Average Error: 58.6 → 0.6
Time: 4.7min
Precision: binary64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(\log 1 + 2 \cdot \left(x \cdot \left(x - \frac{x}{{1}^{2}}\right) + x\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(\log 1 + 2 \cdot \left(x \cdot \left(x - \frac{x}{{1}^{2}}\right) + x\right)\right)
double code(double x) {
	return ((double) ((1.0 / 2.0) * ((double) log((((double) (1.0 + x)) / ((double) (1.0 - x)))))));
}
↓
double code(double x) {
	return ((double) ((1.0 / 2.0) * ((double) (((double) log(1.0)) + ((double) (2.0 * ((double) (((double) (x * ((double) (x - (x / ((double) pow(1.0, 2.0))))))) + x))))))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))