Average Error: 58.6 → 0.6
Time: 9.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(2 \cdot \left(x + x \cdot \left(x - \frac{x}{1 \cdot 1}\right)\right) + \log 1\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(2 \cdot \left(x + x \cdot \left(x - \frac{x}{1 \cdot 1}\right)\right) + \log 1\right)
double f(double x) {
        double r63885 = 1.0;
        double r63886 = 2.0;
        double r63887 = r63885 / r63886;
        double r63888 = x;
        double r63889 = r63885 + r63888;
        double r63890 = r63885 - r63888;
        double r63891 = r63889 / r63890;
        double r63892 = log(r63891);
        double r63893 = r63887 * r63892;
        return r63893;
}
↓
double f(double x) {
        double r63894 = 1.0;
        double r63895 = 2.0;
        double r63896 = r63894 / r63895;
        double r63897 = x;
        double r63898 = r63894 * r63894;
        double r63899 = r63897 / r63898;
        double r63900 = r63897 - r63899;
        double r63901 = r63897 * r63900;
        double r63902 = r63897 + r63901;
        double r63903 = r63895 * r63902;
        double r63904 = log(r63894);
        double r63905 = r63903 + r63904;
        double r63906 = r63896 * r63905;
        return r63906;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.6

    \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 \cdot {x}^{2} + \left(2 \cdot x + \log 1\right)\right) - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)}\]

  3. Simplified0.6

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 \cdot \left(x + x \cdot \left(x - \frac{x}{1 \cdot 1}\right)\right) + \log 1\right)}\]

  4. Final simplification0.6

    \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \left(x + x \cdot \left(x - \frac{x}{1 \cdot 1}\right)\right) + \log 1\right)\]

Reproduce

herbie shell --seed 2019351 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))