Average Error: 0.4 → 0.4
Time: 17.6s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\sqrt{x} \cdot \left(\left(\left(\frac{0.1111111111111111049432054187491303309798}{x} - 1\right) + y\right) \cdot 3\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\sqrt{x} \cdot \left(\left(\left(\frac{0.1111111111111111049432054187491303309798}{x} - 1\right) + y\right) \cdot 3\right)
double f(double x, double y) {
        double r311824 = 3.0;
        double r311825 = x;
        double r311826 = sqrt(r311825);
        double r311827 = r311824 * r311826;
        double r311828 = y;
        double r311829 = 1.0;
        double r311830 = 9.0;
        double r311831 = r311825 * r311830;
        double r311832 = r311829 / r311831;
        double r311833 = r311828 + r311832;
        double r311834 = r311833 - r311829;
        double r311835 = r311827 * r311834;
        return r311835;
}
↓
double f(double x, double y) {
        double r311836 = x;
        double r311837 = sqrt(r311836);
        double r311838 = 0.1111111111111111;
        double r311839 = r311838 / r311836;
        double r311840 = 1.0;
        double r311841 = r311839 - r311840;
        double r311842 = y;
        double r311843 = r311841 + r311842;
        double r311844 = 3.0;
        double r311845 = r311843 * r311844;
        double r311846 = r311837 * r311845;
        return r311846;
}

Error

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Bits error versus y

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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 pow10.4

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

  4. Applied pow10.4

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

  5. Applied pow10.4

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

  6. Applied pow-prod-down0.4

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

  7. Applied pow-prod-down0.4

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

  8. Simplified0.4

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

  9. Taylor expanded around 0 0.4

    \[\leadsto {\left(\left(3 \cdot \left(\left(\color{blue}{\frac{0.1111111111111111049432054187491303309798}{x}} - 1\right) + y\right)\right) \cdot \sqrt{x}\right)}^{1}\]

  10. Final simplification0.4

    \[\leadsto \sqrt{x} \cdot \left(\left(\left(\frac{0.1111111111111111049432054187491303309798}{x} - 1\right) + y\right) \cdot 3\right)\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))