Average Error: 9.1 → 0.1
Time: 10.8s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)
double f(double x, double y) {
        double r637289 = x;
        double r637290 = y;
        double r637291 = r637289 / r637290;
        double r637292 = 1.0;
        double r637293 = r637291 + r637292;
        double r637294 = r637289 * r637293;
        double r637295 = r637289 + r637292;
        double r637296 = r637294 / r637295;
        return r637296;
}
↓
double f(double x, double y) {
        double r637297 = x;
        double r637298 = 1.0;
        double r637299 = r637297 + r637298;
        double r637300 = r637297 / r637299;
        double r637301 = y;
        double r637302 = r637297 / r637301;
        double r637303 = r637302 + r637298;
        double r637304 = r637300 * r637303;
        return r637304;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

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

Derivation

  1. Initial program 9.1

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}}\]
  4. Using strategy rm

  5. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)}\]

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019208 
(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)))