Hyperbolic arc-(co)tangent

Average Error: 58.5 → 0.2

Time: 52.8s

Precision: binary64

Cost: 26752

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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(Float64(fma(2.0, x, Float64(0.6666666666666666 * (x ^ 3.0))) + Float64(Float64(0.4 * (x ^ 5.0)) + Float64(0.2857142857142857 * (x ^ 7.0)))) / 2.0)
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[(N[(N[(2.0 * x + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.4 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.2857142857142857 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 58.5

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

2. Simplified 58.5

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

3. Taylor expanded in x around 0 0.2

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

4. Applied egg-rr 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	20480

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

Alternative 2
Error	0.3
Cost	13760

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

Alternative 3
Error	0.4
Cost	7040

\[\frac{2 \cdot x + 0.6666666666666666 \cdot {x}^{3}}{2} \]

Alternative 4
Error	0.7
Cost	320

\[\frac{x \cdot 2}{2} \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))