Average Error: 0.2 → 0.2
Time: 14.1s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(\left({\left(\sqrt[3]{1}\right)}^{3} - \frac{\frac{{\left(\sqrt[3]{1}\right)}^{3}}{x}}{9}\right) + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{x} \cdot \left(\left(-\frac{\sqrt[3]{1}}{9}\right) + \frac{\sqrt[3]{1}}{9}\right)\right) - \frac{y}{3} \cdot {x}^{\frac{-1}{2}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(\left({\left(\sqrt[3]{1}\right)}^{3} - \frac{\frac{{\left(\sqrt[3]{1}\right)}^{3}}{x}}{9}\right) + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{x} \cdot \left(\left(-\frac{\sqrt[3]{1}}{9}\right) + \frac{\sqrt[3]{1}}{9}\right)\right) - \frac{y}{3} \cdot {x}^{\frac{-1}{2}}
double f(double x, double y) {
        double r360652 = 1.0;
        double r360653 = x;
        double r360654 = 9.0;
        double r360655 = r360653 * r360654;
        double r360656 = r360652 / r360655;
        double r360657 = r360652 - r360656;
        double r360658 = y;
        double r360659 = 3.0;
        double r360660 = sqrt(r360653);
        double r360661 = r360659 * r360660;
        double r360662 = r360658 / r360661;
        double r360663 = r360657 - r360662;
        return r360663;
}
↓
double f(double x, double y) {
        double r360664 = 1.0;
        double r360665 = cbrt(r360664);
        double r360666 = 3.0;
        double r360667 = pow(r360665, r360666);
        double r360668 = x;
        double r360669 = r360667 / r360668;
        double r360670 = 9.0;
        double r360671 = r360669 / r360670;
        double r360672 = r360667 - r360671;
        double r360673 = r360665 * r360665;
        double r360674 = r360673 / r360668;
        double r360675 = r360665 / r360670;
        double r360676 = -r360675;
        double r360677 = r360676 + r360675;
        double r360678 = r360674 * r360677;
        double r360679 = r360672 + r360678;
        double r360680 = y;
        double r360681 = 3.0;
        double r360682 = r360680 / r360681;
        double r360683 = -0.5;
        double r360684 = pow(r360668, r360683);
        double r360685 = r360682 * r360684;
        double r360686 = r360679 - r360685;
        return r360686;
}

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 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 div-inv0.2

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

  7. Applied pow1/20.2

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

  8. Applied pow-flip0.2

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

  9. Simplified0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3} \cdot {x}^{\color{blue}{\frac{-1}{2}}}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt0.2

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

  12. Applied times-frac0.3

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

  13. Applied add-cube-cbrt0.3

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

  14. Applied prod-diff0.3

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

  15. Simplified0.2

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2019306 +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)))))