Average Error: 0.1 → 0.1
Time: 6.8s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), {\left(x \cdot \log \left({y}^{\frac{1}{3}}\right)\right)}^{1}\right) - y\right) - z\right) + \log t\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), {\left(x \cdot \log \left({y}^{\frac{1}{3}}\right)\right)}^{1}\right) - y\right) - z\right) + \log t
double f(double x, double y, double z, double t) {
        double r148853 = x;
        double r148854 = y;
        double r148855 = log(r148854);
        double r148856 = r148853 * r148855;
        double r148857 = r148856 - r148854;
        double r148858 = z;
        double r148859 = r148857 - r148858;
        double r148860 = t;
        double r148861 = log(r148860);
        double r148862 = r148859 + r148861;
        return r148862;
}
↓
double f(double x, double y, double z, double t) {
        double r148863 = x;
        double r148864 = 2.0;
        double r148865 = y;
        double r148866 = cbrt(r148865);
        double r148867 = log(r148866);
        double r148868 = r148864 * r148867;
        double r148869 = 0.3333333333333333;
        double r148870 = pow(r148865, r148869);
        double r148871 = log(r148870);
        double r148872 = r148863 * r148871;
        double r148873 = 1.0;
        double r148874 = pow(r148872, r148873);
        double r148875 = fma(r148863, r148868, r148874);
        double r148876 = r148875 - r148865;
        double r148877 = z;
        double r148878 = r148876 - r148877;
        double r148879 = t;
        double r148880 = log(r148879);
        double r148881 = r148878 + r148880;
        return r148881;
}

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 fma-def0.1

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

  10. Applied pow10.1

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

  11. Applied pow10.1

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

  12. Applied pow-prod-down0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020083 +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)))