintf( stdout, "fpclassify : %d\n", i1);
f1 = frexpf(42.0, &i1);
fprintf( stdout, "frexpf : %f\n", f1);
f1 = hypotf(42.0, 42.0);
fprintf( stdout, "hypotf : %f\n", f1);
i1 = ilogbf(42.0);
fprintf( stdout, "ilogbf : %d\n", i1);
/* no type-specific variant */
i1 = isfinite(3.0);
fprintf( stdout, "isfinite : %d\n", i1);
/* no type-specific variant */
i1 = isgreater(3.0, 3.1);
fprintf( stdout, "isgreater : %d\n", i1);
/* no type-specific variant */
i1 = isgreaterequal(3.0, 3.1);
fprintf( stdout, "isgreaterequal : %d\n", i1);
/* no type-specific variant */
i1 = isinf(3.0);
fprintf( stdout, "isinf : %d\n", i1);
/* no type-specific variant */
i1 = isless(3.0, 3.1);
fprintf( stdout, "isless : %d\n", i1);
/* no type-specific variant */
i1 = islessequal(3.0, 3.1);
fprintf( stdout, "islessequal : %d\n", i1);
/* no type-specific variant */
i1 = islessgreater(3.0, 3.1);
fprintf( stdout, "islessgreater : %d\n", i1);
/* no type-specific variant */
i1 = isnan(0.0);
fprintf( stdout, "isnan : %d\n", i1);
/* no type-specific variant */
i1 = isnormal(3.0);
fprintf( stdout, "isnormal : %d\n", i1);
/* no type-specific variant */
f1 = isunordered(1.0, 2.0);
fprintf( stdout, "isunordered : %d\n", i1);
f1 = j0f(1.2);
fprintf( stdout, "j0f : %f\n", f1);
f1 = j1f(1.2);
fprintf( stdout, "j1f : %f\n", f1);
f1 = jnf(2,1.2);
fprintf( stdout, "jnf : %f\n", f1);
f1 = ldexpf(1.2,3);
fprintf( stdout, "ldexpf : %f\n", f1);
f1 = lgammaf(42.0);
fprintf( stdout, "lgammaf : %f\n", f1);
f1 = llrintf(-0.5);
fprintf( stdout, "llrintf : %f\n", f1);
f1 = llrintf(0.5);
fprintf( stdout, "llrintf : %f\n", f1);
f1 = llroundf(-0.5);
fprintf( stdout, "lroundf : %f\n", f1);
f1 = llroundf(0.5);
fprintf( stdout, "lroundf : %f\n", f1);
f1 = logf(42.0);
fprintf( stdout, "logf : %f\n", f1);
f1 = log10f(42.0);
fprintf( stdout, "log10f : %f\n", f1);
f1 = log1pf(42.0);
fprintf( stdout, "log1pf : %f\n", f1);
f1 = log2f(42.0);
fprintf( stdout, "log2f : %f\n", f1);
f1 = logbf(42.0);
fprintf( stdout, "logbf : %f\n", f1);
f1 = lrintf(-0.5);
fprintf( stdout, "lrintf : %f\n", f1);
f1 = lrintf(0.5);
fprintf( stdout, "lrintf : %f\n", f1);
f1 = lroundf(-0.5);
fprintf( stdout, "lroundf : %f\n", f1);
f1 = lroundf(0.5);
fprintf( stdout, "lroundf : %f\n", f1);
f1 = modff(42.0,&f2);
fprintf( stdout, "lmodff : %f\n", f1);
f1 = nanf("");
fprintf( stdout, "nanf : %f\n", f1);
f1 = nearbyintf(1.5);
fprintf( stdout, "nearbyintf : %f\n", f1);
f1 = nextafterf(1.5,2.0);
fprintf( stdout, "nextafterf : %f\n", f1);
f1 = powf(3.01, 2.0);
fprintf( stdout, "powf : %f\n", f1);
f1 = remainderf(3.01,2.0);
fprintf( stdout, "remainderf : %f\n", f1);
f1 = remquof(29.0,3.0,&i1);
fprintf( stdout, "remquof : %f\n", f1);
f1 = rintf(0.5);
fprintf( stdout, "rintf : %f\n", f1);
f1 = rintf(-0.5);
fprintf( stdout, "rintf : %f\n", f1);
f1 = roundf(0.5);
fprintf( stdout, "roundf : %f\n", f1);
f1 = roundf(-0.5);
fprintf( stdout, "roundf : %f\n", f1);
f1 = scalblnf(1.2,3);
fprintf( stdout, "scalblnf : %f\n", f1);
f1 = scalbnf(1.2,3);
fprintf( stdout, "scalbnf : %f\n", f1);
/* no type-specific variant */
i1 = signbit(1.0);
fprintf( stdout, "signbit : %i\n", i1);
f1 = sinf(M_PI_4);
fprintf( stdout, "sinf : %f\n", f1);
f1 = sinhf(M_PI_4);
fprintf( stdout, "sinhf : %f\n", f1);
f1 = sqrtf(9.0);
fprintf( stdout, "sqrtf : %f\n", f1);
f1 = tanf(M_PI_4);
fprintf( stdout, "tanf : %f\n", f1);
f1 = tanhf(M_PI_4);
fprintf( stdout, "tanhf : %f\n", f1);
f1 = tgammaf(2.1);
fprintf( stdout, "tgammaf : %f\n", f1);
f1 = truncf(3.5);
fprintf( stdout, "truncf : %f\n", f1);
f1 = y0f(1.2);
fprintf( stdout, "y0f : %f\n", f1);
f1 = y1f(1.2);
fprintf( stdout, "y1f : %f\n", f1);
f1 = ynf(3,1.2);
fprintf( stdout, "ynf : %f\n", f1);
#endif
}
int main(int argc, char *argv[])
{
float x = 0.0;
if (argv) x = j0f((float) argc);
return 0;
}
TEST(math, j0f) {
ASSERT_FLOAT_EQ(1.0f, j0f(0.0f));
ASSERT_FLOAT_EQ(0.76519769f, j0f(1.0f));
}
float jnf(int n, float x) {
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) = J(n,-x), use |n|-1 to avoid overflow in -n */
if (n == 0)
return j0f(x);
if (n < 0) {
nm1 = -(n + 1);
x = -x;
sign ^= 1;
} else
nm1 = n - 1;
if (nm1 == 0)
return j1f(x);
sign &= n; /* even n: 0, odd n: signbit(x) */
x = fabsf(x);
if (ix == 0 || ix == 0x7f800000) /* if x is 0 or inf */
b = 0.0f;
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 = 0; i < nm1;) {
i++;
temp = b;
b = b * (2.0f * i / x) - a;
a = temp;
}
} else {
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 (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
* 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 */
float t, q0, q1, w, h, z, tmp, nf;
int k;
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 < 1.0e4f) {
k += 1;
z += h;
tmp = z * q1 - q0;
q0 = q1;
q1 = tmp;
}
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)
* 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 = nf * logf(fabsf(w));
if (tmp < 88.721679688f) {
for (i = nm1; i > 0; i--) {
temp = b;
b = 2.0f * i * b / x - a;
a = temp;
}
} else {
for (i = nm1; i > 0; i--) {
temp = b;
b = 2.0f * i * b / x - a;
a = temp;
/* scale b to avoid spurious overflow */
if (b > 0x1p60f) {
a /= b;
t /= b;
b = 1.0f;
}
}
}
z = j0f(x);
w = j1f(x);
if (fabsf(z) >= fabsf(w))
b = t * z / b;
else
b = t * w / a;
}
}
return sign ? -b : b;
}
__global__ void FloatMathPrecise() {
int iX;
float fX, fY;
acosf(1.0f);
acoshf(1.0f);
asinf(0.0f);
asinhf(0.0f);
atan2f(0.0f, 1.0f);
atanf(0.0f);
atanhf(0.0f);
cbrtf(0.0f);
fX = ceilf(0.0f);
fX = copysignf(1.0f, -2.0f);
cosf(0.0f);
coshf(0.0f);
cospif(0.0f);
cyl_bessel_i0f(0.0f);
cyl_bessel_i1f(0.0f);
erfcf(0.0f);
erfcinvf(2.0f);
erfcxf(0.0f);
erff(0.0f);
erfinvf(1.0f);
exp10f(0.0f);
exp2f(0.0f);
expf(0.0f);
expm1f(0.0f);
fX = fabsf(1.0f);
fdimf(1.0f, 0.0f);
fdividef(0.0f, 1.0f);
fX = floorf(0.0f);
fmaf(1.0f, 2.0f, 3.0f);
fX = fmaxf(0.0f, 0.0f);
fX = fminf(0.0f, 0.0f);
fmodf(0.0f, 1.0f);
frexpf(0.0f, &iX);
hypotf(1.0f, 0.0f);
ilogbf(1.0f);
isfinite(0.0f);
fX = isinf(0.0f);
fX = isnan(0.0f);
j0f(0.0f);
j1f(0.0f);
jnf(-1.0f, 1.0f);
ldexpf(0.0f, 0);
lgammaf(1.0f);
llrintf(0.0f);
llroundf(0.0f);
log10f(1.0f);
log1pf(-1.0f);
log2f(1.0f);
logbf(1.0f);
logf(1.0f);
lrintf(0.0f);
lroundf(0.0f);
modff(0.0f, &fX);
fX = nanf("1");
fX = nearbyintf(0.0f);
nextafterf(0.0f, 0.0f);
norm3df(1.0f, 0.0f, 0.0f);
norm4df(1.0f, 0.0f, 0.0f, 0.0f);
normcdff(0.0f);
normcdfinvf(1.0f);
fX = 1.0f;
normf(1, &fX);
powf(1.0f, 0.0f);
rcbrtf(1.0f);
remainderf(2.0f, 1.0f);
remquof(1.0f, 2.0f, &iX);
rhypotf(0.0f, 1.0f);
fY = rintf(1.0f);
rnorm3df(0.0f, 0.0f, 1.0f);
rnorm4df(0.0f, 0.0f, 0.0f, 1.0f);
fX = 1.0f;
rnormf(1, &fX);
fY = roundf(0.0f);
rsqrtf(1.0f);
scalblnf(0.0f, 1);
scalbnf(0.0f, 1);
signbit(1.0f);
sincosf(0.0f, &fX, &fY);
sincospif(0.0f, &fX, &fY);
sinf(0.0f);
sinhf(0.0f);
sinpif(0.0f);
sqrtf(0.0f);
tanf(0.0f);
tanhf(0.0f);
tgammaf(2.0f);
fY = truncf(0.0f);
y0f(1.0f);
y1f(1.0f);
ynf(1, 1.0f);
}