Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.1 → 0.1

Time: 2.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r849030 = x;
        double r849031 = y;
        double r849032 = r849030 / r849031;
        double r849033 = 1.0;
        double r849034 = r849032 + r849033;
        double r849035 = r849030 * r849034;
        double r849036 = r849030 + r849033;
        double r849037 = r849035 / r849036;
        return r849037;
}

↓

double f(double x, double y) {
        double r849038 = x;
        double r849039 = 1.0;
        double r849040 = r849038 + r849039;
        double r849041 = r849038 / r849040;
        double r849042 = y;
        double r849043 = r849038 / r849042;
        double r849044 = r849041 * r849043;
        double r849045 = r849041 * r849039;
        double r849046 = r849044 + r849045;
        return r849046;
}

Error

Bits error versus x

Bits error versus y

Target

Original	9.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.1

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
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 associate-/r/ 0.1

   \[\leadsto \color{blue}{\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)}\]
6. Using strategy rm

7. Applied distribute-lft-in 0.1

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

8. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020047 
(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)))