Average Error: 0.2 → 0.3
Time: 17.1s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{\frac{1}{x}}{\left(\sqrt[3]{\sqrt[3]{9}} \cdot \sqrt[3]{\sqrt[3]{9}}\right) \cdot {\left(\sqrt[3]{9}\right)}^{2}}}{\sqrt[3]{\sqrt[3]{9}}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{\frac{1}{x}}{\left(\sqrt[3]{\sqrt[3]{9}} \cdot \sqrt[3]{\sqrt[3]{9}}\right) \cdot {\left(\sqrt[3]{9}\right)}^{2}}}{\sqrt[3]{\sqrt[3]{9}}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}
double f(double x, double y) {
        double r400840 = 1.0;
        double r400841 = x;
        double r400842 = 9.0;
        double r400843 = r400841 * r400842;
        double r400844 = r400840 / r400843;
        double r400845 = r400840 - r400844;
        double r400846 = y;
        double r400847 = 3.0;
        double r400848 = sqrt(r400841);
        double r400849 = r400847 * r400848;
        double r400850 = r400846 / r400849;
        double r400851 = r400845 - r400850;
        return r400851;
}
↓
double f(double x, double y) {
        double r400852 = 1.0;
        double r400853 = x;
        double r400854 = r400852 / r400853;
        double r400855 = 9.0;
        double r400856 = cbrt(r400855);
        double r400857 = cbrt(r400856);
        double r400858 = r400857 * r400857;
        double r400859 = 2.0;
        double r400860 = pow(r400856, r400859);
        double r400861 = r400858 * r400860;
        double r400862 = r400854 / r400861;
        double r400863 = r400862 / r400857;
        double r400864 = r400852 - r400863;
        double r400865 = y;
        double r400866 = 3.0;
        double r400867 = r400865 / r400866;
        double r400868 = sqrt(r400853);
        double r400869 = r400867 / r400868;
        double r400870 = r400864 - r400869;
        return r400870;
}

Error

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Results

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Target

Original0.2
Target0.2
Herbie0.3
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

    \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}}\]
  4. Using strategy rm

  5. Applied associate-/r*0.2

    \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{x}}{9}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt0.2

    \[\leadsto \left(1 - \frac{\frac{1}{x}}{\color{blue}{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \sqrt[3]{9}}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]

  8. Applied associate-/r*0.3

    \[\leadsto \left(1 - \color{blue}{\frac{\frac{\frac{1}{x}}{\sqrt[3]{9} \cdot \sqrt[3]{9}}}{\sqrt[3]{9}}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.3

    \[\leadsto \left(1 - \frac{\frac{\frac{1}{x}}{\sqrt[3]{9} \cdot \sqrt[3]{9}}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{9}} \cdot \sqrt[3]{\sqrt[3]{9}}\right) \cdot \sqrt[3]{\sqrt[3]{9}}}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]

  11. Applied associate-/r*0.3

    \[\leadsto \left(1 - \color{blue}{\frac{\frac{\frac{\frac{1}{x}}{\sqrt[3]{9} \cdot \sqrt[3]{9}}}{\sqrt[3]{\sqrt[3]{9}} \cdot \sqrt[3]{\sqrt[3]{9}}}}{\sqrt[3]{\sqrt[3]{9}}}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]

  12. Simplified0.3

    \[\leadsto \left(1 - \frac{\color{blue}{\frac{\frac{1}{x}}{\left(\sqrt[3]{\sqrt[3]{9}} \cdot \sqrt[3]{\sqrt[3]{9}}\right) \cdot {\left(\sqrt[3]{9}\right)}^{2}}}}{\sqrt[3]{\sqrt[3]{9}}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]

  13. Final simplification0.3

    \[\leadsto \left(1 - \frac{\frac{\frac{1}{x}}{\left(\sqrt[3]{\sqrt[3]{9}} \cdot \sqrt[3]{\sqrt[3]{9}}\right) \cdot {\left(\sqrt[3]{9}\right)}^{2}}}{\sqrt[3]{\sqrt[3]{9}}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x))))

  (- (- 1 (/ 1 (* x 9))) (/ y (* 3 (sqrt x)))))