Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.5 → 0.2

Time: 12.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r45463191 = x;
        double r45463192 = y;
        double r45463193 = r45463191 / r45463192;
        double r45463194 = 1.0;
        double r45463195 = r45463193 + r45463194;
        double r45463196 = r45463191 * r45463195;
        double r45463197 = r45463191 + r45463194;
        double r45463198 = r45463196 / r45463197;
        return r45463198;
}

↓

double f(double x, double y) {
        double r45463199 = 1.0;
        double r45463200 = x;
        double r45463201 = 1.0;
        double r45463202 = r45463200 + r45463201;
        double r45463203 = y;
        double r45463204 = r45463200 / r45463203;
        double r45463205 = r45463204 + r45463201;
        double r45463206 = r45463202 / r45463205;
        double r45463207 = r45463206 / r45463200;
        double r45463208 = r45463199 / r45463207;
        return r45463208;
}

Error

Bits error versus x

Bits error versus y

Target

Original	9.5
Target	0.1
Herbie	0.2

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

Derivation

1. Initial program 9.5

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
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. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019170 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

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

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