Average Error: 58.4 → 0.7
Time: 5.3s
Precision: binary64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(x \cdot \left(2 \cdot x\right) + \left(\log 1 + 2 \cdot \left(x - \frac{x}{1} \cdot \frac{x}{1}\right)\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(x \cdot \left(2 \cdot x\right) + \left(\log 1 + 2 \cdot \left(x - \frac{x}{1} \cdot \frac{x}{1}\right)\right)\right)
(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))
↓
(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 2.0)
  (+ (* x (* 2.0 x)) (+ (log 1.0) (* 2.0 (- x (* (/ x 1.0) (/ x 1.0))))))))
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) (x * ((double) (2.0 * x)))) + ((double) (((double) log(1.0)) + ((double) (2.0 * ((double) (x - ((double) ((x / 1.0) * (x / 1.0)))))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.4

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020198 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))