Average Error: 9.4 → 0.1
Time: 12.0s
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 r988741 = x;
        double r988742 = y;
        double r988743 = r988741 / r988742;
        double r988744 = 1.0;
        double r988745 = r988743 + r988744;
        double r988746 = r988741 * r988745;
        double r988747 = r988741 + r988744;
        double r988748 = r988746 / r988747;
        return r988748;
}
↓
double f(double x, double y) {
        double r988749 = x;
        double r988750 = 1.0;
        double r988751 = r988749 + r988750;
        double r988752 = y;
        double r988753 = r988749 / r988752;
        double r988754 = r988753 + r988750;
        double r988755 = r988751 / r988754;
        double r988756 = r988749 / r988755;
        return r988756;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.4
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.4

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020045 +o rules:numerics
(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)))