Average Error: 9.2 → 0.1
Time: 13.1s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{\frac{1 + x}{1 + \frac{x}{y}}}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{\frac{1 + x}{1 + \frac{x}{y}}}
double f(double x, double y) {
        double r40695880 = x;
        double r40695881 = y;
        double r40695882 = r40695880 / r40695881;
        double r40695883 = 1.0;
        double r40695884 = r40695882 + r40695883;
        double r40695885 = r40695880 * r40695884;
        double r40695886 = r40695880 + r40695883;
        double r40695887 = r40695885 / r40695886;
        return r40695887;
}
↓
double f(double x, double y) {
        double r40695888 = x;
        double r40695889 = 1.0;
        double r40695890 = r40695889 + r40695888;
        double r40695891 = y;
        double r40695892 = r40695888 / r40695891;
        double r40695893 = r40695889 + r40695892;
        double r40695894 = r40695890 / r40695893;
        double r40695895 = r40695888 / r40695894;
        return r40695895;
}

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

    \[\leadsto \frac{x}{\frac{1 + x}{1 + \frac{x}{y}}}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))