Average Error: 58.6 → 0.6
Time: 17.9s
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 r68553 = 1.0;
        double r68554 = 2.0;
        double r68555 = r68553 / r68554;
        double r68556 = x;
        double r68557 = r68553 + r68556;
        double r68558 = r68553 - r68556;
        double r68559 = r68557 / r68558;
        double r68560 = log(r68559);
        double r68561 = r68555 * r68560;
        return r68561;
}
↓
double f(double x) {
        double r68562 = 1.0;
        double r68563 = 2.0;
        double r68564 = r68562 / r68563;
        double r68565 = x;
        double r68566 = r68562 * r68562;
        double r68567 = r68565 / r68566;
        double r68568 = r68565 - r68567;
        double r68569 = r68565 * r68568;
        double r68570 = r68565 + r68569;
        double r68571 = r68563 * r68570;
        double r68572 = log(r68562);
        double r68573 = r68571 + r68572;
        double r68574 = r68564 * r68573;
        return r68574;
}

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)))))