Average Error: 0.2 → 0.2
Time: 25.3s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{y}{\sqrt{x} \cdot 3}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{y}{\sqrt{x} \cdot 3}
double f(double x, double y) {
        double r20496483 = 1.0;
        double r20496484 = x;
        double r20496485 = 9.0;
        double r20496486 = r20496484 * r20496485;
        double r20496487 = r20496483 / r20496486;
        double r20496488 = r20496483 - r20496487;
        double r20496489 = y;
        double r20496490 = 3.0;
        double r20496491 = sqrt(r20496484);
        double r20496492 = r20496490 * r20496491;
        double r20496493 = r20496489 / r20496492;
        double r20496494 = r20496488 - r20496493;
        return r20496494;
}
↓
double f(double x, double y) {
        double r20496495 = 1.0;
        double r20496496 = 0.1111111111111111;
        double r20496497 = x;
        double r20496498 = r20496496 / r20496497;
        double r20496499 = r20496495 - r20496498;
        double r20496500 = y;
        double r20496501 = sqrt(r20496497);
        double r20496502 = 3.0;
        double r20496503 = r20496501 * r20496502;
        double r20496504 = r20496500 / r20496503;
        double r20496505 = r20496499 - r20496504;
        return r20496505;
}

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. Taylor expanded around 0 0.2

    \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111049432054187491303309798}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

  3. Final simplification0.2

    \[\leadsto \left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{y}{\sqrt{x} \cdot 3}\]

Reproduce

herbie shell --seed 2019171 
(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)))))