Average Error: 0.2 → 0.3
Time: 21.5s
Precision: 64
\[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
↓
\[\left(1.0 - \frac{1.0}{9.0 \cdot x}\right) - \frac{y}{\sqrt{x}} \cdot \frac{1}{3.0}\]
\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}
↓
\left(1.0 - \frac{1.0}{9.0 \cdot x}\right) - \frac{y}{\sqrt{x}} \cdot \frac{1}{3.0}
double f(double x, double y) {
        double r18357492 = 1.0;
        double r18357493 = x;
        double r18357494 = 9.0;
        double r18357495 = r18357493 * r18357494;
        double r18357496 = r18357492 / r18357495;
        double r18357497 = r18357492 - r18357496;
        double r18357498 = y;
        double r18357499 = 3.0;
        double r18357500 = sqrt(r18357493);
        double r18357501 = r18357499 * r18357500;
        double r18357502 = r18357498 / r18357501;
        double r18357503 = r18357497 - r18357502;
        return r18357503;
}
↓
double f(double x, double y) {
        double r18357504 = 1.0;
        double r18357505 = 9.0;
        double r18357506 = x;
        double r18357507 = r18357505 * r18357506;
        double r18357508 = r18357504 / r18357507;
        double r18357509 = r18357504 - r18357508;
        double r18357510 = y;
        double r18357511 = sqrt(r18357506);
        double r18357512 = r18357510 / r18357511;
        double r18357513 = 1.0;
        double r18357514 = 3.0;
        double r18357515 = r18357513 / r18357514;
        double r18357516 = r18357512 * r18357515;
        double r18357517 = r18357509 - r18357516;
        return r18357517;
}

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 *-un-lft-identity0.2

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

  4. Applied times-frac0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019164 
(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)))))