Hyperbolic arc-(co)tangent

Average Error: 58.7 → 0.2

Time: 7.3s

Precision: binary64

Cost: 13504

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+ (* 0.3333333333333333 (pow x 3.0)) (+ (* 0.2 (pow x 5.0)) x)))

C

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

↓

double code(double x) {
	return (0.3333333333333333 * pow(x, 3.0)) + ((0.2 * pow(x, 5.0)) + x);
}

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 = (0.3333333333333333d0 * (x ** 3.0d0)) + ((0.2d0 * (x ** 5.0d0)) + x)
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 (0.3333333333333333 * Math.pow(x, 3.0)) + ((0.2 * Math.pow(x, 5.0)) + x);
}

Python

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

↓

def code(x):
	return (0.3333333333333333 * math.pow(x, 3.0)) + ((0.2 * math.pow(x, 5.0)) + 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(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(Float64(0.2 * (x ^ 5.0)) + x))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (0.3333333333333333 * (x ^ 3.0)) + ((0.2 * (x ^ 5.0)) + 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[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.7

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

2. Simplified 58.7

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

   [Start] 58.7	\[ \frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
   metadata-eval [=>] 58.7	\[ \color{blue}{0.5} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
   exponential-simplify-13 [=>] 58.7	\[ \color{blue}{\log \left(\sqrt{\frac{1 + x}{1 - x}}\right)} \]

3. Taylor expanded in x around 0 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	6784

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

Alternative 2
Error	0.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023096 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))