Average Error: 9.6 → 0.1
Time: 13.8s
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 r544781 = x;
        double r544782 = y;
        double r544783 = r544781 / r544782;
        double r544784 = 1.0;
        double r544785 = r544783 + r544784;
        double r544786 = r544781 * r544785;
        double r544787 = r544781 + r544784;
        double r544788 = r544786 / r544787;
        return r544788;
}
↓
double f(double x, double y) {
        double r544789 = x;
        double r544790 = 1.0;
        double r544791 = r544789 + r544790;
        double r544792 = y;
        double r544793 = r544789 / r544792;
        double r544794 = r544793 + r544790;
        double r544795 = r544791 / r544794;
        double r544796 = r544789 / r544795;
        return r544796;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.6
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.6

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019304 +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)))