Average Error: 58.6 → 0.6
Time: 19.0s
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 r57697 = 1.0;
        double r57698 = 2.0;
        double r57699 = r57697 / r57698;
        double r57700 = x;
        double r57701 = r57697 + r57700;
        double r57702 = r57697 - r57700;
        double r57703 = r57701 / r57702;
        double r57704 = log(r57703);
        double r57705 = r57699 * r57704;
        return r57705;
}
↓
double f(double x) {
        double r57706 = 1.0;
        double r57707 = 2.0;
        double r57708 = r57706 / r57707;
        double r57709 = x;
        double r57710 = r57706 * r57706;
        double r57711 = r57709 / r57710;
        double r57712 = r57709 - r57711;
        double r57713 = r57709 * r57712;
        double r57714 = r57709 + r57713;
        double r57715 = r57707 * r57714;
        double r57716 = log(r57706);
        double r57717 = r57715 + r57716;
        double r57718 = r57708 * r57717;
        return r57718;
}

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 2019323 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))