Average Error: 0.2 → 0.2
Time: 4.9s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{\sqrt[3]{1}}{9}}{x}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{\sqrt[3]{1}}{9}}{x}\right) - \frac{y}{3 \cdot \sqrt{x}}
double f(double x, double y) {
        double r410901 = 1.0;
        double r410902 = x;
        double r410903 = 9.0;
        double r410904 = r410902 * r410903;
        double r410905 = r410901 / r410904;
        double r410906 = r410901 - r410905;
        double r410907 = y;
        double r410908 = 3.0;
        double r410909 = sqrt(r410902);
        double r410910 = r410908 * r410909;
        double r410911 = r410907 / r410910;
        double r410912 = r410906 - r410911;
        return r410912;
}
↓
double f(double x, double y) {
        double r410913 = 1.0;
        double r410914 = cbrt(r410913);
        double r410915 = r410914 * r410914;
        double r410916 = 9.0;
        double r410917 = r410914 / r410916;
        double r410918 = x;
        double r410919 = r410917 / r410918;
        double r410920 = r410915 * r410919;
        double r410921 = r410913 - r410920;
        double r410922 = y;
        double r410923 = 3.0;
        double r410924 = sqrt(r410918);
        double r410925 = r410923 * r410924;
        double r410926 = r410922 / r410925;
        double r410927 = r410921 - r410926;
        return r410927;
}

Error

Bits error versus x

Bits error versus y

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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-cbrt0.2

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

  4. Applied times-frac0.3

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

  6. Applied div-inv0.3

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

  7. Applied associate-*l*0.3

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020083 
(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)))))