Average Error: 0.2 → 0.2
Time: 11.0s
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 r349652 = 1.0;
        double r349653 = x;
        double r349654 = 9.0;
        double r349655 = r349653 * r349654;
        double r349656 = r349652 / r349655;
        double r349657 = r349652 - r349656;
        double r349658 = y;
        double r349659 = 3.0;
        double r349660 = sqrt(r349653);
        double r349661 = r349659 * r349660;
        double r349662 = r349658 / r349661;
        double r349663 = r349657 - r349662;
        return r349663;
}
↓
double f(double x, double y) {
        double r349664 = 1.0;
        double r349665 = 0.1111111111111111;
        double r349666 = x;
        double r349667 = r349665 / r349666;
        double r349668 = r349664 - r349667;
        double r349669 = y;
        double r349670 = 3.0;
        double r349671 = r349669 / r349670;
        double r349672 = sqrt(r349666);
        double r349673 = r349671 / r349672;
        double r349674 = r349668 - r349673;
        return r349674;
}

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 +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)))))