Average Error: 58.5 → 0.2
Time: 18.2s
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 r3013423 = 1.0;
        double r3013424 = 2.0;
        double r3013425 = r3013423 / r3013424;
        double r3013426 = x;
        double r3013427 = r3013423 + r3013426;
        double r3013428 = r3013423 - r3013426;
        double r3013429 = r3013427 / r3013428;
        double r3013430 = log(r3013429);
        double r3013431 = r3013425 * r3013430;
        return r3013431;
}
↓
double f(double x) {
        double r3013432 = x;
        double r3013433 = 5.0;
        double r3013434 = pow(r3013432, r3013433);
        double r3013435 = 0.4;
        double r3013436 = 2.0;
        double r3013437 = r3013432 * r3013436;
        double r3013438 = 0.6666666666666666;
        double r3013439 = r3013438 * r3013432;
        double r3013440 = r3013439 * r3013432;
        double r3013441 = r3013432 * r3013440;
        double r3013442 = r3013437 + r3013441;
        double r3013443 = fma(r3013434, r3013435, r3013442);
        double r3013444 = 0.5;
        double r3013445 = r3013443 * r3013444;
        return r3013445;
}

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-rgt-in0.2

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

  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)))))