Average Error: 0.4 → 0.4
Time: 3.6s
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 r337992 = 3.0;
        double r337993 = x;
        double r337994 = sqrt(r337993);
        double r337995 = r337992 * r337994;
        double r337996 = y;
        double r337997 = 1.0;
        double r337998 = 9.0;
        double r337999 = r337993 * r337998;
        double r338000 = r337997 / r337999;
        double r338001 = r337996 + r338000;
        double r338002 = r338001 - r337997;
        double r338003 = r337995 * r338002;
        return r338003;
}
↓
double f(double x, double y) {
        double r338004 = 3.0;
        double r338005 = x;
        double r338006 = sqrt(r338005);
        double r338007 = y;
        double r338008 = 0.1111111111111111;
        double r338009 = r338008 / r338005;
        double r338010 = r338007 + r338009;
        double r338011 = 1.0;
        double r338012 = r338010 - r338011;
        double r338013 = r338006 * r338012;
        double r338014 = r338004 * r338013;
        return r338014;
}

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)))