Average Error: 9.2 → 0.1
Time: 5.0s
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 r889883 = x;
        double r889884 = y;
        double r889885 = r889883 / r889884;
        double r889886 = 1.0;
        double r889887 = r889885 + r889886;
        double r889888 = r889883 * r889887;
        double r889889 = r889883 + r889886;
        double r889890 = r889888 / r889889;
        return r889890;
}
↓
double f(double x, double y) {
        double r889891 = x;
        double r889892 = 1.0;
        double r889893 = r889891 + r889892;
        double r889894 = y;
        double r889895 = r889891 / r889894;
        double r889896 = r889895 + r889892;
        double r889897 = r889893 / r889896;
        double r889898 = r889891 / r889897;
        return r889898;
}

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 2020056 
(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)))