Average Error: 58.5 → 0.2
Time: 3.6s
Precision: binary64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[0.5 \cdot \left(\left(x + x\right) + \left({x}^{3} \cdot 0.6666666666666666 + 0.4 \cdot {x}^{5}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
0.5 \cdot \left(\left(x + x\right) + \left({x}^{3} \cdot 0.6666666666666666 + 0.4 \cdot {x}^{5}\right)\right)
(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))
↓
(FPCore (x)
 :precision binary64
 (*
  0.5
  (+ (+ x x) (+ (* (pow x 3.0) 0.6666666666666666) (* 0.4 (pow x 5.0))))))
double code(double x) {
	return (1.0 / 2.0) * log((1.0 + x) / (1.0 - x));
}
↓
double code(double x) {
	return 0.5 * ((x + x) + ((pow(x, 3.0) * 0.6666666666666666) + (0.4 * pow(x, 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 58.5

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

  2. Simplified58.5

    \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)}\]

  3. Taylor expanded around 0 0.2

    \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot x + \left(0.6666666666666666 \cdot {x}^{3} + 0.4 \cdot {x}^{5}\right)\right)}\]

  4. Simplified0.2

    \[\leadsto 0.5 \cdot \color{blue}{\left(\left(x + x\right) + \left({x}^{3} \cdot 0.6666666666666666 + 0.4 \cdot {x}^{5}\right)\right)}\]

  5. Final simplification0.2

    \[\leadsto 0.5 \cdot \left(\left(x + x\right) + \left({x}^{3} \cdot 0.6666666666666666 + 0.4 \cdot {x}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2020253 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))