Average Error: 58.6 → 0.6
Time: 15.2s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}, \log 1\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}, \log 1\right)
double f(double x) {
        double r2114061 = 1.0;
        double r2114062 = 2.0;
        double r2114063 = r2114061 / r2114062;
        double r2114064 = x;
        double r2114065 = r2114061 + r2114064;
        double r2114066 = r2114061 - r2114064;
        double r2114067 = r2114065 / r2114066;
        double r2114068 = log(r2114067);
        double r2114069 = r2114063 * r2114068;
        return r2114069;
}
↓
double f(double x) {
        double r2114070 = 1.0;
        double r2114071 = 2.0;
        double r2114072 = r2114070 / r2114071;
        double r2114073 = x;
        double r2114074 = fma(r2114073, r2114073, r2114073);
        double r2114075 = r2114073 / r2114070;
        double r2114076 = r2114075 * r2114075;
        double r2114077 = r2114074 - r2114076;
        double r2114078 = log(r2114070);
        double r2114079 = fma(r2114071, r2114077, r2114078);
        double r2114080 = r2114072 * r2114079;
        return r2114080;
}

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

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))