Average Error: 0.2 → 0.3
Time: 4.5s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{1}{3} \cdot \frac{y}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{1}{3} \cdot \frac{y}{\sqrt{x}}
double f(double x, double y) {
        double r383877 = 1.0;
        double r383878 = x;
        double r383879 = 9.0;
        double r383880 = r383878 * r383879;
        double r383881 = r383877 / r383880;
        double r383882 = r383877 - r383881;
        double r383883 = y;
        double r383884 = 3.0;
        double r383885 = sqrt(r383878);
        double r383886 = r383884 * r383885;
        double r383887 = r383883 / r383886;
        double r383888 = r383882 - r383887;
        return r383888;
}
↓
double f(double x, double y) {
        double r383889 = 1.0;
        double r383890 = x;
        double r383891 = r383889 / r383890;
        double r383892 = 9.0;
        double r383893 = r383891 / r383892;
        double r383894 = r383889 - r383893;
        double r383895 = 1.0;
        double r383896 = 3.0;
        double r383897 = r383895 / r383896;
        double r383898 = y;
        double r383899 = sqrt(r383890);
        double r383900 = r383898 / r383899;
        double r383901 = r383897 * r383900;
        double r383902 = r383894 - r383901;
        return r383902;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

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

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-frac0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020039 
(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)))))