Average Error: 58.5 → 0.2
Time: 20.7s
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 r2091219 = 1.0;
        double r2091220 = 2.0;
        double r2091221 = r2091219 / r2091220;
        double r2091222 = x;
        double r2091223 = r2091219 + r2091222;
        double r2091224 = r2091219 - r2091222;
        double r2091225 = r2091223 / r2091224;
        double r2091226 = log(r2091225);
        double r2091227 = r2091221 * r2091226;
        return r2091227;
}
↓
double f(double x) {
        double r2091228 = x;
        double r2091229 = 5.0;
        double r2091230 = pow(r2091228, r2091229);
        double r2091231 = 0.4;
        double r2091232 = 2.0;
        double r2091233 = r2091228 * r2091232;
        double r2091234 = 0.6666666666666666;
        double r2091235 = r2091234 * r2091228;
        double r2091236 = r2091235 * r2091228;
        double r2091237 = r2091228 * r2091236;
        double r2091238 = r2091233 + r2091237;
        double r2091239 = fma(r2091230, r2091231, r2091238);
        double r2091240 = 0.5;
        double r2091241 = r2091239 * r2091240;
        return r2091241;
}

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 2019162 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))