Average Error: 0.4 → 0.4
Time: 12.8s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \left(\left(y - 1\right) + \frac{0.1111111111111111049432054187491303309798}{x}\right)\right) \cdot \sqrt{x}\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \left(\left(y - 1\right) + \frac{0.1111111111111111049432054187491303309798}{x}\right)\right) \cdot \sqrt{x}
double f(double x, double y) {
        double r335828 = 3.0;
        double r335829 = x;
        double r335830 = sqrt(r335829);
        double r335831 = r335828 * r335830;
        double r335832 = y;
        double r335833 = 1.0;
        double r335834 = 9.0;
        double r335835 = r335829 * r335834;
        double r335836 = r335833 / r335835;
        double r335837 = r335832 + r335836;
        double r335838 = r335837 - r335833;
        double r335839 = r335831 * r335838;
        return r335839;
}
↓
double f(double x, double y) {
        double r335840 = 3.0;
        double r335841 = y;
        double r335842 = 1.0;
        double r335843 = r335841 - r335842;
        double r335844 = 0.1111111111111111;
        double r335845 = x;
        double r335846 = r335844 / r335845;
        double r335847 = r335843 + r335846;
        double r335848 = r335840 * r335847;
        double r335849 = sqrt(r335845);
        double r335850 = r335848 * r335849;
        return r335850;
}

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

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019298 
(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)))