Average Error: 58.6 → 0.7
Time: 7.1s
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 r93948 = 1.0;
        double r93949 = 2.0;
        double r93950 = r93948 / r93949;
        double r93951 = x;
        double r93952 = r93948 + r93951;
        double r93953 = r93948 - r93951;
        double r93954 = r93952 / r93953;
        double r93955 = log(r93954);
        double r93956 = r93950 * r93955;
        return r93956;
}
↓
double f(double x) {
        double r93957 = 1.0;
        double r93958 = 2.0;
        double r93959 = r93957 / r93958;
        double r93960 = x;
        double r93961 = fma(r93960, r93960, r93960);
        double r93962 = log(r93957);
        double r93963 = 2.0;
        double r93964 = pow(r93960, r93963);
        double r93965 = pow(r93957, r93963);
        double r93966 = r93964 / r93965;
        double r93967 = r93958 * r93966;
        double r93968 = r93962 - r93967;
        double r93969 = fma(r93961, r93958, r93968);
        double r93970 = r93959 * r93969;
        return r93970;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))