Average Error: 0.2 → 0.2
Time: 3.8s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}
double f(double x, double y) {
        double r507662 = 1.0;
        double r507663 = x;
        double r507664 = 9.0;
        double r507665 = r507663 * r507664;
        double r507666 = r507662 / r507665;
        double r507667 = r507662 - r507666;
        double r507668 = y;
        double r507669 = 3.0;
        double r507670 = sqrt(r507663);
        double r507671 = r507669 * r507670;
        double r507672 = r507668 / r507671;
        double r507673 = r507667 - r507672;
        return r507673;
}
↓
double f(double x, double y) {
        double r507674 = 1.0;
        double r507675 = 0.1111111111111111;
        double r507676 = x;
        double r507677 = r507675 / r507676;
        double r507678 = r507674 - r507677;
        double r507679 = y;
        double r507680 = 3.0;
        double r507681 = r507679 / r507680;
        double r507682 = sqrt(r507676);
        double r507683 = r507681 / r507682;
        double r507684 = r507678 - r507683;
        return r507684;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.3
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. Taylor expanded around 0 0.2

    \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]

  5. Final simplification0.2

    \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]

Reproduce

herbie shell --seed 2020047 
(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)))))