Average Error: 0.2 → 0.2
Time: 1.1m
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{\frac{1}{3}}{\sqrt{x}} \cdot y\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{\frac{1}{3}}{\sqrt{x}} \cdot y
double f(double x, double y) {
        double r18894665 = 1.0;
        double r18894666 = x;
        double r18894667 = 9.0;
        double r18894668 = r18894666 * r18894667;
        double r18894669 = r18894665 / r18894668;
        double r18894670 = r18894665 - r18894669;
        double r18894671 = y;
        double r18894672 = 3.0;
        double r18894673 = sqrt(r18894666);
        double r18894674 = r18894672 * r18894673;
        double r18894675 = r18894671 / r18894674;
        double r18894676 = r18894670 - r18894675;
        return r18894676;
}
↓
double f(double x, double y) {
        double r18894677 = 1.0;
        double r18894678 = x;
        double r18894679 = r18894677 / r18894678;
        double r18894680 = 9.0;
        double r18894681 = r18894679 / r18894680;
        double r18894682 = r18894677 - r18894681;
        double r18894683 = 1.0;
        double r18894684 = 3.0;
        double r18894685 = r18894683 / r18894684;
        double r18894686 = sqrt(r18894678);
        double r18894687 = r18894685 / r18894686;
        double r18894688 = y;
        double r18894689 = r18894687 * r18894688;
        double r18894690 = r18894682 - r18894689;
        return r18894690;
}

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

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

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

  5. Applied associate-/r*0.2

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

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

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

  8. Applied sqrt-prod0.2

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

  9. Applied div-inv0.2

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

  10. Applied times-frac0.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 
(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)))))