Average Error: 9.3 → 0.1
Time: 15.3s
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 r561830 = x;
        double r561831 = y;
        double r561832 = r561830 / r561831;
        double r561833 = 1.0;
        double r561834 = r561832 + r561833;
        double r561835 = r561830 * r561834;
        double r561836 = r561830 + r561833;
        double r561837 = r561835 / r561836;
        return r561837;
}
↓
double f(double x, double y) {
        double r561838 = x;
        double r561839 = 1.0;
        double r561840 = r561838 + r561839;
        double r561841 = y;
        double r561842 = r561838 / r561841;
        double r561843 = r561842 + r561839;
        double r561844 = r561840 / r561843;
        double r561845 = r561838 / r561844;
        return r561845;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.3
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.3

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019303 
(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)))