Average Error: 58.5 → 0.2
Time: 18.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, \left(2 + \frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, \left(2 + \frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \frac{1}{2}
double f(double x) {
        double r2690214 = 1.0;
        double r2690215 = 2.0;
        double r2690216 = r2690214 / r2690215;
        double r2690217 = x;
        double r2690218 = r2690214 + r2690217;
        double r2690219 = r2690214 - r2690217;
        double r2690220 = r2690218 / r2690219;
        double r2690221 = log(r2690220);
        double r2690222 = r2690216 * r2690221;
        return r2690222;
}
↓
double f(double x) {
        double r2690223 = 0.4;
        double r2690224 = x;
        double r2690225 = 5.0;
        double r2690226 = pow(r2690224, r2690225);
        double r2690227 = 2.0;
        double r2690228 = 0.6666666666666666;
        double r2690229 = r2690224 * r2690224;
        double r2690230 = r2690228 * r2690229;
        double r2690231 = r2690227 + r2690230;
        double r2690232 = r2690231 * r2690224;
        double r2690233 = fma(r2690223, r2690226, r2690232);
        double r2690234 = 0.5;
        double r2690235 = r2690233 * r2690234;
        return r2690235;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.5

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

  2. Simplified58.5

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))