Average Error: 0.4 → 0.4
Time: 38.5s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{0.1111111111111111049432054187491303309798}{x}\right) - 1\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{0.1111111111111111049432054187491303309798}{x}\right) - 1\right)
double f(double x, double y) {
        double r325362 = 3.0;
        double r325363 = x;
        double r325364 = sqrt(r325363);
        double r325365 = r325362 * r325364;
        double r325366 = y;
        double r325367 = 1.0;
        double r325368 = 9.0;
        double r325369 = r325363 * r325368;
        double r325370 = r325367 / r325369;
        double r325371 = r325366 + r325370;
        double r325372 = r325371 - r325367;
        double r325373 = r325365 * r325372;
        return r325373;
}
↓
double f(double x, double y) {
        double r325374 = 3.0;
        double r325375 = x;
        double r325376 = sqrt(r325375);
        double r325377 = r325374 * r325376;
        double r325378 = y;
        double r325379 = 0.1111111111111111;
        double r325380 = r325379 / r325375;
        double r325381 = r325378 + r325380;
        double r325382 = 1.0;
        double r325383 = r325381 - r325382;
        double r325384 = r325377 * r325383;
        return r325384;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.4
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

Derivation

  1. Initial program 0.4

    \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

  2. Taylor expanded around 0 0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x))))

  (* (* 3 (sqrt x)) (- (+ y (/ 1 (* x 9))) 1)))