Average Error: 8.9 → 0.1
Time: 4.2s
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 r866195 = x;
        double r866196 = y;
        double r866197 = r866195 / r866196;
        double r866198 = 1.0;
        double r866199 = r866197 + r866198;
        double r866200 = r866195 * r866199;
        double r866201 = r866195 + r866198;
        double r866202 = r866200 / r866201;
        return r866202;
}
↓
double f(double x, double y) {
        double r866203 = x;
        double r866204 = 1.0;
        double r866205 = r866203 + r866204;
        double r866206 = y;
        double r866207 = r866203 / r866206;
        double r866208 = r866207 + r866204;
        double r866209 = r866205 / r866208;
        double r866210 = r866203 / r866209;
        return r866210;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original8.9
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 8.9

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019356 
(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)))