Average Error: 9.2 → 0.1
Time: 14.3s
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 r587745 = x;
        double r587746 = y;
        double r587747 = r587745 / r587746;
        double r587748 = 1.0;
        double r587749 = r587747 + r587748;
        double r587750 = r587745 * r587749;
        double r587751 = r587745 + r587748;
        double r587752 = r587750 / r587751;
        return r587752;
}
↓
double f(double x, double y) {
        double r587753 = x;
        double r587754 = 1.0;
        double r587755 = r587753 + r587754;
        double r587756 = y;
        double r587757 = r587753 / r587756;
        double r587758 = r587757 + r587754;
        double r587759 = r587755 / r587758;
        double r587760 = r587753 / r587759;
        return r587760;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.2
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.2

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019323 +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)))