Average Error: 0.3 → 0.3
Time: 39.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\left(\log \left(\sqrt[3]{z}\right) - \mathsf{fma}\left(0.5 - a, \log t, t\right)\right) + \log \left(y + x\right)\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\left(\log \left(\sqrt[3]{z}\right) - \mathsf{fma}\left(0.5 - a, \log t, t\right)\right) + \log \left(y + x\right)\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)
double f(double x, double y, double z, double t, double a) {
        double r17572859 = x;
        double r17572860 = y;
        double r17572861 = r17572859 + r17572860;
        double r17572862 = log(r17572861);
        double r17572863 = z;
        double r17572864 = log(r17572863);
        double r17572865 = r17572862 + r17572864;
        double r17572866 = t;
        double r17572867 = r17572865 - r17572866;
        double r17572868 = a;
        double r17572869 = 0.5;
        double r17572870 = r17572868 - r17572869;
        double r17572871 = log(r17572866);
        double r17572872 = r17572870 * r17572871;
        double r17572873 = r17572867 + r17572872;
        return r17572873;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r17572874 = z;
        double r17572875 = cbrt(r17572874);
        double r17572876 = log(r17572875);
        double r17572877 = 0.5;
        double r17572878 = a;
        double r17572879 = r17572877 - r17572878;
        double r17572880 = t;
        double r17572881 = log(r17572880);
        double r17572882 = fma(r17572879, r17572881, r17572880);
        double r17572883 = r17572876 - r17572882;
        double r17572884 = y;
        double r17572885 = x;
        double r17572886 = r17572884 + r17572885;
        double r17572887 = log(r17572886);
        double r17572888 = r17572883 + r17572887;
        double r17572889 = r17572875 * r17572875;
        double r17572890 = log(r17572889);
        double r17572891 = r17572888 + r17572890;
        return r17572891;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.3
Target0.3
Herbie0.3
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

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

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate--l+0.3

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

  7. Applied associate-+l+0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"

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

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