Average Error: 8.9 → 0.1
Time: 12.7s
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 r623590 = x;
        double r623591 = y;
        double r623592 = r623590 / r623591;
        double r623593 = 1.0;
        double r623594 = r623592 + r623593;
        double r623595 = r623590 * r623594;
        double r623596 = r623590 + r623593;
        double r623597 = r623595 / r623596;
        return r623597;
}
↓
double f(double x, double y) {
        double r623598 = x;
        double r623599 = 1.0;
        double r623600 = r623598 + r623599;
        double r623601 = y;
        double r623602 = r623598 / r623601;
        double r623603 = r623602 + r623599;
        double r623604 = r623600 / r623603;
        double r623605 = r623598 / r623604;
        return r623605;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

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 2019198 +o rules:numerics
(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)))