Average Error: 8.5 → 0.1
Time: 12.1s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
↓
\[\frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}\]
\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}
↓
\frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}
double f(double x, double y) {
        double r32036973 = x;
        double r32036974 = y;
        double r32036975 = r32036973 / r32036974;
        double r32036976 = 1.0;
        double r32036977 = r32036975 + r32036976;
        double r32036978 = r32036973 * r32036977;
        double r32036979 = r32036973 + r32036976;
        double r32036980 = r32036978 / r32036979;
        return r32036980;
}
↓
double f(double x, double y) {
        double r32036981 = x;
        double r32036982 = 1.0;
        double r32036983 = r32036982 + r32036981;
        double r32036984 = y;
        double r32036985 = r32036981 / r32036984;
        double r32036986 = r32036982 + r32036985;
        double r32036987 = r32036983 / r32036986;
        double r32036988 = r32036981 / r32036987;
        return r32036988;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.5
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1.0}{x + 1.0}\]

Derivation

  1. Initial program 8.5

    \[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1.0}{\frac{x}{y} + 1.0}}}\]

  4. Final simplification0.1

    \[\leadsto \frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1) (/ (+ (/ x y) 1.0) (+ x 1.0)))

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