Average Error: 0.2 → 0.2
Time: 4.2s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}
double f(double x, double y) {
        double r324633 = 1.0;
        double r324634 = x;
        double r324635 = 9.0;
        double r324636 = r324634 * r324635;
        double r324637 = r324633 / r324636;
        double r324638 = r324633 - r324637;
        double r324639 = y;
        double r324640 = 3.0;
        double r324641 = sqrt(r324634);
        double r324642 = r324640 * r324641;
        double r324643 = r324639 / r324642;
        double r324644 = r324638 - r324643;
        return r324644;
}
↓
double f(double x, double y) {
        double r324645 = 1.0;
        double r324646 = 0.1111111111111111;
        double r324647 = x;
        double r324648 = r324646 / r324647;
        double r324649 = r324645 - r324648;
        double r324650 = y;
        double r324651 = 3.0;
        double r324652 = r324650 / r324651;
        double r324653 = sqrt(r324647);
        double r324654 = r324652 / r324653;
        double r324655 = r324649 - r324654;
        return r324655;
}

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. Taylor expanded around 0 0.2

    \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]

  5. Final simplification0.2

    \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]

Reproduce

herbie shell --seed 2020024 
(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)))))