Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.4 → 0.1

Time: 2.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r867098 = x;
        double r867099 = y;
        double r867100 = r867098 / r867099;
        double r867101 = 1.0;
        double r867102 = r867100 + r867101;
        double r867103 = r867098 * r867102;
        double r867104 = r867098 + r867101;
        double r867105 = r867103 / r867104;
        return r867105;
}

↓

double f(double x, double y) {
        double r867106 = x;
        double r867107 = y;
        double r867108 = r867106 / r867107;
        double r867109 = 1.0;
        double r867110 = r867108 + r867109;
        double r867111 = r867106 + r867109;
        double r867112 = r867111 / r867106;
        double r867113 = r867110 / r867112;
        return r867113;
}

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 clear-num 9.5

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

5. Applied associate-/r* 0.2

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

7. Applied *-un-lft-identity 0.2

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

8. Applied *-un-lft-identity 0.2

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

9. Applied *-un-lft-identity 0.2

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

10. Applied times-frac 0.2

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

11. Applied times-frac 0.2

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

12. Applied add-sqr-sqrt 0.2

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

13. Applied times-frac 0.2

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

14. Simplified 0.2

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

15. Simplified 0.1

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

16. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020049 
(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)))