Average Error: 0.4 → 0.5
Time: 4.8s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + 0.1111111111111111 \cdot \frac{1}{x}\right) - 1\right)\]
\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 + 0.1111111111111111 \cdot \frac{1}{x}\right) - 1\right)
double f(double x, double y) {
        double r507753 = 3.0;
        double r507754 = x;
        double r507755 = sqrt(r507754);
        double r507756 = r507753 * r507755;
        double r507757 = y;
        double r507758 = 1.0;
        double r507759 = 9.0;
        double r507760 = r507754 * r507759;
        double r507761 = r507758 / r507760;
        double r507762 = r507757 + r507761;
        double r507763 = r507762 - r507758;
        double r507764 = r507756 * r507763;
        return r507764;
}
↓
double f(double x, double y) {
        double r507765 = 3.0;
        double r507766 = x;
        double r507767 = sqrt(r507766);
        double r507768 = r507765 * r507767;
        double r507769 = y;
        double r507770 = 0.1111111111111111;
        double r507771 = 1.0;
        double r507772 = r507771 / r507766;
        double r507773 = r507770 * r507772;
        double r507774 = r507769 + r507773;
        double r507775 = 1.0;
        double r507776 = r507774 - r507775;
        double r507777 = r507768 * r507776;
        return r507777;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.5
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

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-/r*0.4

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

  4. Taylor expanded around 0 0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020089 
(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)))