Average Error: 0.2 → 0.2
Time: 14.1s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{\sqrt{x} \cdot 3}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{\sqrt{x} \cdot 3}
double f(double x, double y) {
        double r261928 = 1.0;
        double r261929 = x;
        double r261930 = 9.0;
        double r261931 = r261929 * r261930;
        double r261932 = r261928 / r261931;
        double r261933 = r261928 - r261932;
        double r261934 = y;
        double r261935 = 3.0;
        double r261936 = sqrt(r261929);
        double r261937 = r261935 * r261936;
        double r261938 = r261934 / r261937;
        double r261939 = r261933 - r261938;
        return r261939;
}
↓
double f(double x, double y) {
        double r261940 = 1.0;
        double r261941 = x;
        double r261942 = r261940 / r261941;
        double r261943 = 9.0;
        double r261944 = r261942 / r261943;
        double r261945 = r261940 - r261944;
        double r261946 = y;
        double r261947 = sqrt(r261941);
        double r261948 = 3.0;
        double r261949 = r261947 * r261948;
        double r261950 = r261946 / r261949;
        double r261951 = r261945 - r261950;
        return r261951;
}

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

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

  5. Applied *-un-lft-identity0.2

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

  6. Applied times-frac0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x))))

  (- (- 1 (/ 1 (* x 9))) (/ y (* 3 (sqrt x)))))