Hyperbolic arc-(co)tangent

Average Error: 58.7 → 0.1

Time: 6.1s

Precision: binary64

Math

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

↓

\[0.5 \cdot \mathsf{fma}\left(0.4, {x}^{5}, \mathsf{fma}\left(0.2857142857142857, {x}^{7}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right) \]

FPCore

(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))

↓

(FPCore (x)
 :precision binary64
 (*
  0.5
  (fma
   0.4
   (pow x 5.0)
   (fma
    0.2857142857142857
    (pow x 7.0)
    (fma 2.0 x (* 0.6666666666666666 (pow x 3.0)))))))

C

double code(double x) {
	return (1.0 / 2.0) * log((1.0 + x) / (1.0 - x));
}

↓

double code(double x) {
	return 0.5 * fma(0.4, pow(x, 5.0), fma(0.2857142857142857, pow(x, 7.0), fma(2.0, x, (0.6666666666666666 * pow(x, 3.0)))));
}

Error

Bits error versus x

Derivation

1. Initial program 58.7

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

2. Simplified 0.0

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

3. Taylor expanded in x around 0 0.1

   \[\leadsto 0.5 \cdot \color{blue}{\left(0.4 \cdot {x}^{5} + \left(0.2857142857142857 \cdot {x}^{7} + \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right)\right)} \]

4. Simplified 0.1

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(0.4, {x}^{5}, \mathsf{fma}\left(0.2857142857142857, {x}^{7}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)} \]

5. Final simplification 0.1

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(0.4, {x}^{5}, \mathsf{fma}\left(0.2857142857142857, {x}^{7}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right) \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))