Average Error: 9.5 → 0.1
Time: 38.8s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\left(1 + \frac{x}{y}\right) \cdot \frac{x}{x + 1}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\left(1 + \frac{x}{y}\right) \cdot \frac{x}{x + 1}
double f(double x, double y) {
        double r24441734 = x;
        double r24441735 = y;
        double r24441736 = r24441734 / r24441735;
        double r24441737 = 1.0;
        double r24441738 = r24441736 + r24441737;
        double r24441739 = r24441734 * r24441738;
        double r24441740 = r24441734 + r24441737;
        double r24441741 = r24441739 / r24441740;
        return r24441741;
}
↓
double f(double x, double y) {
        double r24441742 = 1.0;
        double r24441743 = x;
        double r24441744 = y;
        double r24441745 = r24441743 / r24441744;
        double r24441746 = r24441742 + r24441745;
        double r24441747 = r24441743 + r24441742;
        double r24441748 = r24441743 / r24441747;
        double r24441749 = r24441746 * r24441748;
        return r24441749;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

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

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

Reproduce

herbie shell --seed 2019200 +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)))