Average Error: 0.2 → 0.2
Time: 6.8s
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 r378679 = 1.0;
        double r378680 = x;
        double r378681 = 9.0;
        double r378682 = r378680 * r378681;
        double r378683 = r378679 / r378682;
        double r378684 = r378679 - r378683;
        double r378685 = y;
        double r378686 = 3.0;
        double r378687 = sqrt(r378680);
        double r378688 = r378686 * r378687;
        double r378689 = r378685 / r378688;
        double r378690 = r378684 - r378689;
        return r378690;
}
↓
double f(double x, double y) {
        double r378691 = 1.0;
        double r378692 = 0.1111111111111111;
        double r378693 = x;
        double r378694 = r378692 / r378693;
        double r378695 = r378691 - r378694;
        double r378696 = y;
        double r378697 = 3.0;
        double r378698 = r378696 / r378697;
        double r378699 = sqrt(r378693);
        double r378700 = r378698 / r378699;
        double r378701 = r378695 - r378700;
        return r378701;
}

Error

Bits error versus x

Bits error versus y

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Results

Enter valid numbers for all inputs

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)))))