Average Error: 58.4 → 0.7
Time: 7.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(2 \cdot \left({x}^{2} + x\right) + \left(\log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(2 \cdot \left({x}^{2} + x\right) + \left(\log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\right)
double f(double x) {
        double r79448 = 1.0;
        double r79449 = 2.0;
        double r79450 = r79448 / r79449;
        double r79451 = x;
        double r79452 = r79448 + r79451;
        double r79453 = r79448 - r79451;
        double r79454 = r79452 / r79453;
        double r79455 = log(r79454);
        double r79456 = r79450 * r79455;
        return r79456;
}
↓
double f(double x) {
        double r79457 = 1.0;
        double r79458 = 2.0;
        double r79459 = r79457 / r79458;
        double r79460 = x;
        double r79461 = 2.0;
        double r79462 = pow(r79460, r79461);
        double r79463 = r79462 + r79460;
        double r79464 = r79458 * r79463;
        double r79465 = log(r79457);
        double r79466 = pow(r79457, r79461);
        double r79467 = r79462 / r79466;
        double r79468 = r79458 * r79467;
        double r79469 = r79465 - r79468;
        double r79470 = r79464 + r79469;
        double r79471 = r79459 * r79470;
        return r79471;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.4

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

  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

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

Reproduce

herbie shell --seed 2020100 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))