Average Error: 9.1 → 0.1
Time: 12.2s
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 r548583 = x;
        double r548584 = y;
        double r548585 = r548583 / r548584;
        double r548586 = 1.0;
        double r548587 = r548585 + r548586;
        double r548588 = r548583 * r548587;
        double r548589 = r548583 + r548586;
        double r548590 = r548588 / r548589;
        return r548590;
}
↓
double f(double x, double y) {
        double r548591 = x;
        double r548592 = 1.0;
        double r548593 = r548591 + r548592;
        double r548594 = r548591 / r548593;
        double r548595 = y;
        double r548596 = r548591 / r548595;
        double r548597 = r548596 + r548592;
        double r548598 = r548594 * r548597;
        return r548598;
}

Error

Bits error versus x

Bits error versus y

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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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019208 +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)))