Average Error: 9.0 → 0.1
Time: 3.7s
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 r1192942 = x;
        double r1192943 = y;
        double r1192944 = r1192942 / r1192943;
        double r1192945 = 1.0;
        double r1192946 = r1192944 + r1192945;
        double r1192947 = r1192942 * r1192946;
        double r1192948 = r1192942 + r1192945;
        double r1192949 = r1192947 / r1192948;
        return r1192949;
}
↓
double f(double x, double y) {
        double r1192950 = x;
        double r1192951 = 1.0;
        double r1192952 = r1192950 + r1192951;
        double r1192953 = y;
        double r1192954 = r1192950 / r1192953;
        double r1192955 = r1192954 + r1192951;
        double r1192956 = r1192952 / r1192955;
        double r1192957 = r1192950 / r1192956;
        return r1192957;
}

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 2020060 +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)))