Average Error: 9.2 → 0.1
Time: 3.0s
Precision: binary64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{\frac{x + 1}{1 + \frac{x}{y}}}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{\frac{x + 1}{1 + \frac{x}{y}}}
(FPCore (x y) :precision binary64 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))
↓
(FPCore (x y) :precision binary64 (/ x (/ (+ x 1.0) (+ 1.0 (/ x y)))))
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 (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

Original9.2
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.2

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

  2. Simplified0.1

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

  4. Applied clear-num0.2

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

  6. Applied un-div-inv0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020198 
(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)))