Average Error: 58.5 → 0.7
Time: 5.6s
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 r87404 = 1.0;
        double r87405 = 2.0;
        double r87406 = r87404 / r87405;
        double r87407 = x;
        double r87408 = r87404 + r87407;
        double r87409 = r87404 - r87407;
        double r87410 = r87408 / r87409;
        double r87411 = log(r87410);
        double r87412 = r87406 * r87411;
        return r87412;
}
↓
double f(double x) {
        double r87413 = 1.0;
        double r87414 = 2.0;
        double r87415 = r87413 / r87414;
        double r87416 = x;
        double r87417 = fma(r87416, r87416, r87416);
        double r87418 = log(r87413);
        double r87419 = 2.0;
        double r87420 = pow(r87416, r87419);
        double r87421 = pow(r87413, r87419);
        double r87422 = r87420 / r87421;
        double r87423 = r87414 * r87422;
        double r87424 = r87418 - r87423;
        double r87425 = fma(r87417, r87414, r87424);
        double r87426 = r87415 * r87425;
        return r87426;
}

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

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))