Average Error: 0.4 → 0.5
Time: 4.3s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3\right) \cdot \sqrt{x}\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3\right) \cdot \sqrt{x}
double code(double x, double y) {
	return ((double) (((double) (3.0 * ((double) sqrt(x)))) * ((double) (((double) (y + ((double) (1.0 / ((double) (x * 9.0)))))) - 1.0))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) (((double) (y + ((double) (((double) (1.0 / x)) / 9.0)))) - 1.0)) * 3.0)) * ((double) sqrt(x))));
}

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. Using strategy rm

  5. Applied *-un-lft-identity0.4

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

  6. Applied sqrt-prod0.4

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

  7. Applied associate-*r*0.4

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

  8. Applied associate-*l*0.4

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

  10. Applied *-commutative0.4

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

  11. Applied associate-*r*0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020114 +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)))