Average Error: 58.5 → 0.2
Time: 17.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, x \cdot 2 + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, x \cdot 2 + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \frac{1}{2}
double f(double x) {
        double r3037273 = 1.0;
        double r3037274 = 2.0;
        double r3037275 = r3037273 / r3037274;
        double r3037276 = x;
        double r3037277 = r3037273 + r3037276;
        double r3037278 = r3037273 - r3037276;
        double r3037279 = r3037277 / r3037278;
        double r3037280 = log(r3037279);
        double r3037281 = r3037275 * r3037280;
        return r3037281;
}
↓
double f(double x) {
        double r3037282 = 0.4;
        double r3037283 = x;
        double r3037284 = 5.0;
        double r3037285 = pow(r3037283, r3037284);
        double r3037286 = 2.0;
        double r3037287 = r3037283 * r3037286;
        double r3037288 = 0.6666666666666666;
        double r3037289 = r3037283 * r3037283;
        double r3037290 = r3037288 * r3037289;
        double r3037291 = r3037290 * r3037283;
        double r3037292 = r3037287 + r3037291;
        double r3037293 = fma(r3037282, r3037285, r3037292);
        double r3037294 = 0.5;
        double r3037295 = r3037293 * r3037294;
        return r3037295;
}

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

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

  6. Applied fma-udef0.2

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

  7. Applied distribute-lft-in0.2

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

  8. Final simplification0.2

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

Reproduce

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