Average Error: 58.7 → 0.6
Time: 10.0s
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 r65494 = 1.0;
        double r65495 = 2.0;
        double r65496 = r65494 / r65495;
        double r65497 = x;
        double r65498 = r65494 + r65497;
        double r65499 = r65494 - r65497;
        double r65500 = r65498 / r65499;
        double r65501 = log(r65500);
        double r65502 = r65496 * r65501;
        return r65502;
}
↓
double f(double x) {
        double r65503 = 1.0;
        double r65504 = 2.0;
        double r65505 = r65503 / r65504;
        double r65506 = x;
        double r65507 = r65503 * r65503;
        double r65508 = r65506 / r65507;
        double r65509 = r65506 - r65508;
        double r65510 = r65506 * r65509;
        double r65511 = r65506 + r65510;
        double r65512 = r65504 * r65511;
        double r65513 = log(r65503);
        double r65514 = r65512 + r65513;
        double r65515 = r65505 * r65514;
        return r65515;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.7

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

  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 2019350 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))