Average Error: 0.2 → 0.4
Time: 12.8s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[1 - \left(\frac{1}{{\left(\sqrt[3]{3}\right)}^{3}} \cdot \frac{y}{\sqrt{x}} - \left(-\frac{\frac{1}{x}}{9}\right)\right)\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
1 - \left(\frac{1}{{\left(\sqrt[3]{3}\right)}^{3}} \cdot \frac{y}{\sqrt{x}} - \left(-\frac{\frac{1}{x}}{9}\right)\right)
double f(double x, double y) {
        double r310658 = 1.0;
        double r310659 = x;
        double r310660 = 9.0;
        double r310661 = r310659 * r310660;
        double r310662 = r310658 / r310661;
        double r310663 = r310658 - r310662;
        double r310664 = y;
        double r310665 = 3.0;
        double r310666 = sqrt(r310659);
        double r310667 = r310665 * r310666;
        double r310668 = r310664 / r310667;
        double r310669 = r310663 - r310668;
        return r310669;
}
↓
double f(double x, double y) {
        double r310670 = 1.0;
        double r310671 = 1.0;
        double r310672 = 3.0;
        double r310673 = cbrt(r310672);
        double r310674 = 3.0;
        double r310675 = pow(r310673, r310674);
        double r310676 = r310671 / r310675;
        double r310677 = y;
        double r310678 = x;
        double r310679 = sqrt(r310678);
        double r310680 = r310677 / r310679;
        double r310681 = r310676 * r310680;
        double r310682 = r310670 / r310678;
        double r310683 = 9.0;
        double r310684 = r310682 / r310683;
        double r310685 = -r310684;
        double r310686 = r310681 - r310685;
        double r310687 = r310670 - r310686;
        return r310687;
}

Error

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Results

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Target

Original0.2
Target0.2
Herbie0.4
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

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

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

  5. Applied associate-/r*0.2

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

  7. Applied *-un-lft-identity0.2

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

  8. Applied add-cube-cbrt0.2

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

  9. Applied *-un-lft-identity0.2

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

  10. Applied times-frac0.3

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

  11. Applied times-frac0.3

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

  12. Simplified0.3

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019298 
(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)))))