Average Error: 0.4 → 0.4
Time: 3.8s
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 r437748 = 3.0;
        double r437749 = x;
        double r437750 = sqrt(r437749);
        double r437751 = r437748 * r437750;
        double r437752 = y;
        double r437753 = 1.0;
        double r437754 = 9.0;
        double r437755 = r437749 * r437754;
        double r437756 = r437753 / r437755;
        double r437757 = r437752 + r437756;
        double r437758 = r437757 - r437753;
        double r437759 = r437751 * r437758;
        return r437759;
}
↓
double f(double x, double y) {
        double r437760 = 3.0;
        double r437761 = x;
        double r437762 = sqrt(r437761);
        double r437763 = y;
        double r437764 = 0.1111111111111111;
        double r437765 = r437764 / r437761;
        double r437766 = r437763 + r437765;
        double r437767 = 1.0;
        double r437768 = r437766 - r437767;
        double r437769 = r437762 * r437768;
        double r437770 = r437760 * r437769;
        return r437770;
}

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 +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)))