Average Error: 0.4 → 0.4
Time: 20.4s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right)\right) \cdot \sqrt{x} + \left(3 \cdot \sqrt{x}\right) \cdot \left(1 \cdot 0\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right)\right) \cdot \sqrt{x} + \left(3 \cdot \sqrt{x}\right) \cdot \left(1 \cdot 0\right)
double f(double x, double y) {
        double r540884 = 3.0;
        double r540885 = x;
        double r540886 = sqrt(r540885);
        double r540887 = r540884 * r540886;
        double r540888 = y;
        double r540889 = 1.0;
        double r540890 = 9.0;
        double r540891 = r540885 * r540890;
        double r540892 = r540889 / r540891;
        double r540893 = r540888 + r540892;
        double r540894 = r540893 - r540889;
        double r540895 = r540887 * r540894;
        return r540895;
}
↓
double f(double x, double y) {
        double r540896 = 3.0;
        double r540897 = y;
        double r540898 = 1.0;
        double r540899 = x;
        double r540900 = r540898 / r540899;
        double r540901 = 9.0;
        double r540902 = r540900 / r540901;
        double r540903 = r540897 + r540902;
        double r540904 = r540903 - r540898;
        double r540905 = r540896 * r540904;
        double r540906 = sqrt(r540899);
        double r540907 = r540905 * r540906;
        double r540908 = r540896 * r540906;
        double r540909 = 0.0;
        double r540910 = r540898 * r540909;
        double r540911 = r540908 * r540910;
        double r540912 = r540907 + r540911;
        return r540912;
}

Error

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Results

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Target

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

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 add-cube-cbrt0.4

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

  4. Applied add-sqr-sqrt15.5

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

  5. Applied prod-diff15.5

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

  6. Applied distribute-lft-in15.5

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  10. Applied associate-/r*0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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)))