Average Error: 9.4 → 0.1
Time: 1.9s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[x \cdot \frac{\frac{x}{y} + 1}{x + 1}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
x \cdot \frac{\frac{x}{y} + 1}{x + 1}
double code(double x, double y) {
	return ((x * ((x / y) + 1.0)) / (x + 1.0));
}
↓
double code(double x, double y) {
	return (x * (((x / y) + 1.0) / (x + 1.0)));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.4
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.4

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}}\]
  4. Using strategy rm

  5. Applied clear-num0.2

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{x + 1}{\frac{x}{y} + 1}}{x}}}\]
  6. Using strategy rm

  7. Applied clear-num0.3

    \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{\frac{\frac{x}{y} + 1}{x + 1}}}}{x}}\]

  8. Applied associate-/l/0.3

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}}}}\]

  9. Applied associate-/r/0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

    \[\leadsto x \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1)))

  (/ (* x (+ (/ x y) 1)) (+ x 1)))