Average Error: 0.2 → 0.2
Time: 51.0s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{9 \cdot x}\right) - \frac{y}{3} \cdot {x}^{\frac{-1}{2}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{9 \cdot x}\right) - \frac{y}{3} \cdot {x}^{\frac{-1}{2}}
double f(double x, double y) {
        double r19731911 = 1.0;
        double r19731912 = x;
        double r19731913 = 9.0;
        double r19731914 = r19731912 * r19731913;
        double r19731915 = r19731911 / r19731914;
        double r19731916 = r19731911 - r19731915;
        double r19731917 = y;
        double r19731918 = 3.0;
        double r19731919 = sqrt(r19731912);
        double r19731920 = r19731918 * r19731919;
        double r19731921 = r19731917 / r19731920;
        double r19731922 = r19731916 - r19731921;
        return r19731922;
}
↓
double f(double x, double y) {
        double r19731923 = 1.0;
        double r19731924 = 9.0;
        double r19731925 = x;
        double r19731926 = r19731924 * r19731925;
        double r19731927 = r19731923 / r19731926;
        double r19731928 = r19731923 - r19731927;
        double r19731929 = y;
        double r19731930 = 3.0;
        double r19731931 = r19731929 / r19731930;
        double r19731932 = -0.5;
        double r19731933 = pow(r19731925, r19731932);
        double r19731934 = r19731931 * r19731933;
        double r19731935 = r19731928 - r19731934;
        return r19731935;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.2
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

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

  3. Applied associate-/r*0.2

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

  5. Applied div-inv0.2

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

  7. Applied pow1/20.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3} \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}}\]

  8. Applied pow-flip0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3} \cdot \color{blue}{{x}^{\left(-\frac{1}{2}\right)}}\]

  9. Simplified0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3} \cdot {x}^{\color{blue}{\frac{-1}{2}}}\]

  10. Final simplification0.2

    \[\leadsto \left(1 - \frac{1}{9 \cdot x}\right) - \frac{y}{3} \cdot {x}^{\frac{-1}{2}}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"

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

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