Average Error: 58.5 → 0.7
Time: 15.5s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{\left(\left(2 \cdot x + \log 1\right) - {x}^{2} \cdot \left(\frac{2}{{1}^{2}} - 2\right)\right) \cdot 1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{\left(\left(2 \cdot x + \log 1\right) - {x}^{2} \cdot \left(\frac{2}{{1}^{2}} - 2\right)\right) \cdot 1}{2}
double f(double x) {
        double r59867 = 1.0;
        double r59868 = 2.0;
        double r59869 = r59867 / r59868;
        double r59870 = x;
        double r59871 = r59867 + r59870;
        double r59872 = r59867 - r59870;
        double r59873 = r59871 / r59872;
        double r59874 = log(r59873);
        double r59875 = r59869 * r59874;
        return r59875;
}
↓
double f(double x) {
        double r59876 = 2.0;
        double r59877 = x;
        double r59878 = r59876 * r59877;
        double r59879 = 1.0;
        double r59880 = log(r59879);
        double r59881 = r59878 + r59880;
        double r59882 = 2.0;
        double r59883 = pow(r59877, r59882);
        double r59884 = pow(r59879, r59882);
        double r59885 = r59876 / r59884;
        double r59886 = r59885 - r59876;
        double r59887 = r59883 * r59886;
        double r59888 = r59881 - r59887;
        double r59889 = r59888 * r59879;
        double r59890 = r59889 / r59876;
        return r59890;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))