Average Error: 58.6 → 0.2
Time: 8.3s
Precision: binary64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{x \cdot 2 + \left({x}^{3} \cdot 0.6666666666666666 + 0.4 \cdot {x}^{5}\right)}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{x \cdot 2 + \left({x}^{3} \cdot 0.6666666666666666 + 0.4 \cdot {x}^{5}\right)}{2}
(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))
↓
(FPCore (x)
 :precision binary64
 (/
  (+ (* x 2.0) (+ (* (pow x 3.0) 0.6666666666666666) (* 0.4 (pow x 5.0))))
  2.0))
double code(double x) {
	return (1.0 / 2.0) * log((1.0 + x) / (1.0 - x));
}
↓
double code(double x) {
	return ((x * 2.0) + ((pow(x, 3.0) * 0.6666666666666666) + (0.4 * pow(x, 5.0)))) / 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 58.6

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

  2. Simplified58.6

    \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{1 - x}\right)}{2}}\]
  3. Using strategy rm

  4. Applied log-div_binary64_255258.6

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

  5. Simplified58.6

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

  6. Taylor expanded around 0 0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2021176 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))