Average Error: 9.2 → 0.1
Time: 15.5s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{1 + x} \cdot \left(1 + \frac{x}{y}\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{1 + x} \cdot \left(1 + \frac{x}{y}\right)
double f(double x, double y) {
        double r730681 = x;
        double r730682 = y;
        double r730683 = r730681 / r730682;
        double r730684 = 1.0;
        double r730685 = r730683 + r730684;
        double r730686 = r730681 * r730685;
        double r730687 = r730681 + r730684;
        double r730688 = r730686 / r730687;
        return r730688;
}
↓
double f(double x, double y) {
        double r730689 = x;
        double r730690 = 1.0;
        double r730691 = r730690 + r730689;
        double r730692 = r730689 / r730691;
        double r730693 = y;
        double r730694 = r730689 / r730693;
        double r730695 = r730690 + r730694;
        double r730696 = r730692 * r730695;
        return r730696;
}

Error

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Bits error versus y

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Results

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Target

Original9.2
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.2

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019196 
(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)))