Average Error: 9.1 → 0.1
Time: 7.2s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{y} \cdot \frac{x}{x + 1} + \frac{x}{x + 1} \cdot 1\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{y} \cdot \frac{x}{x + 1} + \frac{x}{x + 1} \cdot 1
double f(double x, double y) {
        double r1413049 = x;
        double r1413050 = y;
        double r1413051 = r1413049 / r1413050;
        double r1413052 = 1.0;
        double r1413053 = r1413051 + r1413052;
        double r1413054 = r1413049 * r1413053;
        double r1413055 = r1413049 + r1413052;
        double r1413056 = r1413054 / r1413055;
        return r1413056;
}
↓
double f(double x, double y) {
        double r1413057 = x;
        double r1413058 = y;
        double r1413059 = r1413057 / r1413058;
        double r1413060 = 1.0;
        double r1413061 = r1413057 + r1413060;
        double r1413062 = r1413057 / r1413061;
        double r1413063 = r1413059 * r1413062;
        double r1413064 = r1413062 * r1413060;
        double r1413065 = r1413063 + r1413064;
        return r1413065;
}

Error

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Results

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Target

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

Derivation

  1. Initial program 9.1

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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. Using strategy rm

  7. Applied distribute-lft-in0.1

    \[\leadsto \color{blue}{\frac{x}{x + 1} \cdot \frac{x}{y} + \frac{x}{x + 1} \cdot 1}\]

  8. Simplified0.1

    \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{x + 1}} + \frac{x}{x + 1} \cdot 1\]

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 
(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)))