Hyperbolic arc-(co)tangent

Average Error: 58.4 → 0.2

Time: 28.3s

Precision: binary64

Cost: 20480

Math

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

↓

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

FPCore

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

↓

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

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 = (((0.6666666666666666d0 * (x ** 3.0d0)) + (0.4d0 * (x ** 5.0d0))) + ((2.0d0 * x) + (0.2857142857142857d0 * (x ** 7.0d0)))) / 2.0d0
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 (((0.6666666666666666 * Math.pow(x, 3.0)) + (0.4 * Math.pow(x, 5.0))) + ((2.0 * x) + (0.2857142857142857 * Math.pow(x, 7.0)))) / 2.0;
}

Python

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

↓

def code(x):
	return (((0.6666666666666666 * math.pow(x, 3.0)) + (0.4 * math.pow(x, 5.0))) + ((2.0 * x) + (0.2857142857142857 * math.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(Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(0.4 * (x ^ 5.0))) + Float64(Float64(2.0 * x) + Float64(0.2857142857142857 * (x ^ 7.0)))) / 2.0)
end

MATLAB

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

↓

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

Derivation

1. Initial program 58.4

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

2. Simplified 58.5

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

   [Start] 58.4	\[ \frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
   metadata-eval [=>] 58.4	\[ \color{blue}{0.5} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
   exponential-simplify-13 [=>] 58.5	\[ \color{blue}{\log \left(\sqrt{\frac{1 + x}{1 - x}}\right)} \]
   rational_best-simplify-3 [=>] 58.5	\[ \log \left(\sqrt{\frac{\color{blue}{x + 1}}{1 - x}}\right) \]
   rational_best-simplify-21 [=>] 58.5	\[ \log \left(\sqrt{\frac{\color{blue}{x - -1}}{1 - x}}\right) \]

3. Applied egg-rr 58.4

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

4. Taylor expanded in x around 0 0.2

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

5. Simplified 0.2

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

   [Start] 0.2	\[ \frac{0.2857142857142857 \cdot {x}^{7} + \left(2 \cdot x + \left(0.6666666666666666 \cdot {x}^{3} + 0.4 \cdot {x}^{5}\right)\right)}{2} \]
   rational_best-simplify-47 [=>] 0.2	\[ \frac{\color{blue}{\left(0.6666666666666666 \cdot {x}^{3} + 0.4 \cdot {x}^{5}\right) + \left(2 \cdot x + 0.2857142857142857 \cdot {x}^{7}\right)}}{2} \]

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	20224

\[0.3333333333333333 \cdot {x}^{3} + \left(\left(x + 0.2 \cdot {x}^{5}\right) + 0.14285714285714285 \cdot {x}^{7}\right) \]

Alternative 2
Error	0.3
Cost	13504

\[0.3333333333333333 \cdot {x}^{3} + \left(0.2 \cdot {x}^{5} + x\right) \]

Alternative 3
Error	0.4
Cost	6784

\[0.3333333333333333 \cdot {x}^{3} + x \]

Alternative 4
Error	0.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023099 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))