Average Error: 58.7 → 0.2
Time: 44.5s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot \left(x \cdot \frac{2}{3}\right) + 2\right) \cdot x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot \left(x \cdot \frac{2}{3}\right) + 2\right) \cdot x\right) \cdot \frac{1}{2}
double f(double x) {
        double r6384630 = 1.0;
        double r6384631 = 2.0;
        double r6384632 = r6384630 / r6384631;
        double r6384633 = x;
        double r6384634 = r6384630 + r6384633;
        double r6384635 = r6384630 - r6384633;
        double r6384636 = r6384634 / r6384635;
        double r6384637 = log(r6384636);
        double r6384638 = r6384632 * r6384637;
        return r6384638;
}
↓
double f(double x) {
        double r6384639 = 0.4;
        double r6384640 = x;
        double r6384641 = 5.0;
        double r6384642 = pow(r6384640, r6384641);
        double r6384643 = r6384639 * r6384642;
        double r6384644 = 0.6666666666666666;
        double r6384645 = r6384640 * r6384644;
        double r6384646 = r6384640 * r6384645;
        double r6384647 = 2.0;
        double r6384648 = r6384646 + r6384647;
        double r6384649 = r6384648 * r6384640;
        double r6384650 = r6384643 + r6384649;
        double r6384651 = 0.5;
        double r6384652 = r6384650 * r6384651;
        return r6384652;
}

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. Simplified58.7

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

  3. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\left(2 \cdot x + \left(\frac{2}{3} \cdot {x}^{3} + \frac{2}{5} \cdot {x}^{5}\right)\right)} \cdot \frac{1}{2}\]

  4. Simplified0.2

    \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{2}{3} \cdot x\right) + 2\right) + {x}^{5} \cdot \frac{2}{5}\right)} \cdot \frac{1}{2}\]

  5. Final simplification0.2

    \[\leadsto \left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot \left(x \cdot \frac{2}{3}\right) + 2\right) \cdot x\right) \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))