Average Error: 9.2 → 0.1
Time: 8.0s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}
double f(double x, double y) {
        double r877742 = x;
        double r877743 = y;
        double r877744 = r877742 / r877743;
        double r877745 = 1.0;
        double r877746 = r877744 + r877745;
        double r877747 = r877742 * r877746;
        double r877748 = r877742 + r877745;
        double r877749 = r877747 / r877748;
        return r877749;
}
↓
double f(double x, double y) {
        double r877750 = x;
        double r877751 = 1.0;
        double r877752 = r877750 + r877751;
        double r877753 = y;
        double r877754 = r877750 / r877753;
        double r877755 = r877754 + r877751;
        double r877756 = r877752 / r877755;
        double r877757 = r877750 / r877756;
        return r877757;
}

Error

Bits error versus x

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. Using strategy rm

  3. Applied associate-/l*0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020042 
(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)))