Average Error: 9.4 → 0.1
Time: 4.0s
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 r788160 = x;
        double r788161 = y;
        double r788162 = r788160 / r788161;
        double r788163 = 1.0;
        double r788164 = r788162 + r788163;
        double r788165 = r788160 * r788164;
        double r788166 = r788160 + r788163;
        double r788167 = r788165 / r788166;
        return r788167;
}
↓
double f(double x, double y) {
        double r788168 = x;
        double r788169 = 1.0;
        double r788170 = r788168 + r788169;
        double r788171 = r788168 / r788170;
        double r788172 = y;
        double r788173 = r788168 / r788172;
        double r788174 = r788173 + r788169;
        double r788175 = r788171 * r788174;
        return r788175;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.4
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.4

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020064 
(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)))