Average Error: 9.5 → 0.1
Time: 3.6s
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 r997932 = x;
        double r997933 = y;
        double r997934 = r997932 / r997933;
        double r997935 = 1.0;
        double r997936 = r997934 + r997935;
        double r997937 = r997932 * r997936;
        double r997938 = r997932 + r997935;
        double r997939 = r997937 / r997938;
        return r997939;
}
↓
double f(double x, double y) {
        double r997940 = x;
        double r997941 = 1.0;
        double r997942 = r997940 + r997941;
        double r997943 = y;
        double r997944 = r997940 / r997943;
        double r997945 = r997944 + r997941;
        double r997946 = r997942 / r997945;
        double r997947 = r997940 / r997946;
        return r997947;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.5
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.5

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019353 +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)))