Average Error: 0.2 → 0.2
Time: 20.4s
Precision: 64
\[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
↓
\[\left(1.0 - \frac{\frac{\frac{1.0}{x}}{\sqrt{9.0}}}{\sqrt{9.0}}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}
↓
\left(1.0 - \frac{\frac{\frac{1.0}{x}}{\sqrt{9.0}}}{\sqrt{9.0}}\right) - \frac{y}{3.0 \cdot \sqrt{x}}
double f(double x, double y) {
        double r22984715 = 1.0;
        double r22984716 = x;
        double r22984717 = 9.0;
        double r22984718 = r22984716 * r22984717;
        double r22984719 = r22984715 / r22984718;
        double r22984720 = r22984715 - r22984719;
        double r22984721 = y;
        double r22984722 = 3.0;
        double r22984723 = sqrt(r22984716);
        double r22984724 = r22984722 * r22984723;
        double r22984725 = r22984721 / r22984724;
        double r22984726 = r22984720 - r22984725;
        return r22984726;
}
↓
double f(double x, double y) {
        double r22984727 = 1.0;
        double r22984728 = x;
        double r22984729 = r22984727 / r22984728;
        double r22984730 = 9.0;
        double r22984731 = sqrt(r22984730);
        double r22984732 = r22984729 / r22984731;
        double r22984733 = r22984732 / r22984731;
        double r22984734 = r22984727 - r22984733;
        double r22984735 = y;
        double r22984736 = 3.0;
        double r22984737 = sqrt(r22984728);
        double r22984738 = r22984736 * r22984737;
        double r22984739 = r22984735 / r22984738;
        double r22984740 = r22984734 - r22984739;
        return r22984740;
}

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

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 add-sqr-sqrt0.2

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

  6. Applied associate-/r*0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 
(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)))))