Average Error: 0.3 → 0.3
Time: 40.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log \left(\left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}\right) \cdot \left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}\right)\right), \left(\log z - t\right) + \log \left(x + y\right)\right) + \left(a - 0.5\right) \cdot \log \left({\left({t}^{\left(\sqrt{\frac{1}{3}}\right)}\right)}^{\left(\sqrt{\frac{1}{3}}\right)}\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(a - 0.5, \log \left(\left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}\right) \cdot \left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}\right)\right), \left(\log z - t\right) + \log \left(x + y\right)\right) + \left(a - 0.5\right) \cdot \log \left({\left({t}^{\left(\sqrt{\frac{1}{3}}\right)}\right)}^{\left(\sqrt{\frac{1}{3}}\right)}\right)
double f(double x, double y, double z, double t, double a) {
        double r1999861 = x;
        double r1999862 = y;
        double r1999863 = r1999861 + r1999862;
        double r1999864 = log(r1999863);
        double r1999865 = z;
        double r1999866 = log(r1999865);
        double r1999867 = r1999864 + r1999866;
        double r1999868 = t;
        double r1999869 = r1999867 - r1999868;
        double r1999870 = a;
        double r1999871 = 0.5;
        double r1999872 = r1999870 - r1999871;
        double r1999873 = log(r1999868);
        double r1999874 = r1999872 * r1999873;
        double r1999875 = r1999869 + r1999874;
        return r1999875;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r1999876 = a;
        double r1999877 = 0.5;
        double r1999878 = r1999876 - r1999877;
        double r1999879 = t;
        double r1999880 = sqrt(r1999879);
        double r1999881 = cbrt(r1999880);
        double r1999882 = r1999881 * r1999881;
        double r1999883 = r1999882 * r1999882;
        double r1999884 = log(r1999883);
        double r1999885 = z;
        double r1999886 = log(r1999885);
        double r1999887 = r1999886 - r1999879;
        double r1999888 = x;
        double r1999889 = y;
        double r1999890 = r1999888 + r1999889;
        double r1999891 = log(r1999890);
        double r1999892 = r1999887 + r1999891;
        double r1999893 = fma(r1999878, r1999884, r1999892);
        double r1999894 = 0.3333333333333333;
        double r1999895 = sqrt(r1999894);
        double r1999896 = pow(r1999879, r1999895);
        double r1999897 = pow(r1999896, r1999895);
        double r1999898 = log(r1999897);
        double r1999899 = r1999878 * r1999898;
        double r1999900 = r1999893 + r1999899;
        return r1999900;
}

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. Using strategy rm

  3. Applied add-cube-cbrt0.3

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

  4. Applied log-prod0.3

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

  5. Applied distribute-lft-in0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

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

  9. Applied pow1/30.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), \log \left(y + x\right) + \left(\log z - t\right)\right) + \left(a - 0.5\right) \cdot \log \color{blue}{\left({t}^{\frac{1}{3}}\right)}\]
  10. Using strategy rm

  11. Applied add-sqr-sqrt0.3

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

  12. Applied cbrt-prod0.3

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

  13. Applied add-sqr-sqrt0.3

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

  14. Applied cbrt-prod0.3

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

  15. Applied swap-sqr0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log \color{blue}{\left(\left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}\right) \cdot \left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}\right)\right)}, \log \left(y + x\right) + \left(\log z - t\right)\right) + \left(a - 0.5\right) \cdot \log \left({t}^{\frac{1}{3}}\right)\]
  16. Using strategy rm

  17. Applied add-sqr-sqrt0.3

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

  18. Applied pow-unpow0.3

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

  19. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +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))))