Average Error: 58.5 → 0.2
Time: 11.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.66666666666666663, {x}^{3}, 0.40000000000000002 \cdot {x}^{5}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.66666666666666663, {x}^{3}, 0.40000000000000002 \cdot {x}^{5}\right)\right)
double f(double x) {
        double r60970 = 1.0;
        double r60971 = 2.0;
        double r60972 = r60970 / r60971;
        double r60973 = x;
        double r60974 = r60970 + r60973;
        double r60975 = r60970 - r60973;
        double r60976 = r60974 / r60975;
        double r60977 = log(r60976);
        double r60978 = r60972 * r60977;
        return r60978;
}
↓
double f(double x) {
        double r60979 = 1.0;
        double r60980 = 2.0;
        double r60981 = r60979 / r60980;
        double r60982 = x;
        double r60983 = 0.6666666666666666;
        double r60984 = 3.0;
        double r60985 = pow(r60982, r60984);
        double r60986 = 0.4;
        double r60987 = 5.0;
        double r60988 = pow(r60982, r60987);
        double r60989 = r60986 * r60988;
        double r60990 = fma(r60983, r60985, r60989);
        double r60991 = fma(r60980, r60982, r60990);
        double r60992 = r60981 * r60991;
        return r60992;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.5

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

  3. Applied div-inv58.5

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

  4. Applied log-prod58.5

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

  5. Simplified58.5

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

  6. Taylor expanded around 0 0.2

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

  7. Simplified0.2

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

  8. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(0.66666666666666663 \cdot {x}^{3} + \left(0.40000000000000002 \cdot {x}^{5} + 2 \cdot x\right)\right)}\]

  9. Simplified0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.66666666666666663, {x}^{3}, 0.40000000000000002 \cdot {x}^{5}\right)\right)}\]

  10. Final simplification0.2

    \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.66666666666666663, {x}^{3}, 0.40000000000000002 \cdot {x}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))