Average Error: 58.6 → 0.6
Time: 6.4s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, x\right), 2, \log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, x\right), 2, \log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)
double f(double x) {
        double r58062 = 1.0;
        double r58063 = 2.0;
        double r58064 = r58062 / r58063;
        double r58065 = x;
        double r58066 = r58062 + r58065;
        double r58067 = r58062 - r58065;
        double r58068 = r58066 / r58067;
        double r58069 = log(r58068);
        double r58070 = r58064 * r58069;
        return r58070;
}
↓
double f(double x) {
        double r58071 = 1.0;
        double r58072 = 2.0;
        double r58073 = r58071 / r58072;
        double r58074 = x;
        double r58075 = fma(r58074, r58074, r58074);
        double r58076 = log(r58071);
        double r58077 = 2.0;
        double r58078 = pow(r58074, r58077);
        double r58079 = pow(r58071, r58077);
        double r58080 = r58078 / r58079;
        double r58081 = r58072 * r58080;
        double r58082 = r58076 - r58081;
        double r58083 = fma(r58075, r58072, r58082);
        double r58084 = r58073 * r58083;
        return r58084;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))