Average Error: 0.4 → 0.5
Time: 14.1s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) \cdot 3\right) + \left(-\left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(1 \cdot \sqrt{x}\right)\right) \cdot \sqrt[3]{3}\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) \cdot 3\right) + \left(-\left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(1 \cdot \sqrt{x}\right)\right) \cdot \sqrt[3]{3}\right)
double f(double x, double y) {
        double r317871 = 3.0;
        double r317872 = x;
        double r317873 = sqrt(r317872);
        double r317874 = r317871 * r317873;
        double r317875 = y;
        double r317876 = 1.0;
        double r317877 = 9.0;
        double r317878 = r317872 * r317877;
        double r317879 = r317876 / r317878;
        double r317880 = r317875 + r317879;
        double r317881 = r317880 - r317876;
        double r317882 = r317874 * r317881;
        return r317882;
}
↓
double f(double x, double y) {
        double r317883 = x;
        double r317884 = sqrt(r317883);
        double r317885 = y;
        double r317886 = 1.0;
        double r317887 = 9.0;
        double r317888 = r317883 * r317887;
        double r317889 = r317886 / r317888;
        double r317890 = r317885 + r317889;
        double r317891 = 3.0;
        double r317892 = r317890 * r317891;
        double r317893 = r317884 * r317892;
        double r317894 = cbrt(r317891);
        double r317895 = r317894 * r317894;
        double r317896 = r317886 * r317884;
        double r317897 = r317895 * r317896;
        double r317898 = r317897 * r317894;
        double r317899 = -r317898;
        double r317900 = r317893 + r317899;
        return r317900;
}

Error

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Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.5
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

Derivation

  1. Initial program 0.4

    \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
  2. Using strategy rm

  3. Applied associate-*l*0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied associate-*l*0.6

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

  8. Applied sub-neg0.6

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

  9. Applied distribute-lft-in0.6

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

  10. Applied distribute-lft-in0.6

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

  11. Applied distribute-lft-in0.6

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

  12. Simplified0.5

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

  13. Simplified0.5

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

  15. Applied add-cube-cbrt0.5

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

  16. Applied associate-*r*0.5

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

  17. Simplified0.5

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

  18. Final simplification0.5

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

Reproduce

herbie shell --seed 2019208 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x))))

  (* (* 3 (sqrt x)) (- (+ y (/ 1 (* x 9))) 1)))