Average Error: 8.2 → 0.1
Time: 12.3s
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 r35649977 = x;
        double r35649978 = y;
        double r35649979 = r35649977 / r35649978;
        double r35649980 = 1.0;
        double r35649981 = r35649979 + r35649980;
        double r35649982 = r35649977 * r35649981;
        double r35649983 = r35649977 + r35649980;
        double r35649984 = r35649982 / r35649983;
        return r35649984;
}
↓
double f(double x, double y) {
        double r35649985 = x;
        double r35649986 = 1.0;
        double r35649987 = r35649986 + r35649985;
        double r35649988 = y;
        double r35649989 = r35649985 / r35649988;
        double r35649990 = r35649986 + r35649989;
        double r35649991 = r35649987 / r35649990;
        double r35649992 = r35649985 / r35649991;
        return r35649992;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.2
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1.0}{x + 1.0}\]

Derivation

  1. Initial program 8.2

    \[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
  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 2019158 +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)))