Average Error: 0.4 → 0.4
Time: 5.1s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[3 \cdot \left(y \cdot \sqrt{x}\right) + \left(3 \cdot \left(\frac{\frac{1}{x}}{9} - 1\right)\right) \cdot \sqrt{x}\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
3 \cdot \left(y \cdot \sqrt{x}\right) + \left(3 \cdot \left(\frac{\frac{1}{x}}{9} - 1\right)\right) \cdot \sqrt{x}
double f(double x, double y) {
        double r488866 = 3.0;
        double r488867 = x;
        double r488868 = sqrt(r488867);
        double r488869 = r488866 * r488868;
        double r488870 = y;
        double r488871 = 1.0;
        double r488872 = 9.0;
        double r488873 = r488867 * r488872;
        double r488874 = r488871 / r488873;
        double r488875 = r488870 + r488874;
        double r488876 = r488875 - r488871;
        double r488877 = r488869 * r488876;
        return r488877;
}
↓
double f(double x, double y) {
        double r488878 = 3.0;
        double r488879 = y;
        double r488880 = x;
        double r488881 = sqrt(r488880);
        double r488882 = r488879 * r488881;
        double r488883 = r488878 * r488882;
        double r488884 = 1.0;
        double r488885 = r488884 / r488880;
        double r488886 = 9.0;
        double r488887 = r488885 / r488886;
        double r488888 = r488887 - r488884;
        double r488889 = r488878 * r488888;
        double r488890 = r488889 * r488881;
        double r488891 = r488883 + r488890;
        return r488891;
}

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 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 associate-/r*0.4

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

  7. Applied associate--l+0.4

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

  8. Applied distribute-lft-in0.4

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

  9. Applied distribute-lft-in0.4

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  13. Applied associate-*r*0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020020 +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)))