Average Error: 9.0 → 0.1
Time: 12.3s
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 r40163695 = x;
        double r40163696 = y;
        double r40163697 = r40163695 / r40163696;
        double r40163698 = 1.0;
        double r40163699 = r40163697 + r40163698;
        double r40163700 = r40163695 * r40163699;
        double r40163701 = r40163695 + r40163698;
        double r40163702 = r40163700 / r40163701;
        return r40163702;
}
↓
double f(double x, double y) {
        double r40163703 = x;
        double r40163704 = 1.0;
        double r40163705 = y;
        double r40163706 = r40163703 / r40163705;
        double r40163707 = r40163704 + r40163706;
        double r40163708 = r40163704 + r40163703;
        double r40163709 = r40163707 / r40163708;
        double r40163710 = r40163703 * r40163709;
        return r40163710;
}

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 *-un-lft-identity9.0

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

  4. Applied times-frac0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019192 
(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)))