Average Error: 9.2 → 0.1
Time: 24.7s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{1 + x} \cdot \left(1 + \frac{x}{y}\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{1 + x} \cdot \left(1 + \frac{x}{y}\right)
double f(double x, double y) {
        double r44504350 = x;
        double r44504351 = y;
        double r44504352 = r44504350 / r44504351;
        double r44504353 = 1.0;
        double r44504354 = r44504352 + r44504353;
        double r44504355 = r44504350 * r44504354;
        double r44504356 = r44504350 + r44504353;
        double r44504357 = r44504355 / r44504356;
        return r44504357;
}
↓
double f(double x, double y) {
        double r44504358 = x;
        double r44504359 = 1.0;
        double r44504360 = r44504359 + r44504358;
        double r44504361 = r44504358 / r44504360;
        double r44504362 = y;
        double r44504363 = r44504358 / r44504362;
        double r44504364 = r44504359 + r44504363;
        double r44504365 = r44504361 * r44504364;
        return r44504365;
}

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. 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

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

Reproduce

herbie shell --seed 2019168 
(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)))