Average Error: 0.3 → 0.3
Time: 10.8s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]
\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(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r466834 = x;
        double r466835 = y;
        double r466836 = r466834 + r466835;
        double r466837 = log(r466836);
        double r466838 = z;
        double r466839 = log(r466838);
        double r466840 = r466837 + r466839;
        double r466841 = t;
        double r466842 = r466840 - r466841;
        double r466843 = a;
        double r466844 = 0.5;
        double r466845 = r466843 - r466844;
        double r466846 = log(r466841);
        double r466847 = r466845 * r466846;
        double r466848 = r466842 + r466847;
        return r466848;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r466849 = t;
        double r466850 = log(r466849);
        double r466851 = a;
        double r466852 = 0.5;
        double r466853 = r466851 - r466852;
        double r466854 = 2.0;
        double r466855 = z;
        double r466856 = cbrt(r466855);
        double r466857 = log(r466856);
        double r466858 = x;
        double r466859 = y;
        double r466860 = r466858 + r466859;
        double r466861 = log(r466860);
        double r466862 = fma(r466854, r466857, r466861);
        double r466863 = r466862 + r466857;
        double r466864 = r466863 - r466849;
        double r466865 = fma(r466850, r466853, r466864);
        return r466865;
}

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

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]

  8. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]

Reproduce

herbie shell --seed 2020057 +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))))