Average Error: 58.7 → 0.6
Time: 7.4s
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 r80958 = 1.0;
        double r80959 = 2.0;
        double r80960 = r80958 / r80959;
        double r80961 = x;
        double r80962 = r80958 + r80961;
        double r80963 = r80958 - r80961;
        double r80964 = r80962 / r80963;
        double r80965 = log(r80964);
        double r80966 = r80960 * r80965;
        return r80966;
}
↓
double f(double x) {
        double r80967 = 1.0;
        double r80968 = 2.0;
        double r80969 = r80967 / r80968;
        double r80970 = x;
        double r80971 = fma(r80970, r80970, r80970);
        double r80972 = log(r80967);
        double r80973 = 2.0;
        double r80974 = pow(r80970, r80973);
        double r80975 = pow(r80967, r80973);
        double r80976 = r80974 / r80975;
        double r80977 = r80968 * r80976;
        double r80978 = r80972 - r80977;
        double r80979 = fma(r80971, r80968, r80978);
        double r80980 = r80969 * r80979;
        return r80980;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.7

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

  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 2020060 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))