Average Error: 9.0 → 0.1
Time: 2.3s
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 r818840 = x;
        double r818841 = y;
        double r818842 = r818840 / r818841;
        double r818843 = 1.0;
        double r818844 = r818842 + r818843;
        double r818845 = r818840 * r818844;
        double r818846 = r818840 + r818843;
        double r818847 = r818845 / r818846;
        return r818847;
}
↓
double f(double x, double y) {
        double r818848 = x;
        double r818849 = 1.0;
        double r818850 = r818848 + r818849;
        double r818851 = r818848 / r818850;
        double r818852 = y;
        double r818853 = r818848 / r818852;
        double r818854 = r818851 * r818853;
        double r818855 = r818851 * r818849;
        double r818856 = r818854 + r818855;
        return r818856;
}

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 
(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)))