Average Error: 58.5 → 0.3
Time: 18.5s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) + 2 \cdot x\right)\right) \cdot \frac{1}{2}\]
\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 \left(x \cdot \frac{2}{3}\right)\right) + 2 \cdot x\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r2240640 = 1.0;
        double r2240641 = 2.0;
        double r2240642 = r2240640 / r2240641;
        double r2240643 = x;
        double r2240644 = r2240640 + r2240643;
        double r2240645 = r2240640 - r2240643;
        double r2240646 = r2240644 / r2240645;
        double r2240647 = log(r2240646);
        double r2240648 = r2240642 * r2240647;
        return r2240648;
}
↓
double f(double x) {
        double r2240649 = 0.4;
        double r2240650 = x;
        double r2240651 = 5.0;
        double r2240652 = pow(r2240650, r2240651);
        double r2240653 = r2240649 * r2240652;
        double r2240654 = 0.6666666666666666;
        double r2240655 = r2240650 * r2240654;
        double r2240656 = r2240650 * r2240655;
        double r2240657 = r2240650 * r2240656;
        double r2240658 = 2.0;
        double r2240659 = r2240658 * r2240650;
        double r2240660 = r2240657 + r2240659;
        double r2240661 = r2240653 + r2240660;
        double r2240662 = 0.5;
        double r2240663 = r2240661 * r2240662;
        return r2240663;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

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

  2. Simplified58.5

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  6. Applied distribute-rgt-in0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))