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

↓

\[\mathsf{fma}\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right)\]

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

↓

\mathsf{fma}\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right)

double f(double x, double y, double z, double t, double a) {
        double r354843 = x;
        double r354844 = y;
        double r354845 = r354843 + r354844;
        double r354846 = log(r354845);
        double r354847 = z;
        double r354848 = log(r354847);
        double r354849 = r354846 + r354848;
        double r354850 = t;
        double r354851 = r354849 - r354850;
        double r354852 = a;
        double r354853 = 0.5;
        double r354854 = r354852 - r354853;
        double r354855 = log(r354850);
        double r354856 = r354854 * r354855;
        double r354857 = r354851 + r354856;
        return r354857;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r354858 = t;
        double r354859 = cbrt(r354858);
        double r354860 = r354859 * r354859;
        double r354861 = log(r354860);
        double r354862 = a;
        double r354863 = 0.5;
        double r354864 = r354862 - r354863;
        double r354865 = x;
        double r354866 = y;
        double r354867 = r354865 + r354866;
        double r354868 = log(r354867);
        double r354869 = z;
        double r354870 = log(r354869);
        double r354871 = r354868 + r354870;
        double r354872 = r354871 - r354858;
        double r354873 = fma(r354861, r354864, r354872);
        double r354874 = 1.0;
        double r354875 = r354874 / r354858;
        double r354876 = -0.3333333333333333;
        double r354877 = pow(r354875, r354876);
        double r354878 = log(r354877);
        double r354879 = r354864 * r354878;
        double r354880 = r354873 + r354879;
        return r354880;
}

Original	0.2
Target	0.2
Herbie	0.3

Derivation

Initial program 0.2
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
Using strategy rm
Applied add-cube-cbrt0.3
\[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}\]
Applied log-prod0.3
\[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)}\]
Applied distribute-lft-in0.3
\[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{\left(\left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + \left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t}\right)\right)}\]
Applied associate-+r+0.3
\[\leadsto \color{blue}{\left(\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t}\right)}\]
Simplified0.3
\[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right)} + \left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t}\right)\]
Taylor expanded around inf 0.3
\[\leadsto \mathsf{fma}\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \color{blue}{\left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right)}\]
Final simplification0.3
\[\leadsto \mathsf{fma}\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right)\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))

Error

Target

Derivation

Reproduce