Average Error: 9.0 → 0.1
Time: 2.5s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{x + 1} \cdot \frac{x}{y} + \frac{x}{x + 1} \cdot 1\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{x + 1} \cdot \frac{x}{y} + \frac{x}{x + 1} \cdot 1
double f(double x, double y) {
        double r778896 = x;
        double r778897 = y;
        double r778898 = r778896 / r778897;
        double r778899 = 1.0;
        double r778900 = r778898 + r778899;
        double r778901 = r778896 * r778900;
        double r778902 = r778896 + r778899;
        double r778903 = r778901 / r778902;
        return r778903;
}
↓
double f(double x, double y) {
        double r778904 = x;
        double r778905 = 1.0;
        double r778906 = r778904 + r778905;
        double r778907 = r778904 / r778906;
        double r778908 = y;
        double r778909 = r778904 / r778908;
        double r778910 = r778907 * r778909;
        double r778911 = r778907 * r778905;
        double r778912 = r778910 + r778911;
        return r778912;
}

Error

Bits error versus x

Bits error versus y

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Results

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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. Using strategy rm

  5. Applied associate-/r/0.1

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

  7. Applied distribute-lft-in0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020062 +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)))