Average Error: 9.5 → 0.1
Time: 3.5s
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 r963155 = x;
        double r963156 = y;
        double r963157 = r963155 / r963156;
        double r963158 = 1.0;
        double r963159 = r963157 + r963158;
        double r963160 = r963155 * r963159;
        double r963161 = r963155 + r963158;
        double r963162 = r963160 / r963161;
        return r963162;
}
↓
double f(double x, double y) {
        double r963163 = x;
        double r963164 = 1.0;
        double r963165 = r963163 + r963164;
        double r963166 = r963163 / r963165;
        double r963167 = y;
        double r963168 = r963163 / r963167;
        double r963169 = r963168 + r963164;
        double r963170 = r963166 * r963169;
        return r963170;
}

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. 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 2020081 +o rules:numerics
(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)))