Average Error: 9.0 → 0.1
Time: 2.8s
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 r830063 = x;
        double r830064 = y;
        double r830065 = r830063 / r830064;
        double r830066 = 1.0;
        double r830067 = r830065 + r830066;
        double r830068 = r830063 * r830067;
        double r830069 = r830063 + r830066;
        double r830070 = r830068 / r830069;
        return r830070;
}
↓
double f(double x, double y) {
        double r830071 = x;
        double r830072 = 1.0;
        double r830073 = r830071 + r830072;
        double r830074 = y;
        double r830075 = r830071 / r830074;
        double r830076 = r830075 + r830072;
        double r830077 = r830073 / r830076;
        double r830078 = r830071 / r830077;
        return r830078;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.0
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.0

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020034 
(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)))