Average Error: 58.5 → 0.6
Time: 6.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, x\right), 2, \log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, x\right), 2, \log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)
double f(double x) {
        double r54328 = 1.0;
        double r54329 = 2.0;
        double r54330 = r54328 / r54329;
        double r54331 = x;
        double r54332 = r54328 + r54331;
        double r54333 = r54328 - r54331;
        double r54334 = r54332 / r54333;
        double r54335 = log(r54334);
        double r54336 = r54330 * r54335;
        return r54336;
}
↓
double f(double x) {
        double r54337 = 1.0;
        double r54338 = 2.0;
        double r54339 = r54337 / r54338;
        double r54340 = x;
        double r54341 = fma(r54340, r54340, r54340);
        double r54342 = log(r54337);
        double r54343 = 2.0;
        double r54344 = pow(r54340, r54343);
        double r54345 = pow(r54337, r54343);
        double r54346 = r54344 / r54345;
        double r54347 = r54338 * r54346;
        double r54348 = r54342 - r54347;
        double r54349 = fma(r54341, r54338, r54348);
        double r54350 = r54339 * r54349;
        return r54350;
}

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 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))