Average Error: 9.2 → 0.1
Time: 15.0s
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 r514255 = x;
        double r514256 = y;
        double r514257 = r514255 / r514256;
        double r514258 = 1.0;
        double r514259 = r514257 + r514258;
        double r514260 = r514255 * r514259;
        double r514261 = r514255 + r514258;
        double r514262 = r514260 / r514261;
        return r514262;
}
↓
double f(double x, double y) {
        double r514263 = x;
        double r514264 = 1.0;
        double r514265 = r514263 + r514264;
        double r514266 = y;
        double r514267 = r514263 / r514266;
        double r514268 = r514267 + r514264;
        double r514269 = r514265 / r514268;
        double r514270 = r514263 / r514269;
        return r514270;
}

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 2019323 
(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)))