Average Error: 9.2 → 0.1
Time: 18.1s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\left(\frac{x}{y} + 1\right) \cdot \frac{x}{x + 1}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\left(\frac{x}{y} + 1\right) \cdot \frac{x}{x + 1}
double f(double x, double y) {
        double r1266906 = x;
        double r1266907 = y;
        double r1266908 = r1266906 / r1266907;
        double r1266909 = 1.0;
        double r1266910 = r1266908 + r1266909;
        double r1266911 = r1266906 * r1266910;
        double r1266912 = r1266906 + r1266909;
        double r1266913 = r1266911 / r1266912;
        return r1266913;
}
↓
double f(double x, double y) {
        double r1266914 = x;
        double r1266915 = y;
        double r1266916 = r1266914 / r1266915;
        double r1266917 = 1.0;
        double r1266918 = r1266916 + r1266917;
        double r1266919 = r1266914 + r1266917;
        double r1266920 = r1266914 / r1266919;
        double r1266921 = r1266918 * r1266920;
        return r1266921;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.2
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.2

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

  2. Simplified0.1

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

  4. Applied div-inv0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(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)))