Average Error: 0.4 → 0.4
Time: 3.4s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) - 1\right)\right)\]
\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(y + \frac{0.1111111111111111}{x}\right) - 1\right)\right)
double f(double x, double y) {
        double r344781 = 3.0;
        double r344782 = x;
        double r344783 = sqrt(r344782);
        double r344784 = r344781 * r344783;
        double r344785 = y;
        double r344786 = 1.0;
        double r344787 = 9.0;
        double r344788 = r344782 * r344787;
        double r344789 = r344786 / r344788;
        double r344790 = r344785 + r344789;
        double r344791 = r344790 - r344786;
        double r344792 = r344784 * r344791;
        return r344792;
}
↓
double f(double x, double y) {
        double r344793 = 3.0;
        double r344794 = x;
        double r344795 = sqrt(r344794);
        double r344796 = y;
        double r344797 = 0.1111111111111111;
        double r344798 = r344797 / r344794;
        double r344799 = r344796 + r344798;
        double r344800 = 1.0;
        double r344801 = r344799 - r344800;
        double r344802 = r344795 * r344801;
        double r344803 = r344793 * r344802;
        return r344803;
}

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. 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. Taylor expanded around 0 0.4

    \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \color{blue}{\frac{0.1111111111111111}{x}}\right) - 1\right)\right)\]

  5. Final simplification0.4

    \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) - 1\right)\right)\]

Reproduce

herbie shell --seed 2020062 
(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)))