Average Error: 58.6 → 0.6
Time: 18.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(2 \cdot \left(\mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}\right) + \log 1\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(2 \cdot \left(\mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}\right) + \log 1\right)
double f(double x) {
        double r2568158 = 1.0;
        double r2568159 = 2.0;
        double r2568160 = r2568158 / r2568159;
        double r2568161 = x;
        double r2568162 = r2568158 + r2568161;
        double r2568163 = r2568158 - r2568161;
        double r2568164 = r2568162 / r2568163;
        double r2568165 = log(r2568164);
        double r2568166 = r2568160 * r2568165;
        return r2568166;
}
↓
double f(double x) {
        double r2568167 = 1.0;
        double r2568168 = 2.0;
        double r2568169 = r2568167 / r2568168;
        double r2568170 = x;
        double r2568171 = fma(r2568170, r2568170, r2568170);
        double r2568172 = r2568170 / r2568167;
        double r2568173 = r2568172 * r2568172;
        double r2568174 = r2568171 - r2568173;
        double r2568175 = r2568168 * r2568174;
        double r2568176 = log(r2568167);
        double r2568177 = r2568175 + r2568176;
        double r2568178 = r2568169 * r2568177;
        return r2568178;
}

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

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

  4. Final simplification0.6

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

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)))))