Average Error: 58.6 → 0.2
Time: 19.2s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[(\frac{2}{5} \cdot \left({x}^{5}\right) + \left(2 \cdot x + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right))_* \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
(\frac{2}{5} \cdot \left({x}^{5}\right) + \left(2 \cdot x + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right))_* \cdot \frac{1}{2}
double f(double x) {
        double r4060733 = 1.0;
        double r4060734 = 2.0;
        double r4060735 = r4060733 / r4060734;
        double r4060736 = x;
        double r4060737 = r4060733 + r4060736;
        double r4060738 = r4060733 - r4060736;
        double r4060739 = r4060737 / r4060738;
        double r4060740 = log(r4060739);
        double r4060741 = r4060735 * r4060740;
        return r4060741;
}
↓
double f(double x) {
        double r4060742 = 0.4;
        double r4060743 = x;
        double r4060744 = 5.0;
        double r4060745 = pow(r4060743, r4060744);
        double r4060746 = 2.0;
        double r4060747 = r4060746 * r4060743;
        double r4060748 = 0.6666666666666666;
        double r4060749 = r4060743 * r4060743;
        double r4060750 = r4060748 * r4060749;
        double r4060751 = r4060750 * r4060743;
        double r4060752 = r4060747 + r4060751;
        double r4060753 = fma(r4060742, r4060745, r4060752);
        double r4060754 = 0.5;
        double r4060755 = r4060753 * r4060754;
        return r4060755;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  2. Simplified58.6

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

  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

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

  6. Applied fma-udef0.2

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

  7. Applied distribute-lft-in0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))