Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.4 → 0.1

Time: 3.3s

Precision: binary64

Math

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

↓

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

C

double code(double x, double y) {
	return ((double) (((double) (x * ((double) (((double) (x / y)) + 1.0)))) / ((double) (x + 1.0))));
}

↓

double code(double x, double y) {
	return ((double) (((double) (((double) (x / y)) + 1.0)) / ((double) (((double) (x + 1.0)) / x))));
}

Error

Bits error versus x

Bits error versus y

Target

Original	9.4
Target	0.1
Herbie	0.1

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

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-num 0.2

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

7. Applied div-inv 0.3

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

8. Applied associate-/r* 0.3

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

9. Simplified 0.2

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

11. Applied div-inv 0.2

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

12. Applied associate-/l* 0.2

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

13. Simplified 0.1

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

14. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020171 
(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)))