Average Error: 58.7 → 0.2
Time: 48.5s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(2 \cdot x + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(2 \cdot x + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r14699518 = 1.0;
        double r14699519 = 2.0;
        double r14699520 = r14699518 / r14699519;
        double r14699521 = x;
        double r14699522 = r14699518 + r14699521;
        double r14699523 = r14699518 - r14699521;
        double r14699524 = r14699522 / r14699523;
        double r14699525 = log(r14699524);
        double r14699526 = r14699520 * r14699525;
        return r14699526;
}
↓
double f(double x) {
        double r14699527 = 0.4;
        double r14699528 = x;
        double r14699529 = 5.0;
        double r14699530 = pow(r14699528, r14699529);
        double r14699531 = 2.0;
        double r14699532 = r14699531 * r14699528;
        double r14699533 = 0.6666666666666666;
        double r14699534 = r14699528 * r14699528;
        double r14699535 = r14699533 * r14699534;
        double r14699536 = r14699535 * r14699528;
        double r14699537 = r14699532 + r14699536;
        double r14699538 = fma(r14699527, r14699530, r14699537);
        double r14699539 = 0.5;
        double r14699540 = r14699538 * r14699539;
        return r14699540;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.7

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

  2. Simplified58.7

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

  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. Simplified0.2

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

  6. Applied fma-udef0.2

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

  7. Applied distribute-rgt-in0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))