Average Error: 9.3 → 0.1
Time: 4.7s
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 r845727 = x;
        double r845728 = y;
        double r845729 = r845727 / r845728;
        double r845730 = 1.0;
        double r845731 = r845729 + r845730;
        double r845732 = r845727 * r845731;
        double r845733 = r845727 + r845730;
        double r845734 = r845732 / r845733;
        return r845734;
}
↓
double f(double x, double y) {
        double r845735 = x;
        double r845736 = 1.0;
        double r845737 = r845735 + r845736;
        double r845738 = y;
        double r845739 = r845735 / r845738;
        double r845740 = r845739 + r845736;
        double r845741 = r845737 / r845740;
        double r845742 = r845735 / r845741;
        return r845742;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.3
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.3

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020021 
(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)))