Average Error: 58.6 → 0.6
Time: 18.4s
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 r55501 = 1.0;
        double r55502 = 2.0;
        double r55503 = r55501 / r55502;
        double r55504 = x;
        double r55505 = r55501 + r55504;
        double r55506 = r55501 - r55504;
        double r55507 = r55505 / r55506;
        double r55508 = log(r55507);
        double r55509 = r55503 * r55508;
        return r55509;
}
↓
double f(double x) {
        double r55510 = 1.0;
        double r55511 = 2.0;
        double r55512 = r55510 / r55511;
        double r55513 = x;
        double r55514 = r55510 * r55510;
        double r55515 = r55513 / r55514;
        double r55516 = r55513 - r55515;
        double r55517 = r55513 * r55516;
        double r55518 = r55513 + r55517;
        double r55519 = r55511 * r55518;
        double r55520 = log(r55510);
        double r55521 = r55519 + r55520;
        double r55522 = r55512 * r55521;
        return r55522;
}

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)))))