Average Error: 58.5 → 0.7
Time: 5.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left({x}^{2}, 2, \mathsf{fma}\left(2, x, \log 1\right) - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left({x}^{2}, 2, \mathsf{fma}\left(2, x, \log 1\right) - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)
double code(double x) {
	return ((double) (((double) (1.0 / 2.0)) * ((double) log(((double) (((double) (1.0 + x)) / ((double) (1.0 - x))))))));
}
↓
double code(double x) {
	return ((double) (((double) (1.0 / 2.0)) * ((double) fma(((double) pow(x, 2.0)), 2.0, ((double) (((double) fma(2.0, x, ((double) log(1.0)))) - ((double) (2.0 * ((double) (((double) pow(x, 2.0)) / ((double) pow(1.0, 2.0))))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

    \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({x}^{2}, 2, \mathsf{fma}\left(2, x, \log 1\right) - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))