Average Error: 58.7 → 0.2
Time: 5.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{{x}^{3}}{{1}^{3}}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{{x}^{3}}{{1}^{3}}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)
double f(double x) {
        double r88051 = 1.0;
        double r88052 = 2.0;
        double r88053 = r88051 / r88052;
        double r88054 = x;
        double r88055 = r88051 + r88054;
        double r88056 = r88051 - r88054;
        double r88057 = r88055 / r88056;
        double r88058 = log(r88057);
        double r88059 = r88053 * r88058;
        return r88059;
}
↓
double f(double x) {
        double r88060 = 1.0;
        double r88061 = 2.0;
        double r88062 = r88060 / r88061;
        double r88063 = 0.6666666666666666;
        double r88064 = x;
        double r88065 = 3.0;
        double r88066 = pow(r88064, r88065);
        double r88067 = pow(r88060, r88065);
        double r88068 = r88066 / r88067;
        double r88069 = 0.4;
        double r88070 = 5.0;
        double r88071 = pow(r88064, r88070);
        double r88072 = pow(r88060, r88070);
        double r88073 = r88071 / r88072;
        double r88074 = r88069 * r88073;
        double r88075 = fma(r88061, r88064, r88074);
        double r88076 = fma(r88063, r88068, r88075);
        double r88077 = r88062 * r88076;
        return r88077;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.7

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

  3. Applied log-div58.7

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

  4. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)}\]

  5. Simplified0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{2}{3}, \frac{{x}^{3}}{{1}^{3}}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)}\]

  6. Final simplification0.2

    \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{{x}^{3}}{{1}^{3}}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))