Average Error: 58.6 → 0.1
Time: 4.5s
Precision: binary64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
↓
\[0.5 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)\right) \]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
0.5 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)\right)

(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))
↓
(FPCore (x)
 :precision binary64
 (* 0.5 (expm1 (log1p (- (log1p x) (log1p (- x)))))))
double code(double x) {
	return (1.0 / 2.0) * log((1.0 + x) / (1.0 - x));
}
↓
double code(double x) {
	return 0.5 * expm1(log1p(log1p(x) - log1p(-x)));
}

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. Simplified0.0

    \[\leadsto \color{blue}{0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]

  3. Applied expm1-log1p-u_binary640.1

    \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)\right)} \]

  4. Final simplification0.1

    \[\leadsto 0.5 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)\right) \]

Reproduce

herbie shell --seed 2021352 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))