Average Error: 58.7 → 0.5
Time: 6.4s
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 r48119 = 1.0;
        double r48120 = 2.0;
        double r48121 = r48119 / r48120;
        double r48122 = x;
        double r48123 = r48119 + r48122;
        double r48124 = r48119 - r48122;
        double r48125 = r48123 / r48124;
        double r48126 = log(r48125);
        double r48127 = r48121 * r48126;
        return r48127;
}
↓
double f(double x) {
        double r48128 = 1.0;
        double r48129 = 2.0;
        double r48130 = r48128 / r48129;
        double r48131 = x;
        double r48132 = fma(r48131, r48131, r48131);
        double r48133 = log(r48128);
        double r48134 = 2.0;
        double r48135 = pow(r48131, r48134);
        double r48136 = pow(r48128, r48134);
        double r48137 = r48135 / r48136;
        double r48138 = r48129 * r48137;
        double r48139 = r48133 - r48138;
        double r48140 = fma(r48132, r48129, r48139);
        double r48141 = r48130 * r48140;
        return r48141;
}

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 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))