Hyperbolic arc-(co)tangent

Average Error: 58.5 → 0.2

Time: 5.2s

Precision: binary64

Cost: 20480

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / 2.0d0) * log(((1.0d0 + x) / (1.0d0 - x)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * ((0.2857142857142857d0 * (x ** 7.0d0)) + ((x * 2.0d0) + ((0.6666666666666666d0 * (x ** 3.0d0)) + (0.4d0 * (x ** 5.0d0)))))
end function

Java

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

↓

public static double code(double x) {
	return 0.5 * ((0.2857142857142857 * Math.pow(x, 7.0)) + ((x * 2.0) + ((0.6666666666666666 * Math.pow(x, 3.0)) + (0.4 * Math.pow(x, 5.0)))));
}

Python

def code(x):
	return (1.0 / 2.0) * math.log(((1.0 + x) / (1.0 - x)))

↓

def code(x):
	return 0.5 * ((0.2857142857142857 * math.pow(x, 7.0)) + ((x * 2.0) + ((0.6666666666666666 * math.pow(x, 3.0)) + (0.4 * math.pow(x, 5.0)))))

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(0.5 * Float64(Float64(0.2857142857142857 * (x ^ 7.0)) + Float64(Float64(x * 2.0) + Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(0.4 * (x ^ 5.0))))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
end

↓

function tmp = code(x)
	tmp = 0.5 * ((0.2857142857142857 * (x ^ 7.0)) + ((x * 2.0) + ((0.6666666666666666 * (x ^ 3.0)) + (0.4 * (x ^ 5.0)))));
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[(0.5 * N[(N[(0.2857142857142857 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] + N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.4 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.5

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

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 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.0
Cost	13184

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

Alternative 2
Error	0.4
Cost	6976

\[0.5 \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right) \]

Alternative 3
Error	0.4
Cost	832

\[0.5 \cdot \left(x \cdot 2 + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right) \]

Alternative 4
Error	0.7
Cost	320

\[0.5 \cdot \left(x \cdot 2\right) \]

Reproduce

herbie shell --seed 2022225 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))