Average Error: 8.5 → 0.1
Time: 15.0s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
↓
\[\frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}\]
\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}
↓
\frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}
double f(double x, double y) {
        double r38073741 = x;
        double r38073742 = y;
        double r38073743 = r38073741 / r38073742;
        double r38073744 = 1.0;
        double r38073745 = r38073743 + r38073744;
        double r38073746 = r38073741 * r38073745;
        double r38073747 = r38073741 + r38073744;
        double r38073748 = r38073746 / r38073747;
        return r38073748;
}
↓
double f(double x, double y) {
        double r38073749 = x;
        double r38073750 = 1.0;
        double r38073751 = r38073750 + r38073749;
        double r38073752 = y;
        double r38073753 = r38073749 / r38073752;
        double r38073754 = r38073750 + r38073753;
        double r38073755 = r38073751 / r38073754;
        double r38073756 = r38073749 / r38073755;
        return r38073756;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.5
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1.0}{x + 1.0}\]

Derivation

  1. Initial program 8.5

    \[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1.0}{\frac{x}{y} + 1.0}}}\]

  4. Final simplification0.1

    \[\leadsto \frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))