Average Error: 9.4 → 0.1
Time: 11.9s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[x \cdot \frac{1 + \frac{x}{y}}{1 + x}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
x \cdot \frac{1 + \frac{x}{y}}{1 + x}
double f(double x, double y) {
        double r694975 = x;
        double r694976 = y;
        double r694977 = r694975 / r694976;
        double r694978 = 1.0;
        double r694979 = r694977 + r694978;
        double r694980 = r694975 * r694979;
        double r694981 = r694975 + r694978;
        double r694982 = r694980 / r694981;
        return r694982;
}
↓
double f(double x, double y) {
        double r694983 = x;
        double r694984 = 1.0;
        double r694985 = y;
        double r694986 = r694983 / r694985;
        double r694987 = r694984 + r694986;
        double r694988 = r694984 + r694983;
        double r694989 = r694987 / r694988;
        double r694990 = r694983 * r694989;
        return r694990;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.4
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.4

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\left(\frac{x}{y} + 1\right) \cdot \frac{x}{1 + x}}\]
  3. Using strategy rm

  4. Applied clear-num0.2

    \[\leadsto \left(\frac{x}{y} + 1\right) \cdot \color{blue}{\frac{1}{\frac{1 + x}{x}}}\]
  5. Using strategy rm

  6. Applied associate-/r/0.1

    \[\leadsto \left(\frac{x}{y} + 1\right) \cdot \color{blue}{\left(\frac{1}{1 + x} \cdot x\right)}\]

  7. Applied associate-*r*0.2

    \[\leadsto \color{blue}{\left(\left(\frac{x}{y} + 1\right) \cdot \frac{1}{1 + x}\right) \cdot x}\]

  8. Simplified0.1

    \[\leadsto \color{blue}{\frac{\frac{x}{y} + 1}{1 + x}} \cdot x\]

  9. Final simplification0.1

    \[\leadsto x \cdot \frac{1 + \frac{x}{y}}{1 + x}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))