Average Error: 8.9 → 0.1
Time: 13.6s
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 r565926 = x;
        double r565927 = y;
        double r565928 = r565926 / r565927;
        double r565929 = 1.0;
        double r565930 = r565928 + r565929;
        double r565931 = r565926 * r565930;
        double r565932 = r565926 + r565929;
        double r565933 = r565931 / r565932;
        return r565933;
}
↓
double f(double x, double y) {
        double r565934 = x;
        double r565935 = 1.0;
        double r565936 = r565934 + r565935;
        double r565937 = y;
        double r565938 = r565934 / r565937;
        double r565939 = r565938 + r565935;
        double r565940 = r565936 / r565939;
        double r565941 = r565934 / r565940;
        return r565941;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original8.9
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 8.9

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019326 
(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)))