Average Error: 0.1 → 0.1
Time: 6.4s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\left(\left(-1 \cdot \left(\log \left(\frac{1}{y}\right) \cdot x\right) - y\right) - z\right) + \log t\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\left(\left(-1 \cdot \left(\log \left(\frac{1}{y}\right) \cdot x\right) - y\right) - z\right) + \log t
double f(double x, double y, double z, double t) {
        double r119884 = x;
        double r119885 = y;
        double r119886 = log(r119885);
        double r119887 = r119884 * r119886;
        double r119888 = r119887 - r119885;
        double r119889 = z;
        double r119890 = r119888 - r119889;
        double r119891 = t;
        double r119892 = log(r119891);
        double r119893 = r119890 + r119892;
        return r119893;
}
↓
double f(double x, double y, double z, double t) {
        double r119894 = -1.0;
        double r119895 = 1.0;
        double r119896 = y;
        double r119897 = r119895 / r119896;
        double r119898 = log(r119897);
        double r119899 = x;
        double r119900 = r119898 * r119899;
        double r119901 = r119894 * r119900;
        double r119902 = r119901 - r119896;
        double r119903 = z;
        double r119904 = r119902 - r119903;
        double r119905 = t;
        double r119906 = log(r119905);
        double r119907 = r119904 + r119906;
        return r119907;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around inf 0.2

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

  10. Simplified0.1

    \[\leadsto \left(\left(\color{blue}{-1 \cdot \left(\log \left(\frac{1}{y}\right) \cdot x\right)} - y\right) - z\right) + \log t\]

  11. Final simplification0.1

    \[\leadsto \left(\left(-1 \cdot \left(\log \left(\frac{1}{y}\right) \cdot x\right) - y\right) - z\right) + \log t\]

Reproduce

herbie shell --seed 2020046 +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)))