Average Error: 0.3 → 0.3
Time: 43.6s
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 r2519542 = x;
        double r2519543 = y;
        double r2519544 = r2519542 + r2519543;
        double r2519545 = log(r2519544);
        double r2519546 = z;
        double r2519547 = log(r2519546);
        double r2519548 = r2519545 + r2519547;
        double r2519549 = t;
        double r2519550 = r2519548 - r2519549;
        double r2519551 = a;
        double r2519552 = 0.5;
        double r2519553 = r2519551 - r2519552;
        double r2519554 = log(r2519549);
        double r2519555 = r2519553 * r2519554;
        double r2519556 = r2519550 + r2519555;
        return r2519556;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r2519557 = t;
        double r2519558 = log(r2519557);
        double r2519559 = a;
        double r2519560 = 0.5;
        double r2519561 = r2519559 - r2519560;
        double r2519562 = y;
        double r2519563 = x;
        double r2519564 = r2519562 + r2519563;
        double r2519565 = cbrt(r2519564);
        double r2519566 = log(r2519565);
        double r2519567 = z;
        double r2519568 = log(r2519567);
        double r2519569 = r2519557 - r2519568;
        double r2519570 = r2519566 - r2519569;
        double r2519571 = r2519565 * r2519565;
        double r2519572 = log(r2519571);
        double r2519573 = r2519570 + r2519572;
        double r2519574 = fma(r2519558, r2519561, r2519573);
        return r2519574;
}

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 0.3

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

  2. Simplified0.3

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

  4. Applied add-cube-cbrt0.3

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

  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 2019133 +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))))