Average Error: 0.4 → 0.4
Time: 10.3s
Precision: binary64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\frac{0.3333333333333333}{\sqrt{x}} + \left(3 \cdot \left(\sqrt{x} \cdot y\right) - \sqrt{x} \cdot 3\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\frac{0.3333333333333333}{\sqrt{x}} + \left(3 \cdot \left(\sqrt{x} \cdot y\right) - \sqrt{x} \cdot 3\right)
(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))
↓
(FPCore (x y)
 :precision binary64
 (+
  (/ 0.3333333333333333 (sqrt x))
  (- (* 3.0 (* (sqrt x) y)) (* (sqrt x) 3.0))))
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 (0.3333333333333333 / sqrt(x)) + ((3.0 * (sqrt(x) * y)) - (sqrt(x) * 3.0));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \sqrt{\frac{1}{x}} + 3 \cdot \left(\sqrt{x} \cdot y\right)\right) - 3 \cdot \sqrt{x}}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}} + \left(3 \cdot \left(y \cdot \sqrt{x}\right) - 3 \cdot \sqrt{x}\right)}\]
  4. Using strategy rm

  5. Applied sqrt-div_binary64_191910.4

    \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} + \left(3 \cdot \left(y \cdot \sqrt{x}\right) - 3 \cdot \sqrt{x}\right)\]

  6. Applied associate-*r/_binary64_191160.4

    \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \sqrt{1}}{\sqrt{x}}} + \left(3 \cdot \left(y \cdot \sqrt{x}\right) - 3 \cdot \sqrt{x}\right)\]

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2021176 
(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)))