Average Error: 0.2 → 0.2
Time: 4.9s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3} \cdot \frac{1}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3} \cdot \frac{1}{\sqrt{x}}
double f(double x, double y) {
        double r354688 = 1.0;
        double r354689 = x;
        double r354690 = 9.0;
        double r354691 = r354689 * r354690;
        double r354692 = r354688 / r354691;
        double r354693 = r354688 - r354692;
        double r354694 = y;
        double r354695 = 3.0;
        double r354696 = sqrt(r354689);
        double r354697 = r354695 * r354696;
        double r354698 = r354694 / r354697;
        double r354699 = r354693 - r354698;
        return r354699;
}
↓
double f(double x, double y) {
        double r354700 = 1.0;
        double r354701 = x;
        double r354702 = r354700 / r354701;
        double r354703 = 9.0;
        double r354704 = r354702 / r354703;
        double r354705 = r354700 - r354704;
        double r354706 = y;
        double r354707 = 3.0;
        double r354708 = r354706 / r354707;
        double r354709 = 1.0;
        double r354710 = sqrt(r354701);
        double r354711 = r354709 / r354710;
        double r354712 = r354708 * r354711;
        double r354713 = r354705 - r354712;
        return r354713;
}

Error

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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-frac0.3

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

  8. Applied div-inv0.3

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

  9. Applied associate-*r*0.3

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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)))))