Average Error: 9.3 → 0.1
Time: 14.8s
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 r570874 = x;
        double r570875 = y;
        double r570876 = r570874 / r570875;
        double r570877 = 1.0;
        double r570878 = r570876 + r570877;
        double r570879 = r570874 * r570878;
        double r570880 = r570874 + r570877;
        double r570881 = r570879 / r570880;
        return r570881;
}
↓
double f(double x, double y) {
        double r570882 = x;
        double r570883 = 1.0;
        double r570884 = r570882 + r570883;
        double r570885 = y;
        double r570886 = r570882 / r570885;
        double r570887 = r570886 + r570883;
        double r570888 = r570884 / r570887;
        double r570889 = r570882 / r570888;
        return r570889;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.3
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.3

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019212 
(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)))