Average Error: 58.6 → 0.2
Time: 17.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(2 \cdot x + \left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(2 \cdot x + \left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r1006562 = 1.0;
        double r1006563 = 2.0;
        double r1006564 = r1006562 / r1006563;
        double r1006565 = x;
        double r1006566 = r1006562 + r1006565;
        double r1006567 = r1006562 - r1006565;
        double r1006568 = r1006566 / r1006567;
        double r1006569 = log(r1006568);
        double r1006570 = r1006564 * r1006569;
        return r1006570;
}
↓
double f(double x) {
        double r1006571 = 2.0;
        double r1006572 = x;
        double r1006573 = r1006571 * r1006572;
        double r1006574 = 0.4;
        double r1006575 = 5.0;
        double r1006576 = pow(r1006572, r1006575);
        double r1006577 = r1006574 * r1006576;
        double r1006578 = r1006572 * r1006572;
        double r1006579 = 0.6666666666666666;
        double r1006580 = r1006578 * r1006579;
        double r1006581 = r1006572 * r1006580;
        double r1006582 = r1006577 + r1006581;
        double r1006583 = r1006573 + r1006582;
        double r1006584 = 0.5;
        double r1006585 = r1006583 * r1006584;
        return r1006585;
}

Error

Bits error versus x

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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. Simplified58.6

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  6. Applied distribute-rgt-in0.2

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

  7. Applied associate-+r+0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))