Average Error: 0.3 → 0.3
Time: 35.8s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt[3]{y + x}\right) + \left(\log z - t\right)\right) + \log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right)\right)\]
\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(\log \left(\sqrt[3]{y + x}\right) + \left(\log z - t\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 r3349831 = x;
        double r3349832 = y;
        double r3349833 = r3349831 + r3349832;
        double r3349834 = log(r3349833);
        double r3349835 = z;
        double r3349836 = log(r3349835);
        double r3349837 = r3349834 + r3349836;
        double r3349838 = t;
        double r3349839 = r3349837 - r3349838;
        double r3349840 = a;
        double r3349841 = 0.5;
        double r3349842 = r3349840 - r3349841;
        double r3349843 = log(r3349838);
        double r3349844 = r3349842 * r3349843;
        double r3349845 = r3349839 + r3349844;
        return r3349845;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r3349846 = t;
        double r3349847 = log(r3349846);
        double r3349848 = a;
        double r3349849 = 0.5;
        double r3349850 = r3349848 - r3349849;
        double r3349851 = y;
        double r3349852 = x;
        double r3349853 = r3349851 + r3349852;
        double r3349854 = cbrt(r3349853);
        double r3349855 = log(r3349854);
        double r3349856 = z;
        double r3349857 = log(r3349856);
        double r3349858 = r3349857 - r3349846;
        double r3349859 = r3349855 + r3349858;
        double r3349860 = r3349854 * r3349854;
        double r3349861 = log(r3349860);
        double r3349862 = r3349859 + r3349861;
        double r3349863 = fma(r3349847, r3349850, r3349862);
        return r3349863;
}

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

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

  4. Applied add-cube-cbrt0.3

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 +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))))