Average Error: 58.5 → 0.2
Time: 21.5s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, 2 \cdot x + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, 2 \cdot x + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{2}
double f(double x) {
        double r2260046 = 1.0;
        double r2260047 = 2.0;
        double r2260048 = r2260046 / r2260047;
        double r2260049 = x;
        double r2260050 = r2260046 + r2260049;
        double r2260051 = r2260046 - r2260049;
        double r2260052 = r2260050 / r2260051;
        double r2260053 = log(r2260052);
        double r2260054 = r2260048 * r2260053;
        return r2260054;
}
↓
double f(double x) {
        double r2260055 = 0.4;
        double r2260056 = x;
        double r2260057 = 5.0;
        double r2260058 = pow(r2260056, r2260057);
        double r2260059 = 2.0;
        double r2260060 = r2260059 * r2260056;
        double r2260061 = 0.6666666666666666;
        double r2260062 = r2260056 * r2260061;
        double r2260063 = r2260062 * r2260056;
        double r2260064 = r2260063 * r2260056;
        double r2260065 = r2260060 + r2260064;
        double r2260066 = fma(r2260055, r2260058, r2260065);
        double r2260067 = 0.5;
        double r2260068 = r2260066 * r2260067;
        return r2260068;
}

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

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

  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

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

  6. Applied distribute-lft-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))