Average Error: 0.3 → 0.3
Time: 37.7s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\left(\log z - t\right) + \log \left(\sqrt[3]{y + x}\right)\right) + \log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log t, a - 0.5, \left(\left(\log z - t\right) + \log \left(\sqrt[3]{y + x}\right)\right) + \log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r90664 = x;
        double r90665 = y;
        double r90666 = r90664 + r90665;
        double r90667 = log(r90666);
        double r90668 = z;
        double r90669 = log(r90668);
        double r90670 = r90667 + r90669;
        double r90671 = t;
        double r90672 = r90670 - r90671;
        double r90673 = a;
        double r90674 = 0.5;
        double r90675 = r90673 - r90674;
        double r90676 = log(r90671);
        double r90677 = r90675 * r90676;
        double r90678 = r90672 + r90677;
        return r90678;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r90679 = t;
        double r90680 = log(r90679);
        double r90681 = a;
        double r90682 = 0.5;
        double r90683 = r90681 - r90682;
        double r90684 = z;
        double r90685 = log(r90684);
        double r90686 = r90685 - r90679;
        double r90687 = y;
        double r90688 = x;
        double r90689 = r90687 + r90688;
        double r90690 = cbrt(r90689);
        double r90691 = log(r90690);
        double r90692 = r90686 + r90691;
        double r90693 = r90690 * r90690;
        double r90694 = log(r90693);
        double r90695 = r90692 + r90694;
        double r90696 = fma(r90680, r90683, r90695);
        return r90696;
}

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. Simplified0.2

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

  4. Applied add-cube-cbrt0.2

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

  5. Applied log-prod0.3

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

  6. Applied associate-+l+0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 +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))))