Average Error: 58.6 → 0.6
Time: 19.7s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\log 1 + \left(\left(x + x \cdot x\right) - \frac{x \cdot x}{1 \cdot 1}\right) \cdot 2\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\log 1 + \left(\left(x + x \cdot x\right) - \frac{x \cdot x}{1 \cdot 1}\right) \cdot 2\right) \cdot \frac{1}{2}
double f(double x) {
        double r3162818 = 1.0;
        double r3162819 = 2.0;
        double r3162820 = r3162818 / r3162819;
        double r3162821 = x;
        double r3162822 = r3162818 + r3162821;
        double r3162823 = r3162818 - r3162821;
        double r3162824 = r3162822 / r3162823;
        double r3162825 = log(r3162824);
        double r3162826 = r3162820 * r3162825;
        return r3162826;
}
↓
double f(double x) {
        double r3162827 = 1.0;
        double r3162828 = log(r3162827);
        double r3162829 = x;
        double r3162830 = r3162829 * r3162829;
        double r3162831 = r3162829 + r3162830;
        double r3162832 = r3162827 * r3162827;
        double r3162833 = r3162830 / r3162832;
        double r3162834 = r3162831 - r3162833;
        double r3162835 = 2.0;
        double r3162836 = r3162834 * r3162835;
        double r3162837 = r3162828 + r3162836;
        double r3162838 = r3162827 / r3162835;
        double r3162839 = r3162837 * r3162838;
        return r3162839;
}

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

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019170 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))