Average Error: 8.7 → 0.1
Time: 7.2s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
↓
\[\frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}
↓
\frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)
double f(double x, double y) {
        double r17694690 = x;
        double r17694691 = y;
        double r17694692 = r17694690 / r17694691;
        double r17694693 = 1.0;
        double r17694694 = r17694692 + r17694693;
        double r17694695 = r17694690 * r17694694;
        double r17694696 = r17694690 + r17694693;
        double r17694697 = r17694695 / r17694696;
        return r17694697;
}
↓
double f(double x, double y) {
        double r17694698 = x;
        double r17694699 = 1.0;
        double r17694700 = r17694699 + r17694698;
        double r17694701 = r17694698 / r17694700;
        double r17694702 = y;
        double r17694703 = r17694698 / r17694702;
        double r17694704 = r17694699 + r17694703;
        double r17694705 = r17694701 * r17694704;
        return r17694705;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original8.7
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1.0}{x + 1.0}\]

Derivation

  1. Initial program 8.7

    \[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1.0}{\frac{x}{y} + 1.0}}}\]
  4. Using strategy rm

  5. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{x}{x + 1.0} \cdot \left(\frac{x}{y} + 1.0\right)}\]

  6. Final simplification0.1

    \[\leadsto \frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1) (/ (+ (/ x y) 1.0) (+ x 1.0)))

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