Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.1 → 0.1

Time: 2.6s

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 r997568 = x;
        double r997569 = y;
        double r997570 = r997568 / r997569;
        double r997571 = 1.0;
        double r997572 = r997570 + r997571;
        double r997573 = r997568 * r997572;
        double r997574 = r997568 + r997571;
        double r997575 = r997573 / r997574;
        return r997575;
}

↓

double f(double x, double y) {
        double r997576 = x;
        double r997577 = 1.0;
        double r997578 = r997576 + r997577;
        double r997579 = r997576 / r997578;
        double r997580 = y;
        double r997581 = r997576 / r997580;
        double r997582 = r997579 * r997581;
        double r997583 = r997579 * r997577;
        double r997584 = r997582 + r997583;
        return r997584;
}

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 +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)))