Average Error: 0.2 → 0.3
Time: 48.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\left(\log t\right), \left(a - 0.5\right), \left(\left(\log \left(\sqrt[3]{y + x}\right) - \left(t - \log z\right)\right) + \log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right)\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\left(\log t\right), \left(a - 0.5\right), \left(\left(\log \left(\sqrt[3]{y + x}\right) - \left(t - \log z\right)\right) + \log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right)\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r2363507 = x;
        double r2363508 = y;
        double r2363509 = r2363507 + r2363508;
        double r2363510 = log(r2363509);
        double r2363511 = z;
        double r2363512 = log(r2363511);
        double r2363513 = r2363510 + r2363512;
        double r2363514 = t;
        double r2363515 = r2363513 - r2363514;
        double r2363516 = a;
        double r2363517 = 0.5;
        double r2363518 = r2363516 - r2363517;
        double r2363519 = log(r2363514);
        double r2363520 = r2363518 * r2363519;
        double r2363521 = r2363515 + r2363520;
        return r2363521;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r2363522 = t;
        double r2363523 = log(r2363522);
        double r2363524 = a;
        double r2363525 = 0.5;
        double r2363526 = r2363524 - r2363525;
        double r2363527 = y;
        double r2363528 = x;
        double r2363529 = r2363527 + r2363528;
        double r2363530 = cbrt(r2363529);
        double r2363531 = log(r2363530);
        double r2363532 = z;
        double r2363533 = log(r2363532);
        double r2363534 = r2363522 - r2363533;
        double r2363535 = r2363531 - r2363534;
        double r2363536 = r2363530 * r2363530;
        double r2363537 = log(r2363536);
        double r2363538 = r2363535 + r2363537;
        double r2363539 = fma(r2363523, r2363526, r2363538);
        return r2363539;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Initial program 0.2

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\log t\right), \left(a - 0.5\right), \left(\log \left(y + x\right) - \left(t - \log z\right)\right)\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.2

    \[\leadsto \mathsf{fma}\left(\left(\log t\right), \left(a - 0.5\right), \left(\log \color{blue}{\left(\left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right) \cdot \sqrt[3]{y + x}\right)} - \left(t - \log z\right)\right)\right)\]

  5. Applied log-prod0.3

    \[\leadsto \mathsf{fma}\left(\left(\log t\right), \left(a - 0.5\right), \left(\color{blue}{\left(\log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right) + \log \left(\sqrt[3]{y + x}\right)\right)} - \left(t - \log z\right)\right)\right)\]

  6. Applied associate--l+0.3

    \[\leadsto \mathsf{fma}\left(\left(\log t\right), \left(a - 0.5\right), \color{blue}{\left(\log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right) + \left(\log \left(\sqrt[3]{y + x}\right) - \left(t - \log z\right)\right)\right)}\right)\]

  7. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\left(\log t\right), \left(a - 0.5\right), \left(\left(\log \left(\sqrt[3]{y + x}\right) - \left(t - \log z\right)\right) + \log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))