Average Error: 58.4 → 0.7
Time: 14.5s
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 r2416098 = 1.0;
        double r2416099 = 2.0;
        double r2416100 = r2416098 / r2416099;
        double r2416101 = x;
        double r2416102 = r2416098 + r2416101;
        double r2416103 = r2416098 - r2416101;
        double r2416104 = r2416102 / r2416103;
        double r2416105 = log(r2416104);
        double r2416106 = r2416100 * r2416105;
        return r2416106;
}
↓
double f(double x) {
        double r2416107 = 1.0;
        double r2416108 = 2.0;
        double r2416109 = r2416107 / r2416108;
        double r2416110 = x;
        double r2416111 = fma(r2416110, r2416110, r2416110);
        double r2416112 = r2416110 / r2416107;
        double r2416113 = r2416112 * r2416112;
        double r2416114 = r2416111 - r2416113;
        double r2416115 = log(r2416107);
        double r2416116 = fma(r2416108, r2416114, r2416115);
        double r2416117 = r2416109 * r2416116;
        return r2416117;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))