Average Error: 0.2 → 0.2
Time: 19.0s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{y}{\sqrt{x}}}{3}\right) - \frac{1}{9 \cdot x}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{y}{\sqrt{x}}}{3}\right) - \frac{1}{9 \cdot x}
double f(double x, double y) {
        double r267850 = 1.0;
        double r267851 = x;
        double r267852 = 9.0;
        double r267853 = r267851 * r267852;
        double r267854 = r267850 / r267853;
        double r267855 = r267850 - r267854;
        double r267856 = y;
        double r267857 = 3.0;
        double r267858 = sqrt(r267851);
        double r267859 = r267857 * r267858;
        double r267860 = r267856 / r267859;
        double r267861 = r267855 - r267860;
        return r267861;
}
↓
double f(double x, double y) {
        double r267862 = 1.0;
        double r267863 = y;
        double r267864 = x;
        double r267865 = sqrt(r267864);
        double r267866 = r267863 / r267865;
        double r267867 = 3.0;
        double r267868 = r267866 / r267867;
        double r267869 = r267862 - r267868;
        double r267870 = 9.0;
        double r267871 = r267870 * r267864;
        double r267872 = r267862 / r267871;
        double r267873 = r267869 - r267872;
        return r267873;
}

Error

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Results

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Target

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

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 add-cube-cbrt0.5

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

  4. Applied add-sqr-sqrt30.4

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

  5. Applied prod-diff30.4

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

  6. Simplified0.2

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

  7. Simplified0.2

    \[\leadsto \left(\left(1 - \frac{\frac{y}{\sqrt{x}}}{3}\right) - \frac{\frac{1}{x}}{9}\right) + \color{blue}{0}\]
  8. Using strategy rm

  9. Applied div-inv0.2

    \[\leadsto \left(\left(1 - \frac{\frac{y}{\sqrt{x}}}{3}\right) - \frac{\color{blue}{1 \cdot \frac{1}{x}}}{9}\right) + 0\]

  10. Applied associate-/l*0.2

    \[\leadsto \left(\left(1 - \frac{\frac{y}{\sqrt{x}}}{3}\right) - \color{blue}{\frac{1}{\frac{9}{\frac{1}{x}}}}\right) + 0\]

  11. Simplified0.2

    \[\leadsto \left(\left(1 - \frac{\frac{y}{\sqrt{x}}}{3}\right) - \frac{1}{\color{blue}{9 \cdot x}}\right) + 0\]

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019326 +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)))))