Average Error: 58.5 → 0.7
Time: 16.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right) - \frac{{x}^{2}}{{1}^{2}}, \log 1\right)\]
\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}^{2}}{{1}^{2}}, \log 1\right)
double f(double x) {
        double r59493 = 1.0;
        double r59494 = 2.0;
        double r59495 = r59493 / r59494;
        double r59496 = x;
        double r59497 = r59493 + r59496;
        double r59498 = r59493 - r59496;
        double r59499 = r59497 / r59498;
        double r59500 = log(r59499);
        double r59501 = r59495 * r59500;
        return r59501;
}
↓
double f(double x) {
        double r59502 = 1.0;
        double r59503 = 2.0;
        double r59504 = r59502 / r59503;
        double r59505 = x;
        double r59506 = fma(r59505, r59505, r59505);
        double r59507 = 2.0;
        double r59508 = pow(r59505, r59507);
        double r59509 = pow(r59502, r59507);
        double r59510 = r59508 / r59509;
        double r59511 = r59506 - r59510;
        double r59512 = log(r59502);
        double r59513 = fma(r59503, r59511, r59512);
        double r59514 = r59504 * r59513;
        return r59514;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.5

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))