Average Error: 58.5 → 0.2
Time: 24.2s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[(\frac{2}{5} \cdot \left({x}^{5}\right) + \left(2 \cdot x + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right))_* \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
(\frac{2}{5} \cdot \left({x}^{5}\right) + \left(2 \cdot x + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right))_* \cdot \frac{1}{2}
double f(double x) {
        double r5318941 = 1.0;
        double r5318942 = 2.0;
        double r5318943 = r5318941 / r5318942;
        double r5318944 = x;
        double r5318945 = r5318941 + r5318944;
        double r5318946 = r5318941 - r5318944;
        double r5318947 = r5318945 / r5318946;
        double r5318948 = log(r5318947);
        double r5318949 = r5318943 * r5318948;
        return r5318949;
}
↓
double f(double x) {
        double r5318950 = 0.4;
        double r5318951 = x;
        double r5318952 = 5.0;
        double r5318953 = pow(r5318951, r5318952);
        double r5318954 = 2.0;
        double r5318955 = r5318954 * r5318951;
        double r5318956 = 0.6666666666666666;
        double r5318957 = r5318951 * r5318951;
        double r5318958 = r5318956 * r5318957;
        double r5318959 = r5318958 * r5318951;
        double r5318960 = r5318955 + r5318959;
        double r5318961 = fma(r5318950, r5318953, r5318960);
        double r5318962 = 0.5;
        double r5318963 = r5318961 * r5318962;
        return r5318963;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.5

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

  2. Simplified58.5

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.3

    \[\leadsto \color{blue}{(\frac{2}{5} \cdot \left({x}^{5}\right) + \left(x \cdot (\frac{2}{3} \cdot \left(x \cdot x\right) + 2)_*\right))_*} \cdot \frac{1}{2}\]
  5. Using strategy rm

  6. Applied fma-udef0.3

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

  7. Applied distribute-lft-in0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))