Average Error: 0.2 → 0.2
Time: 16.0s
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 r6453394 = 1.0;
        double r6453395 = x;
        double r6453396 = 9.0;
        double r6453397 = r6453395 * r6453396;
        double r6453398 = r6453394 / r6453397;
        double r6453399 = r6453394 - r6453398;
        double r6453400 = y;
        double r6453401 = 3.0;
        double r6453402 = sqrt(r6453395);
        double r6453403 = r6453401 * r6453402;
        double r6453404 = r6453400 / r6453403;
        double r6453405 = r6453399 - r6453404;
        return r6453405;
}
↓
double f(double x, double y) {
        double r6453406 = 1.0;
        double r6453407 = 0.1111111111111111;
        double r6453408 = x;
        double r6453409 = r6453407 / r6453408;
        double r6453410 = r6453406 - r6453409;
        double r6453411 = y;
        double r6453412 = sqrt(r6453408);
        double r6453413 = 3.0;
        double r6453414 = r6453412 * r6453413;
        double r6453415 = r6453411 / r6453414;
        double r6453416 = r6453410 - r6453415;
        return r6453416;
}

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 2019179 
(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)))))