Average Error: 0.1 → 0.1
Time: 7.4s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\left(\left(\left(x \cdot \mathsf{fma}\left(\log \left({y}^{\frac{1}{3}}\right), 2, \log \left(\sqrt{\sqrt[3]{y}}\right)\right) + x \cdot \log \left(\sqrt{\sqrt[3]{y}}\right)\right) - y\right) - z\right) + \log t\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\left(\left(\left(x \cdot \mathsf{fma}\left(\log \left({y}^{\frac{1}{3}}\right), 2, \log \left(\sqrt{\sqrt[3]{y}}\right)\right) + x \cdot \log \left(\sqrt{\sqrt[3]{y}}\right)\right) - y\right) - z\right) + \log t
double f(double x, double y, double z, double t) {
        double r115783 = x;
        double r115784 = y;
        double r115785 = log(r115784);
        double r115786 = r115783 * r115785;
        double r115787 = r115786 - r115784;
        double r115788 = z;
        double r115789 = r115787 - r115788;
        double r115790 = t;
        double r115791 = log(r115790);
        double r115792 = r115789 + r115791;
        return r115792;
}
↓
double f(double x, double y, double z, double t) {
        double r115793 = x;
        double r115794 = y;
        double r115795 = 0.3333333333333333;
        double r115796 = pow(r115794, r115795);
        double r115797 = log(r115796);
        double r115798 = 2.0;
        double r115799 = cbrt(r115794);
        double r115800 = sqrt(r115799);
        double r115801 = log(r115800);
        double r115802 = fma(r115797, r115798, r115801);
        double r115803 = r115793 * r115802;
        double r115804 = r115793 * r115801;
        double r115805 = r115803 + r115804;
        double r115806 = r115805 - r115794;
        double r115807 = z;
        double r115808 = r115806 - r115807;
        double r115809 = t;
        double r115810 = log(r115809);
        double r115811 = r115808 + r115810;
        return r115811;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.1

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

  4. Applied log-prod0.1

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

  5. Applied distribute-lft-in0.1

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

  6. Simplified0.1

    \[\leadsto \left(\left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) - y\right) - z\right) + \log t\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \color{blue}{\left(\sqrt{\sqrt[3]{y}} \cdot \sqrt{\sqrt[3]{y}}\right)}\right) - y\right) - z\right) + \log t\]

  9. Applied log-prod0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \color{blue}{\left(\log \left(\sqrt{\sqrt[3]{y}}\right) + \log \left(\sqrt{\sqrt[3]{y}}\right)\right)}\right) - y\right) - z\right) + \log t\]

  10. Applied distribute-lft-in0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\left(x \cdot \log \left(\sqrt{\sqrt[3]{y}}\right) + x \cdot \log \left(\sqrt{\sqrt[3]{y}}\right)\right)}\right) - y\right) - z\right) + \log t\]

  11. Applied associate-+r+0.1

    \[\leadsto \left(\left(\color{blue}{\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \left(\sqrt{\sqrt[3]{y}}\right)\right) + x \cdot \log \left(\sqrt{\sqrt[3]{y}}\right)\right)} - y\right) - z\right) + \log t\]

  12. Simplified0.1

    \[\leadsto \left(\left(\left(\color{blue}{x \cdot \mathsf{fma}\left(\log \left({y}^{\frac{1}{3}}\right), 2, \log \left(\sqrt{\sqrt[3]{y}}\right)\right)} + x \cdot \log \left(\sqrt{\sqrt[3]{y}}\right)\right) - y\right) - z\right) + \log t\]

  13. Final simplification0.1

    \[\leadsto \left(\left(\left(x \cdot \mathsf{fma}\left(\log \left({y}^{\frac{1}{3}}\right), 2, \log \left(\sqrt{\sqrt[3]{y}}\right)\right) + x \cdot \log \left(\sqrt{\sqrt[3]{y}}\right)\right) - y\right) - z\right) + \log t\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))