Hyperbolic arc-(co)tangent

Average Error: 58.6 → 0.2

Time: 6.0s

Precision: binary64

Cost: 20480

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

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

Derivation

1. Initial program 58.6

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

2. Simplified 58.6

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

   [Start] 58.6	\[ \frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
   metadata-eval [=>] 58.6	\[ \color{blue}{0.5} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 58.6	\[ 0.5 \cdot \log \left(\frac{\color{blue}{x + 1}}{1 - x}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-11 [=>] 58.6	\[ 0.5 \cdot \log \left(\frac{\color{blue}{x - -1}}{1 - x}\right) \]

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. Simplified 0.2

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

   [Start] 0.2	\[ 0.5 \cdot \left(0.2857142857142857 \cdot {x}^{7} + \left(2 \cdot x + \left(0.6666666666666666 \cdot {x}^{3} + 0.4 \cdot {x}^{5}\right)\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.2	\[ 0.5 \cdot \color{blue}{\left(2 \cdot x + \left(0.2857142857142857 \cdot {x}^{7} + \left(0.6666666666666666 \cdot {x}^{3} + 0.4 \cdot {x}^{5}\right)\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.2	\[ 0.5 \cdot \left(2 \cdot x + \color{blue}{\left(\left(0.6666666666666666 \cdot {x}^{3} + 0.4 \cdot {x}^{5}\right) + 0.2857142857142857 \cdot {x}^{7}\right)}\right) \]

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	13760

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

Alternative 2
Error	0.3
Cost	7040

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

Alternative 3
Error	0.6
Cost	320

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

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))