Average Error: 0.4 → 0.4
Time: 37.1s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[3 \cdot \left(\sqrt{x} \cdot \left(\left(\frac{0.1111111111111111049432054187491303309798}{x} - 1\right) + y\right)\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
3 \cdot \left(\sqrt{x} \cdot \left(\left(\frac{0.1111111111111111049432054187491303309798}{x} - 1\right) + y\right)\right)
double f(double x, double y) {
        double r259652 = 3.0;
        double r259653 = x;
        double r259654 = sqrt(r259653);
        double r259655 = r259652 * r259654;
        double r259656 = y;
        double r259657 = 1.0;
        double r259658 = 9.0;
        double r259659 = r259653 * r259658;
        double r259660 = r259657 / r259659;
        double r259661 = r259656 + r259660;
        double r259662 = r259661 - r259657;
        double r259663 = r259655 * r259662;
        return r259663;
}
↓
double f(double x, double y) {
        double r259664 = 3.0;
        double r259665 = x;
        double r259666 = sqrt(r259665);
        double r259667 = 0.1111111111111111;
        double r259668 = r259667 / r259665;
        double r259669 = 1.0;
        double r259670 = r259668 - r259669;
        double r259671 = y;
        double r259672 = r259670 + r259671;
        double r259673 = r259666 * r259672;
        double r259674 = r259664 * r259673;
        return r259674;
}

Error

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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. Using strategy rm

  3. Applied associate-*l*0.4

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

  4. Simplified0.4

    \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(\frac{1}{x \cdot 9} - 1\right) + y\right)\right)}\]
  5. Using strategy rm

  6. Applied associate-/r*0.4

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

  7. Taylor expanded around 0 0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019325 +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)))