Average Error: 9.2 → 0.1
Time: 15.0s
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 r571776 = x;
        double r571777 = y;
        double r571778 = r571776 / r571777;
        double r571779 = 1.0;
        double r571780 = r571778 + r571779;
        double r571781 = r571776 * r571780;
        double r571782 = r571776 + r571779;
        double r571783 = r571781 / r571782;
        return r571783;
}
↓
double f(double x, double y) {
        double r571784 = x;
        double r571785 = 1.0;
        double r571786 = r571784 + r571785;
        double r571787 = y;
        double r571788 = r571784 / r571787;
        double r571789 = r571788 + r571785;
        double r571790 = r571786 / r571789;
        double r571791 = r571784 / r571790;
        return r571791;
}

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 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)))