* a = (1-w)½ = √2 sin z/2
* b = (1+w)½ = √2 cos z/2
*
* Then, with z = x+iy, a = a1+ia2, and b = * b1+ib2, *
* a1 / b1 = tan x/2
* a2b1 - a1b2 = sinh y
*
* and we use {@link Math#atan2(double, double)} and {@link math#asinh(double)} to obtain * x and y. *
* For w sufficiently large that w2≫1, cos-1w * ≈ -i ln(2w). * * @param w * @return cos-1w */ private static PyComplex _acos(PyComplex w) { // Let z = x + iy and w = u + iv. double x, y, u = w.real, v = w.imag; if (Math.abs(u) > 0x1p27 || Math.abs(v) > 0x1p27) { /* * w is large: approximate 2cos(z) by exp(i(x+iy)) or exp(-i(x+iy)), whichever * dominates. Hence, z = x+iy = i ln(2(u+iv)) or -i ln(2(u+iv)) */ x = Math.atan2(Math.abs(v), u); y = Math.copySign(logHypot(u, v) + math.LN2, -v); } else if (Double.isNaN(v)) { // Special cases x = (u == 0.) ? Math.PI / 2. : v; y = v; } else { // Normal case, without risk of overflow. PyComplex a = sqrt(new PyComplex(1. - u, -v)); // a = sqrt(1-w) = sqrt(2) sin(z/2) PyComplex b = sqrt(new PyComplex(1 + u, v)); // b = sqrt(1+w) = sqrt(2) cos(z/2) // Arguments here are sin(x/2)cosh(y/2), cos(x/2)cosh(y/2) giving tan(x/2) x = 2. * Math.atan2(a.real, b.real); // 2 (cos(x/2)**2+sin(x/2)**2) sinh(y/2)cosh(y/2) = sinh y y = math.asinh(a.imag * b.real - a.real * b.imag); } // If that generated a nan, and there wasn't one in the argument, raise a domain error. return exceptNaN(new PyComplex(x, y), w); } /** * Return the hyperbolic arc cosine of w. There is one branch cut, extending left from 1 along * the real axis to -∞, continuous from above. * * @param w * @return cosh-1w */ public static PyComplex acosh(PyObject w) { return _acosh(complexFromPyObject(w)); } /** * Helper to compute z = cosh-1w. The method used is as in CPython: *
* a = (w-1)½ = √2 sinh z/2
* b = (w+1)½ = √2 cosh z/2
*
* Then, with z = x+iy, a = a1+ia2, and b = * b1+ib2, *
* a1b1 + a2b2 = sinh x
* a2 / b1 = tan y/2
*
* and we use {@link math#asinh(double)} and {@link Math#atan2(double, double)} to obtain * x and y. *
* For w sufficiently large that w2≫1, * cosh-1w ≈ ln(2w). We do not use this method also to compute * cos-1w, because the branch cuts do not correspond. * * @param w * @return cosh-1w */ private static PyComplex _acosh(PyComplex w) { // Let z = x + iy and w = u + iv. double x, y, u = w.real, v = w.imag; if (Math.abs(u) > 0x1p27 || Math.abs(v) > 0x1p27) { /* * w is large: approximate 2cosh(z) by exp(x+iy) or exp(-x-iy), whichever dominates. * Hence, z = x+iy = ln(2(u+iv)) or -ln(2(u+iv)) */ x = logHypot(u, v) + math.LN2; y = Math.atan2(v, u); } else if (v == 0. && !Double.isNaN(u)) { /* * We're on the real axis (and maybe the branch cut). u = cosh x cos y. In all cases, * the sign of y follows v. */ if (u >= 1.) { // As real library, cos y = 1, u = cosh x. x = math.acosh(u); y = v; } else if (u < -1.) { // Left part of cut: cos y = -1, u = -cosh x x = math.acosh(-u); y = Math.copySign(Math.PI, v); } else { // -1 <= u <= 1: cosh x = 1, u = cos y. x = 0.; y = Math.copySign(Math.acos(u), v); } } else { // Normal case, without risk of overflow. PyComplex a = sqrt(new PyComplex(u - 1., v)); // a = sqrt(w-1) = sqrt(2) sinh(z/2) PyComplex b = sqrt(new PyComplex(u + 1., v)); // b = sqrt(w+1) = sqrt(2) cosh(z/2) // 2 sinh(x/2)cosh(x/2) (cos(y/2)**2+sin(y/2)**2) = sinh x x = math.asinh(a.real * b.real + a.imag * b.imag); // Arguments here are cosh(x/2)sin(y/2) and cosh(x/2)cos(y/2) giving tan y/2 y = 2. * Math.atan2(a.imag, b.real); } // If that generated a nan, and there wasn't one in the argument, raise a domain error. return exceptNaN(new PyComplex(x, y), w); } /** * Return the arc sine of w. There are two branch cuts. One extends right from 1 along the real * axis to ∞, continuous from below. The other extends left from -1 along the real axis to * -∞, continuous from above. * * @param w * @return sin-1w */ public static PyComplex asin(PyObject w) { return asinOrAsinh(complexFromPyObject(w), false); } /** * Return the hyperbolic arc sine of w. There are two branch cuts. One extends from 1j along the * imaginary axis to ∞j, continuous from the right. The other extends from -1j along the * imaginary axis to -∞j, continuous from the left. * * @param w * @return sinh-1w */ public static PyComplex asinh(PyObject w) { return asinOrAsinh(complexFromPyObject(w), true); } /** * Helper to compute either sin-1w or sinh-1w. The method * used is as in CPython: *
* a = (1-iw)½ = √2 sin(π/4-iz/2)
* b = (1+iw)½ = √2 cos(π/4-iz/2)
*
* Then, with w = u+iv, z = x+iy, a = a1+ia2, and * b = b1+ib2, *
* a1b2 - a2b2 = sinh x
* v / (a2b1 - a1b2) = tan y
*
* and we use {@link math#asinh(double)} and {@link Math#atan2(double, double)} to obtain * x and y. *
* For w sufficiently large that w2≫1,
* sinh-1w ≈ ln(2w). When computing sin-1w, we
* evaluate -i sinh-1iw instead.
*
* @param w
* @param h true to compute sinh-1w, false to
* compute sin-1w.
* @return sinh-1w or sin-1w
*/
private static PyComplex asinOrAsinh(PyComplex w, boolean h) {
double u, v, x, y;
PyComplex z;
if (h) {
// We compute z = asinh(w). Let z = x + iy and w = u + iv.
u = w.real;
v = w.imag;
// Then the function body computes x + iy = asinh(w).
} else {
// We compute w = asin(z). Unusually, let w = u - iv, so u + iv = iw.
v = w.real;
u = -w.imag;
// Then as before, the function body computes asinh(u+iv) = asinh(iw) = i asin(w),
// but we finally return z = y - ix = -i asinh(iw) = asin(w).
}
if (Double.isNaN(u)) {
// Special case for nan in real part. Default clause deals naturally with v=nan.
if (v == 0.) {
x = u;
y = v;
} else if (Double.isInfinite(v)) {
x = Double.POSITIVE_INFINITY;
y = u;
} else { // Any other value of v -> nan+nanj
x = y = u;
}
} else if (Math.abs(u) > 0x1p27 || Math.abs(v) > 0x1p27) {
/*
* w is large: approximate 2sinh(z) by exp(x+iy) or -exp(-x-iy), whichever dominates.
* Hence, z = x+iy = ln(2(u+iv)) or -ln(-2(u+iv))
*/
x = logHypot(u, v) + math.LN2;
if (Math.copySign(1., u) > 0.) {
y = Math.atan2(v, u);
} else {
// Adjust for sign, choosing the angle so that -pi/2 < y < pi/2
x = -x;
y = Math.atan2(v, -u);
}
} else {
// Normal case, without risk of overflow.
PyComplex a = sqrt(new PyComplex(1. + v, -u)); // a = sqrt(1-iw)
PyComplex b = sqrt(new PyComplex(1. - v, u)); // b = sqrt(1+iw)
// Combine the parts so as that terms in y cancel, leaving us with sinh x:
x = math.asinh(a.real * b.imag - a.imag * b.real);
// The arguments are v = cosh x sin y, and cosh x cos y
y = Math.atan2(v, a.real * b.real - a.imag * b.imag);
}
// Compose the result w according to whether we're computing asin(w) or asinh(w).
if (h) {
z = new PyComplex(x, y); // z = x + iy = asinh(u+iv).
} else {
z = new PyComplex(y, -x); // z = y - ix = -i asinh(v-iu) = asin(w)
}
// If that generated a nan, and there wasn't one in the argument, raise a domain error.
return exceptNaN(z, w);
}
/**
* Return the arc tangent of w. There are two branch cuts. One extends from 1j along the
* imaginary axis to ∞j, continuous from the right. The other extends from -1j along the
* imaginary axis to -∞j, continuous from the left.
*
* @param w
* @return tan-1w
*/
public static PyComplex atan(PyObject w) {
return atanOrAtanh(complexFromPyObject(w), false);
}
/**
* Return the hyperbolic arc tangent of w. There are two branch cuts. One extends from 1 along
* the real axis to ∞, continuous from below. The other extends from -1 along the real
* axis to -∞, continuous from above.
*
* @param w
* @return tanh-1w
*/
public static PyComplex atanh(PyObject w) {
return atanOrAtanh(complexFromPyObject(w), true);
}
/**
* Helper to compute either tan-1w or tanh-1w. The method
* used is close to that used in CPython. For z = tanh-1w:
*
* z = ½ln(1 + 2w/(1-w)) *
* Then, letting z = x+iy, and w = u+iv, *
* x = ¼ln(1 + 4u/((1-u)2+v2)) =
* -¼ln(1 - 4u/((1+u)2+v2))
* y = ½tan-1(2v / ((1+u)(1-u)-v2))
*
* We use {@link math#log1p(double)} and {@link Math#atan2(double, double)} to obtain x
* and y. The second expression for x is used when u<0. For
* w sufficiently large that w2≫1, tanh-1w
* ≈ 1/w ± iπ/2). For small w, tanh-1w ≈
* w. When computing tan-1w, we evaluate -i
* tanh-1iw instead.
*
* @param w
* @param h true to compute tanh-1w, false to
* compute tan-1w.
* @return tanh-1w or tan-1w
*/
private static PyComplex atanOrAtanh(PyComplex w, boolean h) {
double u, v, x, y;
PyComplex z;
if (h) {
// We compute z = atanh(w). Let z = x + iy and w = u + iv.
u = w.real;
v = w.imag;
// Then the function body computes x + iy = atanh(w).
} else {
// We compute w = atan(z). Unusually, let w = u - iv, so u + iv = iw.
v = w.real;
u = -w.imag;
// Then as before, the function body computes atanh(u+iv) = atanh(iw) = i atan(w),
// but we finally return z = y - ix = -i atanh(iw) = atan(w).
}
double absu = Math.abs(u), absv = Math.abs(v);
if (absu >= 0x1p511 || absv >= 0x1p511) {
// w is large: approximate atanh(w) by 1/w + i pi/2. 1/w = conjg(w)/|w|**2.
if (Double.isInfinite(absu) || Double.isInfinite(absv)) {
x = Math.copySign(0., u);
} else {
// w is also too big to square, carry a 2**-N scaling factor.
int N = 520;
double uu = Math.scalb(u, -N), vv = Math.scalb(v, -N);
double mod2w = uu * uu + vv * vv;
x = Math.scalb(uu / mod2w, -N);
}
// We don't need the imaginary part of 1/z. Just pi/2 with the sign of v. (If not nan.)
if (Double.isNaN(v)) {
y = v;
} else {
y = Math.copySign(Math.PI / 2., v);
}
} else if (absu < 0x1p-53) {
// u is small enough that u**2 may be neglected relative to 1.
if (absv > 0x1p-27) {
// v is not small, but is not near overflow either.
double v2 = v * v;
double d = 1. + v2;
x = Math.copySign(Math.log1p(4. * absu / d), u) * 0.25;
y = Math.atan2(2. * v, 1. - v2) * 0.5;
} else {
// v is also small enough that v**2 may be neglected (or is nan). So z = w.
x = u;
y = v;
}
} else if (absu == 1. && absv < 0x1p-27) {
// w is close to +1 or -1: needs a different expression, good as v->0
x = Math.copySign(Math.log(absv) - math.LN2, u) * 0.5;
if (v == 0.) {
y = Double.NaN;
} else {
y = Math.copySign(Math.atan2(2., absv), v) * 0.5;
}
} else {
/*
* Normal case, without risk of overflow. The basic expression is z =
* 0.5*ln((1+w)/(1-w)), which for positive u we rearrange as 0.5*ln(1+2w/(1-w)) and for
* negative u as -0.5*ln(1-2w/(1+w)). By use of absu, we reduce the difference between
* the expressions for u>=0 and u<0 to a sign transfer.
*/
double lmu = (1. - absu), lpu = (1. + absu), v2 = v * v;
double d = lmu * lmu + v2;
x = Math.copySign(Math.log1p(4. * absu / d), u) * 0.25;
y = Math.atan2(2. * v, lmu * lpu - v2) * 0.5;
}
// Compose the result w according to whether we're computing atan(w) or atanh(w).
if (h) {
z = new PyComplex(x, y); // z = x + iy = atanh(u+iv).
} else {
z = new PyComplex(y, -x); // z = y - ix = -i atanh(v-iu) = atan(w)
}
// If that generated a nan, and there wasn't one in the argument, raise a domain error.
return exceptNaN(z, w);
}
/**
* Return the cosine of z.
*
* @param z
* @return cos z
*/
public static PyComplex cos(PyObject z) {
return cosOrCosh(complexFromPyObject(z), false);
}
/**
* Return the hyperbolic cosine of z.
*
* @param z
* @return cosh z
*/
public static PyComplex cosh(PyObject z) {
return cosOrCosh(complexFromPyObject(z), true);
}
/**
* Helper to compute either cos z or cosh z.
*
* @param z
* @param h true for cosh, false for cos.
* @return cos z or cosh z
*/
private static PyComplex cosOrCosh(PyComplex z, boolean h) {
double x, y, u, v;
PyComplex w;
if (h) {
// We compute w = cosh(z). Let w = u + iv and z = x + iy.
x = z.real;
y = z.imag;
// Then the function body computes cosh(x+iy), according to:
// u = cosh(x) cos(y),
// v = sinh(x) sin(y),
// And we return w = u + iv.
} else {
// We compute w = sin(z). Unusually, let z = y - ix, so x + iy = iz.
y = z.real;
x = -z.imag;
// Then the function body computes cosh(x+iy) = cosh(iz) = cos(z) as before.
}
if (y == 0.) {
// Real argument for cosh (or imaginary for cos): use real library.
u = math.cosh(x); // This will raise a range error on overflow.
// v is zero but follows the sign of x*y (in which y could be -0.0).
v = Math.copySign(1., x) * y;
} else if (x == 0.) {
// Imaginary argument for cosh (or real for cos): imaginary result at this point.
u = Math.cos(y);
// v is zero but follows the sign of x*y (in which x could be -0.0).
v = x * Math.copySign(1., y);
} else {
// The trig calls will not throw, although if y is infinite, they return nan.
double cosy = Math.cos(y), siny = Math.sin(y), absx = Math.abs(x);
if (absx == Double.POSITIVE_INFINITY) {
if (!Double.isNaN(cosy)) {
// w = (inf,inf), but "rotated" by the direction cosines.
u = absx * cosy;
v = x * siny;
} else {
// Provisionally w = (inf,nan), which will raise domain error if y!=nan.
u = absx;
v = Double.NaN;
}
} else if (absx > ATLEAST_27LN2) {
// Use 0.5*e**x approximation. This is also the region where we risk overflow.
double r = Math.exp(absx - 2.);
// r approximates 2cosh(x)/e**2: multiply in this order to avoid inf:
u = r * cosy * HALF_E2;
// r approximates 2sinh(|x|)/e**2: put back the proper sign of x in passing.
v = Math.copySign(r, x) * siny * HALF_E2;
if (Double.isInfinite(u) || Double.isInfinite(v)) {
// A finite x gave rise to an infinite u or v.
throw math.mathRangeError();
}
} else {
// Normal case, without risk of overflow.
u = Math.cosh(x) * cosy;
v = Math.sinh(x) * siny;
}
}
// Compose the result w = u + iv.
w = new PyComplex(u, v);
// If that generated a nan, and there wasn't one in the argument, raise a domain error.
return exceptNaN(w, z);
}
/**
* Return the exponential value ez.
*
* @param z
* @return ez
*/
public static PyComplex exp(PyObject z) {
PyComplex zz = complexFromPyObject(z);
double x = zz.real, y = zz.imag, r, u, v;
/*
* This has a lot of corner-cases, and some of them make little sense sense, but it matches
* CPython and passes the regression tests.
*/
if (y == 0.) {
// Real value: use a real solution. (This may raise a range error.)
u = math.exp(x);
// v follows sign of y.
v = y;
} else {
// The trig calls will not throw, although if y is infinite, they return nan.
double cosy = Math.cos(y), siny = Math.sin(y);
if (x == Double.NEGATIVE_INFINITY) {
// w = (0,0) but "signed" by the direction cosines (even in they are nan).
u = Math.copySign(0., cosy);
v = Math.copySign(0., siny);
} else if (x == Double.POSITIVE_INFINITY) {
if (!Double.isNaN(cosy)) {
// w = (inf,inf), but "signed" by the direction cosines.
u = Math.copySign(x, cosy);
v = Math.copySign(x, siny);
} else {
// Provisionally w = (inf,nan), which will raise domain error if y!=nan.
u = x;
v = Double.NaN;
}
} else if (x > NEARLY_LN_DBL_MAX) {
// r = e**x would overflow but maybe not r*cos(y) and r*sin(y).
r = Math.exp(x - 1); // = r / e
u = r * cosy * Math.E;
v = r * siny * Math.E;
if (Double.isInfinite(u) || Double.isInfinite(v)) {
// A finite x gave rise to an infinite u or v.
throw math.mathRangeError();
}
} else {
// Normal case, without risk of overflow.
// Compute r = exp(x), and return w = u + iv = r (cos(y) + i*sin(y))
r = Math.exp(x);
u = r * cosy;
v = r * siny;
}
}
// If that generated a nan, and there wasn't one in the argument, raise domain error.
return exceptNaN(new PyComplex(u, v), zz);
}
public static double phase(PyObject in) {
PyComplex x = complexFromPyObject(in);
return Math.atan2(x.imag, x.real);
}
public static PyTuple polar(PyObject in) {
PyComplex z = complexFromPyObject(in);
double phi = Math.atan2(z.imag, z.real);
double r = math.hypot(z.real, z.imag);
return new PyTuple(new PyFloat(r), new PyFloat(phi));
}
/**
* Return the complex number x with polar coordinates r and phi. Equivalent to
* r * (math.cos(phi) + math.sin(phi)*1j).
*
* @param r radius
* @param phi angle
* @return reiφ
*/
public static PyComplex rect(double r, double phi) {
double x, y;
if (Double.isInfinite(r) && (Double.isInfinite(phi) || Double.isNaN(phi))) {
x = Double.POSITIVE_INFINITY;
y = Double.NaN;
} else if (phi == 0.0) {
// cos(phi)=1, sin(phi)=phi: finesse oddball r in computing y, but not x.
x = r;
if (Double.isNaN(r)) {
y = phi;
} else if (Double.isInfinite(r)) {
y = phi * Math.copySign(1., r);
} else {
y = phi * r;
}
} else if (r == 0.0 && (Double.isInfinite(phi) || Double.isNaN(phi))) {
// Ignore any problems (inf, nan) with phi
x = y = 0.;
} else {
// Text-book case, using the trig functions.
x = r * Math.cos(phi);
y = r * Math.sin(phi);
}
return exceptNaN(new PyComplex(x, y), r, phi);
}
/**
* @param in
*
* @return true if in.real or in.imag is positive or negative infinity
*/
public static boolean isinf(PyObject in) {
PyComplex x = complexFromPyObject(in);
return Double.isInfinite(x.real) || Double.isInfinite(x.imag);
}
/**
* @param in
*
* @return true if in.real or in.imag is nan.
*/
public static boolean isnan(PyObject in) {
PyComplex x = complexFromPyObject(in);
return Double.isNaN(x.real) || Double.isNaN(x.imag);
}
/**
* Returns the natural logarithm of w.
*
* @param w
* @return ln w
*/
public static PyComplex log(PyObject w) {
PyComplex ww = complexFromPyObject(w);
double u = ww.real, v = ww.imag;
// The real part of the result is the log of the magnitude.
double lnr = logHypot(u, v);
// The imaginary part of the result is the arg. This may result in a nan.
double theta = Math.atan2(v, u);
PyComplex z = new PyComplex(lnr, theta);
return exceptNaN(z, ww);
}
/**
* Returns the common logarithm of w (base 10 logarithm).
*
* @param w
* @return log10w
*/
public static PyComplex log10(PyObject w) {
PyComplex ww = complexFromPyObject(w);
double u = ww.real, v = ww.imag;
// The expression is the same as for base e, scaled in magnitude.
double logr = logHypot(u, v) * LOG10E;
double theta = Math.atan2(v, u) * LOG10E;
PyComplex z = new PyComplex(logr, theta);
return exceptNaN(z, ww);
}
/**
* Returns the logarithm of w to the given base. If the base is not specified, returns
* the natural logarithm of w. There is one branch cut, from 0 along the negative real
* axis to -∞, continuous from above.
*
* @param w
* @param b
* @return logbw
*/
public static PyComplex log(PyObject w, PyObject b) {
PyComplex ww = complexFromPyObject(w), bb = complexFromPyObject(b), z;
double u = ww.real, v = ww.imag, br = bb.real, bi = bb.imag, x, y;
// Natural log of w is (x,y)
x = logHypot(u, v);
y = Math.atan2(v, u);
if (bi != 0. || br <= 0.) {
// Complex or negative real base requires complex log: general case.
PyComplex lnb = log(bb);
z = (PyComplex)(new PyComplex(x, y)).__div__(lnb);
} else {
// Real positive base: frequent case. (b = inf or nan ends up here too.)
double lnb = Math.log(br);
z = new PyComplex(x / lnb, y / lnb);
}
return exceptNaN(z, ww);
}
/**
* Helper function for the log of a complex number, dealing with the log magnitude, and without
* intermediate overflow or underflow. It returns ln r, where r2 =
* u2+v2. To do this it computes
* ½ln(u2+v2). Special cases are handled as follows:
*
Math.log(Math.hypot(u,v)) because a valid
* result is still possible even when hypot(u,v) overflows, and because there's no
* point in taking a square root when a log is to follow.
*
* @param u
* @param v
* @return ½ln(u2+v2)
*/
private static double logHypot(double u, double v) {
if (Double.isInfinite(u) || Double.isInfinite(v)) {
return Double.POSITIVE_INFINITY;
} else {
// Cannot overflow, but if u=v=0 will return -inf.
int scale = 0, ue = Math.getExponent(u), ve = Math.getExponent(v);
double lnr;
if (ue < -511 && ve < -511) {
// Both u and v are too small to square, or zero. (Just one would be ok.)
scale = 600;
} else if (ue > 510 || ve > 510) {
// One of these is too big to square and double (or is nan or inf).
scale = -600;
}
if (scale == 0) {
// Normal case: there is no risk of overflow or log of zero.
lnr = 0.5 * Math.log(u * u + v * v);
} else {
// We must work with scaled values, us = u * 2**n etc..
double us = Math.scalb(u, scale);
double vs = Math.scalb(v, scale);
// rs**2 = r**2 * 2**2n
double rs2 = us * us + vs * vs;
// So ln(r) = ln(u**2+v**2)/2 = ln(us**2+vs**2)/2 - n ln(2)
lnr = 0.5 * Math.log(rs2) - scale * math.LN2;
}
// (u,v) = 0 leads to ln(r) = -inf, but that's a domain error
if (lnr == Double.NEGATIVE_INFINITY) {
throw math.mathDomainError();
} else {
return lnr;
}
}
}
/**
* Return the sine of z.
*
* @param z
* @return sin z
*/
public static PyComplex sin(PyObject z) {
return sinOrSinh(complexFromPyObject(z), false);
}
/**
* Return the hyperbolic sine of z.
*
* @param z
* @return sinh z
*/
public static PyComplex sinh(PyObject z) {
return sinOrSinh(complexFromPyObject(z), true);
}
/**
* Helper to compute either sin z or sinh z.
*
* @param z
* @param h true for sinh, false for sin.
* @return sinh z or sin z.
*/
private static PyComplex sinOrSinh(PyComplex z, boolean h) {
double x, y, u, v;
PyComplex w;
if (h) {
// We compute w = sinh(z). Let w = u + iv and z = x + iy.
x = z.real;
y = z.imag;
// Then the function body computes sinh(x+iy), according to:
// u = sinh(x) cos(y),
// v = cosh(x) sin(y),
// And we return w = u + iv.
} else {
// We compute w = sin(z). Unusually, let z = y - ix, so x + iy = iz.
y = z.real;
x = -z.imag;
// Then as before, the function body computes sinh(x+iy) = sinh(iz) = i sin(z),
// but we finally return w = v - iu = sin(z).
}
if (y == 0.) {
// Real argument for sinh (or imaginary for sin): use real library.
u = math.sinh(x); // This will raise a range error on overflow.
// v follows the sign of y (which could be -0.0).
v = y;
} else if (x == 0.) {
// Imaginary argument for sinh (or real for sin): imaginary result at this point.
v = Math.sin(y);
// u follows sign of x (which could be -0.0).
u = x;
} else {
// The trig calls will not throw, although if y is infinite, they return nan.
double cosy = Math.cos(y), siny = Math.sin(y), absx = Math.abs(x);
if (absx == Double.POSITIVE_INFINITY) {
if (!Double.isNaN(cosy)) {
// w = (inf,inf), but "rotated" by the direction cosines.
u = x * cosy;
v = absx * siny;
} else {
// Provisionally w = (inf,nan), which will raise domain error if y!=nan.
u = x;
v = Double.NaN;
}
} else if (absx > ATLEAST_27LN2) {
// Use 0.5*e**x approximation. This is also the region where we risk overflow.
double r = Math.exp(absx - 2.);
// r approximates 2cosh(x)/e**2: multiply in this order to avoid inf:
v = r * siny * HALF_E2;
// r approximates 2sinh(|x|)/e**2: put back the proper sign of x in passing.
u = Math.copySign(r, x) * cosy * HALF_E2;
if (Double.isInfinite(u) || Double.isInfinite(v)) {
// A finite x gave rise to an infinite u or v.
throw math.mathRangeError();
}
} else {
// Normal case, without risk of overflow.
u = Math.sinh(x) * cosy;
v = Math.cosh(x) * siny;
}
}
// Compose the result w according to whether we're computing sin(z) or sinh(z).
if (h) {
w = new PyComplex(u, v); // w = u + iv = sinh(x+iy).
} else {
w = new PyComplex(v, -u); // w = v - iu = sin(y-ix) = sin(z)
}
// If that generated a nan, and there wasn't one in the argument, raise a domain error.
return exceptNaN(w, z);
}
/**
* Calculate z = x+iy, such that z2 = w. In taking the square roots to
* get x and y, we choose to have x≥0 always, and y the same sign
* as v.
*
* @param w to square-root
* @return w½
*/
public static PyComplex sqrt(PyObject w) {
/*
* All the difficult parts are written for the first quadrant only (+,+), then the true sign
* of the parts of w are factored in at the end, by flipping the result around the
* diagonals.
*/
PyComplex ww = complexFromPyObject(w);
double u = Math.abs(ww.real), v = Math.abs(ww.imag), x, y;
if (Double.isInfinite(u)) {
// Special cases: u = inf
x = Double.POSITIVE_INFINITY;
y = (Double.isNaN(v) || Double.isInfinite(v)) ? v : 0.;
} else if (Double.isInfinite(v)) {
// Special cases: v = inf, u != inf
x = y = Double.POSITIVE_INFINITY;
} else if (Double.isNaN(u)) {
// In the remaining cases, u == nan infects all.
x = y = u;
} else {
if (v == 0.) {
// Pure real (and positive since in first quadrant).
x = (u == 0.) ? 0. : Math.sqrt(u);
y = 0.;
} else if (u == 0.) {
// Pure imaginary, and v is positive.
x = y = ROOT_HALF * Math.sqrt(v);
} else {
/*
* Let w = u + iv = 2a + 2ib, and define s**2 = a**2 + b**2. Then z = x + iy is
* computed as x**2 = s + a, and y = b /x. Most of the logic here is about managing
* the scaling.
*/
int ue = Math.getExponent(u), ve = Math.getExponent(v);
int diff = ue - ve;
if (diff > 27) {
// u is so much bigger than v we can ignore v in the square: s = u/2.
x = Math.sqrt(u);
} else if (diff < -27) {
// v is so much bigger than u we can ignore u in the square: s = v/2.
if (ve >= Double.MAX_EXPONENT) {
x = Math.sqrt(0.5 * u + 0.5 * v); // Avoid overflow in u+v
} else {
x = Math.sqrt(0.5 * (u + v));
}
} else {
/*
* Use the full-fat formula: s = Math.sqrt(a * a + b * b). During calculation,
* we will be squaring the components, so we scale by 2**n (small values up and
* large values down).
*/
double s, a, b;
final int LARGE = 510; // 1.999... * 2**LARGE is safe to square and double
final int SMALL = -510; // 1.0 * 2**(SMALL-1) may squared with full precision
final int SCALE = 600; // EVEN and > (52+SMALL-Double.MIN_EXPONENT)
int n = 0;
if (ue > LARGE || ve > LARGE) {
// One of these is too big to square without overflow.
a = Math.scalb(u, -(SCALE + 1)); // a = (u/2) * 2**n
b = Math.scalb(v, -(SCALE + 1));
n = -SCALE;
} else if (ue < SMALL && ve < SMALL) {
// Both of these are too small to square without loss of bits.
a = Math.scalb(u, SCALE - 1); // a = (u/2) * 2**n
b = Math.scalb(v, SCALE - 1);
n = SCALE;
} else {
a = 0.5 * u; // a = u/2
b = 0.5 * v;
}
s = Math.sqrt(a * a + b * b);
x = Math.sqrt(s + a);
// Restore x through the square root of the scale 2**(-n/2)
if (n != 0) {
x = Math.scalb(x, -n / 2);
}
}
// Finally, use y = v/2x
y = v / (x + x);
}
}
// Flip according to the signs of the components of w.
if (ww.real < 0.) {
return new PyComplex(y, Math.copySign(x, ww.imag));
} else {
return new PyComplex(x, Math.copySign(y, ww.imag));
}
}
/**
* Return the tangent of z.
*
* @param z
* @return tan z
*/
public static PyComplex tan(PyObject z) {
return tanOrTanh(complexFromPyObject(z), false);
}
/**
* Return the hyperbolic tangent of z.
*
* @param z
* @return tanh z
*/
public static PyComplex tanh(PyObject z) {
return tanOrTanh(complexFromPyObject(z), true);
}
/**
* Helper to compute either tan z or tanh z. The expression used is:
* * tanh(x+iy) = (sinh x cosh x + i sin y cos y) / * (sinh2x + cos2y) *
* A simplification is made for x sufficiently large that e2|x|≫1 that
* deals satisfactorily with large or infinite x. When computing tan, we evaluate
* i tan iz instead, and the approximation applies to
* e2|y|≫1.
*
* @param z
* @param h true to compute tanh z, false to compute tan
* z.
* @return tan or tanh z
*/
private static PyComplex tanOrTanh(PyComplex z, boolean h) {
double x, y, u, v, s;
PyComplex w;
if (h) {
// We compute w = tanh(z). Let w = u + iv and z = x + iy.
x = z.real;
y = z.imag;
// Then the function body computes tanh(x+iy), according to:
// s = sinh**2 x + cos**2 y
// u = sinh x cosh x / s,
// v = sin y cos y / s,
// And we return w = u + iv.
} else {
// We compute w = tan(z). Unusually, let z = y - ix, so x + iy = iz.
y = z.real;
x = -z.imag;
// Then the function body computes tanh(x+iy) = tanh(iz) = i tan(z) as before,
// but we finally return w = v - iu = tan(z).
}
if (y == 0.) {
// Real argument for tanh (or imaginary for tan).
u = Math.tanh(x);
// v is zero but follows the sign of y (which could be -0.0).
v = y;
} else if (x == 0. && !Double.isNaN(y)) {
// Imaginary argument for tanh (or real for tan): imaginary result at this point.
v = Math.tan(y); // May raise domain error
// u is zero but follows sign of x (which could be -0.0).
u = x;
} else {
// The trig calls will not throw, although if y is infinite, they return nan.
double cosy = Math.cos(y), siny = Math.sin(y), absx = Math.abs(x);
if (absx > ATLEAST_27LN2) {
// e**2x is much greater than 1: exponential approximation to sinh and cosh.
s = 0.25 * Math.exp(2 * absx);
u = Math.copySign(1., x);
if (s == Double.POSITIVE_INFINITY) {
// Either x is inf or 2x is large enough to overflow exp(). v=0, but signed:
v = Math.copySign(0., siny * cosy);
} else {
v = siny * cosy / s;
}
} else {
// Normal case: possible overflow in s near (x,y) = (0,pi/2) is harmless.
double sinhx = Math.sinh(x), coshx = Math.cosh(x);
s = sinhx * sinhx + cosy * cosy;
u = sinhx * coshx / s;
v = siny * cosy / s;
}
}
// Compose the result w according to whether we're computing tan(z) or tanh(z).
if (h) {
w = new PyComplex(u, v); // w = u + iv = tanh(x+iy).
} else {
w = new PyComplex(v, -u); // w = v - iu = tan(y-ix) = tan(z)
}
// If that generated a nan, and there wasn't one in the argument, raise a domain error.
return exceptNaN(w, z);
}
/**
* Turn a NaN result into a thrown ValueError, a math domain error, if
* the original argument was not itself NaN. A PyComplex is a
* NaN if either component is a NaN.
*
* @param result to return (if we return)
* @param arg to include in check
* @return result if arg was NaN or result was not
* NaN
* @throws PyException {@code ValueError} if result was NaN and
* arg was not NaN
*/
private static PyComplex exceptNaN(PyComplex result, PyComplex arg) throws PyException {
if ((Double.isNaN(result.real) || Double.isNaN(result.imag))
&& !(Double.isNaN(arg.real) || Double.isNaN(arg.imag))) {
throw math.mathDomainError();
} else {
return result;
}
}
/**
* Raise ValueError if result is a NaN, but neither
* a nor b is NaN. Same as
* {@link #exceptNaN(PyComplex, PyComplex)}.
*/
private static PyComplex exceptNaN(PyComplex result, double a, double b) throws PyException {
if ((Double.isNaN(result.real) || Double.isNaN(result.imag))
&& !(Double.isNaN(a) || Double.isNaN(b))) {
throw math.mathDomainError();
} else {
return result;
}
}
}