Average Error: 9.5 → 0.1
Time: 12.0s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\left(1 + \frac{x}{y}\right) \cdot \left(\frac{1}{1 + x} \cdot x\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\left(1 + \frac{x}{y}\right) \cdot \left(\frac{1}{1 + x} \cdot x\right)
double f(double x, double y) {
        double r721278 = x;
        double r721279 = y;
        double r721280 = r721278 / r721279;
        double r721281 = 1.0;
        double r721282 = r721280 + r721281;
        double r721283 = r721278 * r721282;
        double r721284 = r721278 + r721281;
        double r721285 = r721283 / r721284;
        return r721285;
}
↓
double f(double x, double y) {
        double r721286 = 1.0;
        double r721287 = x;
        double r721288 = y;
        double r721289 = r721287 / r721288;
        double r721290 = r721286 + r721289;
        double r721291 = 1.0;
        double r721292 = r721286 + r721287;
        double r721293 = r721291 / r721292;
        double r721294 = r721293 * r721287;
        double r721295 = r721290 * r721294;
        return r721295;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.5
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.5

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\left(\frac{x}{y} + 1\right) \cdot \frac{x}{1 + x}}\]
  3. Using strategy rm

  4. Applied div-inv0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))