Average Error: 58.5 → 0.2
Time: 16.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left({x}^{5}, \frac{2}{5}, x \cdot 2 + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\mathsf{fma}\left({x}^{5}, \frac{2}{5}, x \cdot 2 + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r2006905 = 1.0;
        double r2006906 = 2.0;
        double r2006907 = r2006905 / r2006906;
        double r2006908 = x;
        double r2006909 = r2006905 + r2006908;
        double r2006910 = r2006905 - r2006908;
        double r2006911 = r2006909 / r2006910;
        double r2006912 = log(r2006911);
        double r2006913 = r2006907 * r2006912;
        return r2006913;
}
↓
double f(double x) {
        double r2006914 = x;
        double r2006915 = 5.0;
        double r2006916 = pow(r2006914, r2006915);
        double r2006917 = 0.4;
        double r2006918 = 2.0;
        double r2006919 = r2006914 * r2006918;
        double r2006920 = 0.6666666666666666;
        double r2006921 = r2006920 * r2006914;
        double r2006922 = r2006921 * r2006914;
        double r2006923 = r2006914 * r2006922;
        double r2006924 = r2006919 + r2006923;
        double r2006925 = fma(r2006916, r2006917, r2006924);
        double r2006926 = 0.5;
        double r2006927 = r2006925 * r2006926;
        return r2006927;
}

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

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  6. Applied fma-udef0.2

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

  7. Applied distribute-lft-in0.2

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

  8. Final simplification0.2

    \[\leadsto \mathsf{fma}\left({x}^{5}, \frac{2}{5}, x \cdot 2 + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))