Average Error: 0.4 → 0.4
Time: 3.2s
Precision: binary64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
↓
\[3 \cdot \left(\sqrt{x} \cdot y\right) + \sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right) \]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
3 \cdot \left(\sqrt{x} \cdot y\right) + \sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))
↓
(FPCore (x y)
 :precision binary64
 (+ (* 3.0 (* (sqrt x) y)) (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 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)) + (sqrt(x) * (-3.0 + (0.3333333333333333 / 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. Applied associate-*l*_binary640.4

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

  3. Applied associate--l+_binary640.4

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

  4. Applied distribute-rgt-in_binary640.4

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

  5. Applied distribute-rgt-in_binary640.4

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

  6. Simplified0.4

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

  7. Simplified0.4

    \[\leadsto 3 \cdot \left(\sqrt{x} \cdot y\right) + \color{blue}{\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)} \]

  8. Final simplification0.4

    \[\leadsto 3 \cdot \left(\sqrt{x} \cdot y\right) + \sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right) \]

Reproduce

herbie shell --seed 2021313 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))