Average Error: 58.4 → 0.2
Time: 4.6s
Precision: binary64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(0.6666666666666666 \cdot {\left(\frac{x}{1}\right)}^{3} + \left(2 \cdot x + 0.4 \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(0.6666666666666666 \cdot {\left(\frac{x}{1}\right)}^{3} + \left(2 \cdot x + 0.4 \cdot \frac{{x}^{5}}{{1}^{5}}\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)
  (+
   (* 0.6666666666666666 (pow (/ x 1.0) 3.0))
   (+ (* 2.0 x) (* 0.4 (/ (pow x 5.0) (pow 1.0 5.0)))))))
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) (0.6666666666666666 * ((double) pow((x / 1.0), 3.0)))) + ((double) (((double) (2.0 * x)) + ((double) (0.4 * (((double) pow(x, 5.0)) / ((double) pow(1.0, 5.0)))))))))));
}

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. Using strategy rm

  3. Applied log-div_binary6458.4

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

  4. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(0.6666666666666666 \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + 0.4 \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)}\]

  5. Simplified0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\frac{x}{1}\right)}^{3} + \left(x \cdot 2 + 0.4 \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)}\]

  6. Final simplification0.2

    \[\leadsto \frac{1}{2} \cdot \left(0.6666666666666666 \cdot {\left(\frac{x}{1}\right)}^{3} + \left(2 \cdot x + 0.4 \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))