Optimizations for pow() with const non-integer exponent?

Question

I have hot spots in my code where I\'m doing pow() taking up around 10-20% of my execution time.

My input to pow(x,y) is very specific, so

Answer 1

I shall answer the question you really wanted to ask, which is how to do fast sRGB <-> linear RGB conversion. To do this precisely and efficiently we can use polynomial approximations. The following polynomial approximations have been generated with sollya, and have a worst case relative error of 0.0144%.

inline double poly7(double x, double a, double b, double c, double d,
                              double e, double f, double g, double h) {
    double ab, cd, ef, gh, abcd, efgh, x2, x4;
    x2 = x*x; x4 = x2*x2;
    ab = a*x + b; cd = c*x + d;
    ef = e*x + f; gh = g*x + h;
    abcd = ab*x2 + cd; efgh = ef*x2 + gh;
    return abcd*x4 + efgh;
}

inline double srgb_to_linear(double x) {
    if (x <= 0.04045) return x / 12.92;

    // Polynomial approximation of ((x+0.055)/1.055)^2.4.
    return poly7(x, 0.15237971711927983387,
                   -0.57235993072870072762,
                    0.92097986411523535821,
                   -0.90208229831912012386,
                    0.88348956209696805075,
                    0.48110797889132134175,
                    0.03563925285274562038,
                    0.00084585397227064120);
}

inline double linear_to_srgb(double x) {
    if (x <= 0.0031308) return x * 12.92;

    // Piecewise polynomial approximation (divided by x^3)
    // of 1.055 * x^(1/2.4) - 0.055.
    if (x <= 0.0523) return poly7(x, -6681.49576364495442248881,
                                      1224.97114922729451791383,
                                      -100.23413743425112443219,
                                         6.60361150127077944916,
                                         0.06114808961060447245,
                                        -0.00022244138470139442,
                                         0.00000041231840827815,
                                        -0.00000000035133685895) / (x*x*x);

    return poly7(x, -0.18730034115395793881,
                     0.64677431008037400417,
                    -0.99032868647877825286,
                     1.20939072663263713636,
                     0.33433459165487383613,
                    -0.01345095746411287783,
                     0.00044351684288719036,
                    -0.00000664263587520855) / (x*x*x);
}

And the sollya input used to generate the polynomials:

suppressmessage(174);
f = ((x+0.055)/1.055)^2.4;
p0 = fpminimax(f, 7, [|D...|], [0.04045;1], relative);
p = fpminimax(f/(p0(1)+1e-18), 7, [|D...|], [0.04045;1], relative);
print("relative:", dirtyinfnorm((f-p)/f, [s;1]));
print("absolute:", dirtyinfnorm((f-p), [s;1]));
print(canonical(p));

s = 0.0523;
z = 3;
f = 1.055 * x^(1/2.4) - 0.055;

p = fpminimax(1.055 * (x^(z+1/2.4) - 0.055*x^z/1.055), 7, [|D...|], [0.0031308;s], relative)/x^z;
print("relative:", dirtyinfnorm((f-p)/f, [0.0031308;s]));
print("absolute:", dirtyinfnorm((f-p), [0.0031308;s]));
print(canonical(p));

p = fpminimax(1.055 * (x^(z+1/2.4) - 0.055*x^z/1.055), 7, [|D...|], [s;1], relative)/x^z;
print("relative:", dirtyinfnorm((f-p)/f, [s;1]));
print("absolute:", dirtyinfnorm((f-p), [s;1]));
print(canonical(p));