Average Error: 0.2 → 0.2
Time: 32.2s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[0 \cdot \frac{\frac{y}{3}}{\sqrt{x}} + \left(\left(1 - \frac{\frac{1}{\sqrt{9}}}{\sqrt{\sqrt[3]{9}}} \cdot \frac{\frac{1}{\left|\sqrt[3]{9}\right|}}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right)\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
0 \cdot \frac{\frac{y}{3}}{\sqrt{x}} + \left(\left(1 - \frac{\frac{1}{\sqrt{9}}}{\sqrt{\sqrt[3]{9}}} \cdot \frac{\frac{1}{\left|\sqrt[3]{9}\right|}}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right)
double f(double x, double y) {
        double r15863821 = 1.0;
        double r15863822 = x;
        double r15863823 = 9.0;
        double r15863824 = r15863822 * r15863823;
        double r15863825 = r15863821 / r15863824;
        double r15863826 = r15863821 - r15863825;
        double r15863827 = y;
        double r15863828 = 3.0;
        double r15863829 = sqrt(r15863822);
        double r15863830 = r15863828 * r15863829;
        double r15863831 = r15863827 / r15863830;
        double r15863832 = r15863826 - r15863831;
        return r15863832;
}
↓
double f(double x, double y) {
        double r15863833 = 0.0;
        double r15863834 = y;
        double r15863835 = 3.0;
        double r15863836 = r15863834 / r15863835;
        double r15863837 = x;
        double r15863838 = sqrt(r15863837);
        double r15863839 = r15863836 / r15863838;
        double r15863840 = r15863833 * r15863839;
        double r15863841 = 1.0;
        double r15863842 = 1.0;
        double r15863843 = 9.0;
        double r15863844 = sqrt(r15863843);
        double r15863845 = r15863842 / r15863844;
        double r15863846 = cbrt(r15863843);
        double r15863847 = sqrt(r15863846);
        double r15863848 = r15863845 / r15863847;
        double r15863849 = fabs(r15863846);
        double r15863850 = r15863841 / r15863849;
        double r15863851 = r15863850 / r15863837;
        double r15863852 = r15863848 * r15863851;
        double r15863853 = r15863841 - r15863852;
        double r15863854 = r15863853 - r15863839;
        double r15863855 = r15863840 + r15863854;
        return r15863855;
}

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-sqrt29.8

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

  5. Applied prod-diff29.8

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

  6. Simplified0.2

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

  7. Simplified0.2

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

  9. Applied add-sqr-sqrt0.2

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

  10. Applied associate-/r*0.2

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

  12. Applied add-cube-cbrt0.2

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

  13. Applied sqrt-prod0.2

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

  14. Applied div-inv0.3

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

  15. Applied times-frac0.3

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))