Average Error: 0.4 → 0.4
Time: 4.6s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot y + \left(\left(\frac{\frac{1}{9}}{x} + \left(-1\right)\right) \cdot 3\right) \cdot \sqrt{x}\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \sqrt{x}\right) \cdot y + \left(\left(\frac{\frac{1}{9}}{x} + \left(-1\right)\right) \cdot 3\right) \cdot \sqrt{x}
double code(double x, double y) {
	return ((3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0));
}
↓
double code(double x, double y) {
	return (((3.0 * sqrt(x)) * y) + (((((1.0 / 9.0) / x) + -1.0) * 3.0) * 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
Herbie0.4
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

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 add-sqr-sqrt0.4

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

  4. Applied times-frac0.4

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

  6. Applied associate--l+0.4

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

  7. Applied distribute-lft-in0.5

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020058 +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)))