Hyperbolic arc-(co)tangent

Average Error: 58.4 → 0.3

Time: 7.2s

Precision: binary64

Math

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

↓

\[0.5 \cdot \left(2 \cdot \mathsf{fma}\left(0.3333333333333333, {x}^{3}, x\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* 0.5 (* 2.0 (fma 0.3333333333333333 (pow x 3.0) x))))

C

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

↓

double code(double x) {
	return 0.5 * (2.0 * fma(0.3333333333333333, pow(x, 3.0), x));
}

Julia

function code(x)
	return Float64(Float64(1.0 / 2.0) * log(Float64(Float64(1.0 + x) / Float64(1.0 - x))))
end

↓

function code(x)
	return Float64(0.5 * Float64(2.0 * fma(0.3333333333333333, (x ^ 3.0), x)))
end

Wolfram

code[x_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[Log[N[(N[(1.0 + x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(0.5 * N[(2.0 * N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Derivation

1. Initial program 58.4

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

2. Simplified 50.4

   \[\leadsto \color{blue}{0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \log \left(1 - x\right)\right)} \]

3. Taylor expanded in x around 0 0.3

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

4. Simplified 0.3

   \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot \mathsf{fma}\left(0.3333333333333333, {x}^{3}, x\right)\right)} \]

5. Final simplification 0.3

   \[\leadsto 0.5 \cdot \left(2 \cdot \mathsf{fma}\left(0.3333333333333333, {x}^{3}, x\right)\right) \]

Reproduce

herbie shell --seed 2022155 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))