Average Error: 0.2 → 0.2
Time: 16.9s
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 r23789318 = 1.0;
        double r23789319 = x;
        double r23789320 = 9.0;
        double r23789321 = r23789319 * r23789320;
        double r23789322 = r23789318 / r23789321;
        double r23789323 = r23789318 - r23789322;
        double r23789324 = y;
        double r23789325 = 3.0;
        double r23789326 = sqrt(r23789319);
        double r23789327 = r23789325 * r23789326;
        double r23789328 = r23789324 / r23789327;
        double r23789329 = r23789323 - r23789328;
        return r23789329;
}
↓
double f(double x, double y) {
        double r23789330 = 1.0;
        double r23789331 = 0.1111111111111111;
        double r23789332 = x;
        double r23789333 = r23789331 / r23789332;
        double r23789334 = r23789330 - r23789333;
        double r23789335 = y;
        double r23789336 = sqrt(r23789332);
        double r23789337 = 3.0;
        double r23789338 = r23789336 * r23789337;
        double r23789339 = r23789335 / r23789338;
        double r23789340 = r23789334 - r23789339;
        return r23789340;
}

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)))))