static double calc_t_div(const SkDCubic& cubic, double precision, double start) {
const double adjust = sqrt(3.) / 36;
SkDCubic sub;
const SkDCubic* cPtr;
if (start == 0) {
cPtr = &cubic;
} else {
// OPTIMIZE: special-case half-split ?
sub = cubic.subDivide(start, 1);
cPtr = ⊂
}
const SkDCubic& c = *cPtr;
double dx = c[3].fX - 3 * (c[2].fX - c[1].fX) - c[0].fX;
double dy = c[3].fY - 3 * (c[2].fY - c[1].fY) - c[0].fY;
double dist = sqrt(dx * dx + dy * dy);
double tDiv3 = precision / (adjust * dist);
double t = SkDCubeRoot(tDiv3);
if (start > 0) {
t = start + (1 - start) * t;
}
return t;
}
int SkDCubic::RootsReal(double A, double B, double C, double D, double s[3]) {
#ifdef SK_DEBUG
// create a string mathematica understands
// GDB set print repe 15 # if repeated digits is a bother
// set print elements 400 # if line doesn't fit
char str[1024];
sk_bzero(str, sizeof(str));
SK_SNPRINTF(str, sizeof(str), "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]",
A, B, C, D);
SkPathOpsDebug::MathematicaIze(str, sizeof(str));
#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA
SkDebugf("%s\n", str);
#endif
#endif
if (approximately_zero(A)
&& approximately_zero_when_compared_to(A, B)
&& approximately_zero_when_compared_to(A, C)
&& approximately_zero_when_compared_to(A, D)) { // we're just a quadratic
return SkDQuad::RootsReal(B, C, D, s);
}
if (approximately_zero_when_compared_to(D, A)
&& approximately_zero_when_compared_to(D, B)
&& approximately_zero_when_compared_to(D, C)) { // 0 is one root
int num = SkDQuad::RootsReal(A, B, C, s);
for (int i = 0; i < num; ++i) {
if (approximately_zero(s[i])) {
return num;
}
}
s[num++] = 0;
return num;
}
if (approximately_zero(A + B + C + D)) { // 1 is one root
int num = SkDQuad::RootsReal(A, A + B, -D, s);
for (int i = 0; i < num; ++i) {
if (AlmostDequalUlps(s[i], 1)) {
return num;
}
}
s[num++] = 1;
return num;
}
double a, b, c;
{
double invA = 1 / A;
a = B * invA;
b = C * invA;
c = D * invA;
}
double a2 = a * a;
double Q = (a2 - b * 3) / 9;
double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54;
double R2 = R * R;
double Q3 = Q * Q * Q;
double R2MinusQ3 = R2 - Q3;
double adiv3 = a / 3;
double r;
double* roots = s;
if (R2MinusQ3 < 0) { // we have 3 real roots
double theta = acos(R / sqrt(Q3));
double neg2RootQ = -2 * sqrt(Q);
r = neg2RootQ * cos(theta / 3) - adiv3;
*roots++ = r;
r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3;
if (!AlmostDequalUlps(s[0], r)) {
*roots++ = r;
}
r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3;
if (!AlmostDequalUlps(s[0], r) && (roots - s == 1 || !AlmostDequalUlps(s[1], r))) {
*roots++ = r;
}
} else { // we have 1 real root
double sqrtR2MinusQ3 = sqrt(R2MinusQ3);
double A = fabs(R) + sqrtR2MinusQ3;
A = SkDCubeRoot(A);
if (R > 0) {
A = -A;
}
if (A != 0) {
A += Q / A;
}
r = A - adiv3;
*roots++ = r;
if (AlmostDequalUlps(R2, Q3)) {
r = -A / 2 - adiv3;
if (!AlmostDequalUlps(s[0], r)) {
*roots++ = r;
}
}
}
return static_cast<int>(roots - s);
}
