Average Error: 0.3 → 0.3
Time: 35.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt[3]{z}\right) - t\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(y + x\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt[3]{z}\right) - t\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(y + x\right)
double f(double x, double y, double z, double t, double a) {
        double r978594 = x;
        double r978595 = y;
        double r978596 = r978594 + r978595;
        double r978597 = log(r978596);
        double r978598 = z;
        double r978599 = log(r978598);
        double r978600 = r978597 + r978599;
        double r978601 = t;
        double r978602 = r978600 - r978601;
        double r978603 = a;
        double r978604 = 0.5;
        double r978605 = r978603 - r978604;
        double r978606 = log(r978601);
        double r978607 = r978605 * r978606;
        double r978608 = r978602 + r978607;
        return r978608;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r978609 = t;
        double r978610 = log(r978609);
        double r978611 = a;
        double r978612 = 0.5;
        double r978613 = r978611 - r978612;
        double r978614 = z;
        double r978615 = cbrt(r978614);
        double r978616 = log(r978615);
        double r978617 = r978616 - r978609;
        double r978618 = r978615 * r978615;
        double r978619 = log(r978618);
        double r978620 = r978617 + r978619;
        double r978621 = fma(r978610, r978613, r978620);
        double r978622 = y;
        double r978623 = x;
        double r978624 = r978622 + r978623;
        double r978625 = log(r978624);
        double r978626 = r978621 + r978625;
        return r978626;
}

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. Using strategy rm

  3. Applied associate--l+0.3

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

  4. Applied associate-+l+0.3

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

  5. Simplified0.3

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

  7. Applied add-cube-cbrt0.3

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

  8. Applied log-prod0.3

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

  9. Applied associate--l+0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019153 +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))))