Average Error: 58.7 → 0.6
Time: 15.9s
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 r2250238 = 1.0;
        double r2250239 = 2.0;
        double r2250240 = r2250238 / r2250239;
        double r2250241 = x;
        double r2250242 = r2250238 + r2250241;
        double r2250243 = r2250238 - r2250241;
        double r2250244 = r2250242 / r2250243;
        double r2250245 = log(r2250244);
        double r2250246 = r2250240 * r2250245;
        return r2250246;
}
↓
double f(double x) {
        double r2250247 = 1.0;
        double r2250248 = 2.0;
        double r2250249 = r2250247 / r2250248;
        double r2250250 = x;
        double r2250251 = fma(r2250250, r2250250, r2250250);
        double r2250252 = r2250250 / r2250247;
        double r2250253 = r2250252 * r2250252;
        double r2250254 = r2250251 - r2250253;
        double r2250255 = log(r2250247);
        double r2250256 = fma(r2250248, r2250254, r2250255);
        double r2250257 = r2250249 * r2250256;
        return r2250257;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.7

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

  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 2019171 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))