Average Error: 58.6 → 0.7
Time: 15.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{\mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}, \log 1\right)}{\frac{2}{1}}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{\mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}, \log 1\right)}{\frac{2}{1}}
double f(double x) {
        double r106185 = 1.0;
        double r106186 = 2.0;
        double r106187 = r106185 / r106186;
        double r106188 = x;
        double r106189 = r106185 + r106188;
        double r106190 = r106185 - r106188;
        double r106191 = r106189 / r106190;
        double r106192 = log(r106191);
        double r106193 = r106187 * r106192;
        return r106193;
}
↓
double f(double x) {
        double r106194 = 2.0;
        double r106195 = x;
        double r106196 = fma(r106195, r106195, r106195);
        double r106197 = 1.0;
        double r106198 = r106195 / r106197;
        double r106199 = r106198 * r106198;
        double r106200 = r106196 - r106199;
        double r106201 = log(r106197);
        double r106202 = fma(r106194, r106200, r106201);
        double r106203 = r106194 / r106197;
        double r106204 = r106202 / r106203;
        return r106204;
}

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

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))