Average Error: 9.2 → 0.1
Time: 11.8s
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 r648776 = x;
        double r648777 = y;
        double r648778 = r648776 / r648777;
        double r648779 = 1.0;
        double r648780 = r648778 + r648779;
        double r648781 = r648776 * r648780;
        double r648782 = r648776 + r648779;
        double r648783 = r648781 / r648782;
        return r648783;
}
↓
double f(double x, double y) {
        double r648784 = x;
        double r648785 = 1.0;
        double r648786 = r648784 + r648785;
        double r648787 = y;
        double r648788 = r648784 / r648787;
        double r648789 = r648788 + r648785;
        double r648790 = r648786 / r648789;
        double r648791 = r648784 / r648790;
        return r648791;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

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 2019325 +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)))