Average Error: 58.4 → 0.3
Time: 19.1s
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 r3140645 = 1.0;
        double r3140646 = 2.0;
        double r3140647 = r3140645 / r3140646;
        double r3140648 = x;
        double r3140649 = r3140645 + r3140648;
        double r3140650 = r3140645 - r3140648;
        double r3140651 = r3140649 / r3140650;
        double r3140652 = log(r3140651);
        double r3140653 = r3140647 * r3140652;
        return r3140653;
}
↓
double f(double x) {
        double r3140654 = 0.5;
        double r3140655 = x;
        double r3140656 = 5.0;
        double r3140657 = pow(r3140655, r3140656);
        double r3140658 = 0.4;
        double r3140659 = 0.6666666666666666;
        double r3140660 = r3140659 * r3140655;
        double r3140661 = 2.0;
        double r3140662 = fma(r3140660, r3140655, r3140661);
        double r3140663 = r3140662 * r3140655;
        double r3140664 = fma(r3140657, r3140658, r3140663);
        double r3140665 = r3140654 * r3140664;
        return r3140665;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.4

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

  2. Simplified58.4

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

  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

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))