Average Error: 58.6 → 0.2
Time: 31.9s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(2 \cdot x + \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot x\right)\right)\]
double f(double x) {
        double r10075566 = 1.0;
        double r10075567 = 2.0;
        double r10075568 = r10075566 / r10075567;
        double r10075569 = x;
        double r10075570 = r10075566 + r10075569;
        double r10075571 = r10075566 - r10075569;
        double r10075572 = r10075570 / r10075571;
        double r10075573 = log(r10075572);
        double r10075574 = r10075568 * r10075573;
        return r10075574;
}
↓
double f(double x) {
        double r10075575 = 0.5;
        double r10075576 = x;
        double r10075577 = 5.0;
        double r10075578 = pow(r10075576, r10075577);
        double r10075579 = 0.4;
        double r10075580 = r10075578 * r10075579;
        double r10075581 = 2.0;
        double r10075582 = r10075581 * r10075576;
        double r10075583 = 0.6666666666666666;
        double r10075584 = r10075576 * r10075583;
        double r10075585 = r10075576 * r10075584;
        double r10075586 = r10075585 * r10075576;
        double r10075587 = r10075582 + r10075586;
        double r10075588 = r10075580 + r10075587;
        double r10075589 = r10075575 * r10075588;
        return r10075589;
}


\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(2 \cdot x + \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot x\right)\right)

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  2. Simplified58.6

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

  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. Using strategy rm

  6. Applied distribute-lft-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019102 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))