Average Error: 8.6 → 0.1
Time: 2.6s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}
double f(double x, double y) {
        double r915714 = x;
        double r915715 = y;
        double r915716 = r915714 / r915715;
        double r915717 = 1.0;
        double r915718 = r915716 + r915717;
        double r915719 = r915714 * r915718;
        double r915720 = r915714 + r915717;
        double r915721 = r915719 / r915720;
        return r915721;
}
↓
double f(double x, double y) {
        double r915722 = x;
        double r915723 = 1.0;
        double r915724 = r915722 + r915723;
        double r915725 = y;
        double r915726 = r915722 / r915725;
        double r915727 = r915726 + r915723;
        double r915728 = r915724 / r915727;
        double r915729 = r915722 / r915728;
        return r915729;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.6
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 8.6

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020033 +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)))