Average Error: 58.6 → 0.6
Time: 22.5s
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 r3167594 = 1.0;
        double r3167595 = 2.0;
        double r3167596 = r3167594 / r3167595;
        double r3167597 = x;
        double r3167598 = r3167594 + r3167597;
        double r3167599 = r3167594 - r3167597;
        double r3167600 = r3167598 / r3167599;
        double r3167601 = log(r3167600);
        double r3167602 = r3167596 * r3167601;
        return r3167602;
}
↓
double f(double x) {
        double r3167603 = 1.0;
        double r3167604 = 2.0;
        double r3167605 = r3167603 / r3167604;
        double r3167606 = x;
        double r3167607 = fma(r3167606, r3167606, r3167606);
        double r3167608 = r3167606 / r3167603;
        double r3167609 = r3167608 * r3167608;
        double r3167610 = r3167607 - r3167609;
        double r3167611 = r3167604 * r3167610;
        double r3167612 = log(r3167603);
        double r3167613 = r3167611 + r3167612;
        double r3167614 = r3167605 * r3167613;
        return r3167614;
}

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 2019170 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))