Average Error: 58.6 → 0.6
Time: 17.4s
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 r74467 = 1.0;
        double r74468 = 2.0;
        double r74469 = r74467 / r74468;
        double r74470 = x;
        double r74471 = r74467 + r74470;
        double r74472 = r74467 - r74470;
        double r74473 = r74471 / r74472;
        double r74474 = log(r74473);
        double r74475 = r74469 * r74474;
        return r74475;
}
↓
double f(double x) {
        double r74476 = 1.0;
        double r74477 = 2.0;
        double r74478 = r74476 / r74477;
        double r74479 = x;
        double r74480 = r74476 * r74476;
        double r74481 = r74479 / r74480;
        double r74482 = r74479 - r74481;
        double r74483 = r74479 * r74482;
        double r74484 = r74479 + r74483;
        double r74485 = r74477 * r74484;
        double r74486 = log(r74476);
        double r74487 = r74485 + r74486;
        double r74488 = r74478 * r74487;
        return r74488;
}

Error

Bits error versus x

Try it out

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 2019326 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))