Average Error: 9.4 → 0.1
Time: 11.0s
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 r1057287 = x;
        double r1057288 = y;
        double r1057289 = r1057287 / r1057288;
        double r1057290 = 1.0;
        double r1057291 = r1057289 + r1057290;
        double r1057292 = r1057287 * r1057291;
        double r1057293 = r1057287 + r1057290;
        double r1057294 = r1057292 / r1057293;
        return r1057294;
}
↓
double f(double x, double y) {
        double r1057295 = x;
        double r1057296 = 1.0;
        double r1057297 = r1057295 + r1057296;
        double r1057298 = r1057295 / r1057297;
        double r1057299 = y;
        double r1057300 = r1057295 / r1057299;
        double r1057301 = r1057300 + r1057296;
        double r1057302 = r1057298 * r1057301;
        return r1057302;
}

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. 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 2020046 
(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)))