Average Error: 58.8 → 0.6
Time: 6.3s
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 r85110 = 1.0;
        double r85111 = 2.0;
        double r85112 = r85110 / r85111;
        double r85113 = x;
        double r85114 = r85110 + r85113;
        double r85115 = r85110 - r85113;
        double r85116 = r85114 / r85115;
        double r85117 = log(r85116);
        double r85118 = r85112 * r85117;
        return r85118;
}
↓
double f(double x) {
        double r85119 = 1.0;
        double r85120 = 2.0;
        double r85121 = r85119 / r85120;
        double r85122 = x;
        double r85123 = fma(r85122, r85122, r85122);
        double r85124 = log(r85119);
        double r85125 = 2.0;
        double r85126 = pow(r85122, r85125);
        double r85127 = pow(r85119, r85125);
        double r85128 = r85126 / r85127;
        double r85129 = r85120 * r85128;
        double r85130 = r85124 - r85129;
        double r85131 = fma(r85123, r85120, r85130);
        double r85132 = r85121 * r85131;
        return r85132;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.8

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

  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 2020001 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))