Average Error: 9.1 → 0.1
Time: 2.8s
Precision: binary64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{x + 1} \cdot \left(1 + \frac{x}{y}\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{x + 1} \cdot \left(1 + \frac{x}{y}\right)
double code(double x, double y) {
	return (((double) (x * ((double) ((x / y) + 1.0)))) / ((double) (x + 1.0)));
}
↓
double code(double x, double y) {
	return ((double) ((x / ((double) (x + 1.0))) * ((double) (1.0 + (x / y)))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program Error: 9.1 bits

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

  2. SimplifiedError: 0.1 bits

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

  4. Applied clear-numError: 0.2 bits

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

  6. Applied associate-/r/Error: 0.2 bits

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

  7. Applied associate-*r*Error: 0.1 bits

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

  8. SimplifiedError: 0.1 bits

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

  9. Final simplificationError: 0.1 bits

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

Reproduce

herbie shell --seed 2020200 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

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

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))