Average Error: 9.5 → 0.1
Time: 22.1s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{\frac{1 + x}{1 + \frac{x}{y}}}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{\frac{1 + x}{1 + \frac{x}{y}}}
double f(double x, double y) {
        double r45757323 = x;
        double r45757324 = y;
        double r45757325 = r45757323 / r45757324;
        double r45757326 = 1.0;
        double r45757327 = r45757325 + r45757326;
        double r45757328 = r45757323 * r45757327;
        double r45757329 = r45757323 + r45757326;
        double r45757330 = r45757328 / r45757329;
        return r45757330;
}
↓
double f(double x, double y) {
        double r45757331 = x;
        double r45757332 = 1.0;
        double r45757333 = r45757332 + r45757331;
        double r45757334 = y;
        double r45757335 = r45757331 / r45757334;
        double r45757336 = r45757332 + r45757335;
        double r45757337 = r45757333 / r45757336;
        double r45757338 = r45757331 / r45757337;
        return r45757338;
}

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. 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

    \[\leadsto \frac{x}{\frac{1 + x}{1 + \frac{x}{y}}}\]

Reproduce

herbie shell --seed 2019171 
(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)))