Average Error: 9.2 → 0.1
Time: 11.1s
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 r626188 = x;
        double r626189 = y;
        double r626190 = r626188 / r626189;
        double r626191 = 1.0;
        double r626192 = r626190 + r626191;
        double r626193 = r626188 * r626192;
        double r626194 = r626188 + r626191;
        double r626195 = r626193 / r626194;
        return r626195;
}
↓
double f(double x, double y) {
        double r626196 = x;
        double r626197 = 1.0;
        double r626198 = r626196 + r626197;
        double r626199 = y;
        double r626200 = r626196 / r626199;
        double r626201 = r626200 + r626197;
        double r626202 = r626198 / r626201;
        double r626203 = r626196 / r626202;
        return r626203;
}

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 2019325 
(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)))