Hyperbolic secant

Average Error: 0.0 → 0.0

Time: 4.9s

Precision: binary64

Cost: 19392

Math

\[\frac{2}{e^{x} + e^{-x}} \]

↓

\[{\left({\cosh x}^{-0.3333333333333333}\right)}^{3} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

↓

(FPCore (x) :precision binary64 (pow (pow (cosh x) -0.3333333333333333) 3.0))

C

double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}

↓

double code(double x) {
	return pow(pow(cosh(x), -0.3333333333333333), 3.0);
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (cosh(x) ** (-0.3333333333333333d0)) ** 3.0d0
end function

Java

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

↓

public static double code(double x) {
	return Math.pow(Math.pow(Math.cosh(x), -0.3333333333333333), 3.0);
}

Python

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))

↓

def code(x):
	return math.pow(math.pow(math.cosh(x), -0.3333333333333333), 3.0)

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end

↓

function code(x)
	return (cosh(x) ^ -0.3333333333333333) ^ 3.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + exp(-x));
end

↓

function tmp = code(x)
	tmp = (cosh(x) ^ -0.3333333333333333) ^ 3.0;
end

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[Power[N[Power[N[Cosh[x], $MachinePrecision], -0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 0.0

   \[\frac{2}{e^{x} + e^{-x}} \]

2. Applied egg-rr 0.0

   \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{2}{2 \cdot \cosh x}} \cdot \sqrt[3]{\frac{2}{2 \cdot \cosh x}}\right) \cdot \sqrt[3]{\frac{2}{2 \cdot \cosh x}}} \]

3. Applied egg-rr 0.0

   \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{\cosh x}}\right)}^{3}} \]

4. Applied egg-rr 0.0

   \[\leadsto {\color{blue}{\left({\cosh x}^{-0.3333333333333333}\right)}}^{3} \]

5. Final simplification 0.0

   \[\leadsto {\left({\cosh x}^{-0.3333333333333333}\right)}^{3} \]

Alternatives

Alternative 1
Error	0.0
Cost	6592

\[\frac{1}{\cosh x} \]

Alternative 2
Error	1.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -135000000 \lor \neg \left(x \leq 126000\right):\\ \;\;\;\;\left(1 + \frac{2}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \end{array} \]

Alternative 3
Error	15.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;\frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \]

Alternative 4
Error	15.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;\frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	15.3
Cost	448

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

Alternative 6
Error	31.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023067 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))