Average Error: 58.5 → 0.3
Time: 19.9s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, x \cdot 2 + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, x \cdot 2 + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{2}
double f(double x) {
        double r2360565 = 1.0;
        double r2360566 = 2.0;
        double r2360567 = r2360565 / r2360566;
        double r2360568 = x;
        double r2360569 = r2360565 + r2360568;
        double r2360570 = r2360565 - r2360568;
        double r2360571 = r2360569 / r2360570;
        double r2360572 = log(r2360571);
        double r2360573 = r2360567 * r2360572;
        return r2360573;
}
↓
double f(double x) {
        double r2360574 = 0.4;
        double r2360575 = x;
        double r2360576 = 5.0;
        double r2360577 = pow(r2360575, r2360576);
        double r2360578 = 2.0;
        double r2360579 = r2360575 * r2360578;
        double r2360580 = 0.6666666666666666;
        double r2360581 = r2360575 * r2360580;
        double r2360582 = r2360581 * r2360575;
        double r2360583 = r2360582 * r2360575;
        double r2360584 = r2360579 + r2360583;
        double r2360585 = fma(r2360574, r2360577, r2360584);
        double r2360586 = 0.5;
        double r2360587 = r2360585 * r2360586;
        return r2360587;
}

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 0.3

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

  4. Simplified0.3

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

  6. Applied distribute-lft-in0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))