Average Error: 0.4 → 0.5
Time: 4.5s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \left(\left(\frac{\frac{1}{x}}{9} + y\right) - 1\right)\right) \cdot \sqrt{x}\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \left(\left(\frac{\frac{1}{x}}{9} + y\right) - 1\right)\right) \cdot \sqrt{x}
double f(double x, double y) {
        double r430014 = 3.0;
        double r430015 = x;
        double r430016 = sqrt(r430015);
        double r430017 = r430014 * r430016;
        double r430018 = y;
        double r430019 = 1.0;
        double r430020 = 9.0;
        double r430021 = r430015 * r430020;
        double r430022 = r430019 / r430021;
        double r430023 = r430018 + r430022;
        double r430024 = r430023 - r430019;
        double r430025 = r430017 * r430024;
        return r430025;
}
↓
double f(double x, double y) {
        double r430026 = 3.0;
        double r430027 = 1.0;
        double r430028 = x;
        double r430029 = r430027 / r430028;
        double r430030 = 9.0;
        double r430031 = r430029 / r430030;
        double r430032 = y;
        double r430033 = r430031 + r430032;
        double r430034 = r430033 - r430027;
        double r430035 = r430026 * r430034;
        double r430036 = sqrt(r430028);
        double r430037 = r430035 * r430036;
        return r430037;
}

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-sqr-sqrt0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied associate-*l*0.4

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

  9. Simplified0.5

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

  11. Applied associate-/r*0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2020003 
(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)))