Average Error: 58.6 → 0.2
Time: 17.7s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left({x}^{5}, \frac{2}{5}, \mathsf{fma}\left(\frac{2}{3} \cdot x, x, 2\right) \cdot x\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left({x}^{5}, \frac{2}{5}, \mathsf{fma}\left(\frac{2}{3} \cdot x, x, 2\right) \cdot x\right)
double f(double x) {
        double r3181556 = 1.0;
        double r3181557 = 2.0;
        double r3181558 = r3181556 / r3181557;
        double r3181559 = x;
        double r3181560 = r3181556 + r3181559;
        double r3181561 = r3181556 - r3181559;
        double r3181562 = r3181560 / r3181561;
        double r3181563 = log(r3181562);
        double r3181564 = r3181558 * r3181563;
        return r3181564;
}
↓
double f(double x) {
        double r3181565 = 0.5;
        double r3181566 = x;
        double r3181567 = 5.0;
        double r3181568 = pow(r3181566, r3181567);
        double r3181569 = 0.4;
        double r3181570 = 0.6666666666666666;
        double r3181571 = r3181570 * r3181566;
        double r3181572 = 2.0;
        double r3181573 = fma(r3181571, r3181566, r3181572);
        double r3181574 = r3181573 * r3181566;
        double r3181575 = fma(r3181568, r3181569, r3181574);
        double r3181576 = r3181565 * r3181575;
        return r3181576;
}

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

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))