Average Error: 0.2 → 0.2
Time: 13.4s
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 r493665 = 1.0;
        double r493666 = x;
        double r493667 = 9.0;
        double r493668 = r493666 * r493667;
        double r493669 = r493665 / r493668;
        double r493670 = r493665 - r493669;
        double r493671 = y;
        double r493672 = 3.0;
        double r493673 = sqrt(r493666);
        double r493674 = r493672 * r493673;
        double r493675 = r493671 / r493674;
        double r493676 = r493670 - r493675;
        return r493676;
}
↓
double f(double x, double y) {
        double r493677 = 1.0;
        double r493678 = x;
        double r493679 = r493677 / r493678;
        double r493680 = 9.0;
        double r493681 = r493679 / r493680;
        double r493682 = r493677 - r493681;
        double r493683 = y;
        double r493684 = 3.0;
        double r493685 = r493683 / r493684;
        double r493686 = 1.0;
        double r493687 = sqrt(r493678);
        double r493688 = r493686 / r493687;
        double r493689 = r493685 * r493688;
        double r493690 = r493682 - r493689;
        return r493690;
}

Error

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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

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

  5. Applied associate-/r*0.2

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

  7. Applied div-inv0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019351 +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)))))