Average Error: 0.2 → 0.3
Time: 15.9s
Precision: 64
\[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
↓
\[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{\sqrt{x}} \cdot \frac{1}{3.0}\]
\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}
↓
\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{\sqrt{x}} \cdot \frac{1}{3.0}
double f(double x, double y) {
        double r17337395 = 1.0;
        double r17337396 = x;
        double r17337397 = 9.0;
        double r17337398 = r17337396 * r17337397;
        double r17337399 = r17337395 / r17337398;
        double r17337400 = r17337395 - r17337399;
        double r17337401 = y;
        double r17337402 = 3.0;
        double r17337403 = sqrt(r17337396);
        double r17337404 = r17337402 * r17337403;
        double r17337405 = r17337401 / r17337404;
        double r17337406 = r17337400 - r17337405;
        return r17337406;
}
↓
double f(double x, double y) {
        double r17337407 = 1.0;
        double r17337408 = x;
        double r17337409 = 9.0;
        double r17337410 = r17337408 * r17337409;
        double r17337411 = r17337407 / r17337410;
        double r17337412 = r17337407 - r17337411;
        double r17337413 = y;
        double r17337414 = sqrt(r17337408);
        double r17337415 = r17337413 / r17337414;
        double r17337416 = 1.0;
        double r17337417 = 3.0;
        double r17337418 = r17337416 / r17337417;
        double r17337419 = r17337415 * r17337418;
        double r17337420 = r17337412 - r17337419;
        return r17337420;
}

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
\[\left(1.0 - \frac{\frac{1.0}{x}}{9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

    \[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

    \[\leadsto \left(1.0 - \color{blue}{\frac{\frac{1.0}{x}}{9.0}}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
  4. Using strategy rm

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

    \[\leadsto \left(1.0 - \frac{\frac{1.0}{x}}{9.0}\right) - \frac{\color{blue}{1 \cdot y}}{3.0 \cdot \sqrt{x}}\]

  6. Applied times-frac0.3

    \[\leadsto \left(1.0 - \frac{\frac{1.0}{x}}{9.0}\right) - \color{blue}{\frac{1}{3.0} \cdot \frac{y}{\sqrt{x}}}\]
  7. Using strategy rm

  8. Applied div-inv0.3

    \[\leadsto \left(1.0 - \frac{\color{blue}{1.0 \cdot \frac{1}{x}}}{9.0}\right) - \frac{1}{3.0} \cdot \frac{y}{\sqrt{x}}\]

  9. Applied associate-/l*0.3

    \[\leadsto \left(1.0 - \color{blue}{\frac{1.0}{\frac{9.0}{\frac{1}{x}}}}\right) - \frac{1}{3.0} \cdot \frac{y}{\sqrt{x}}\]

  10. Simplified0.3

    \[\leadsto \left(1.0 - \frac{1.0}{\color{blue}{9.0 \cdot x}}\right) - \frac{1}{3.0} \cdot \frac{y}{\sqrt{x}}\]

  11. Final simplification0.3

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{\sqrt{x}} \cdot \frac{1}{3.0}\]

Reproduce

herbie shell --seed 2019165 
(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)))))