Hyperbolic secant

Average Error: 0.0 → 0.0

Time: 3.1s

Precision: binary64

Cost: 19392

Math

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

↓

\[{\left({\cosh x}^{2}\right)}^{-0.5} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (pow (pow (cosh x) 2.0) -0.5))

C

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

↓

double code(double x) {
	return pow(pow(cosh(x), 2.0), -0.5);
}

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) ** 2.0d0) ** (-0.5d0)
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), 2.0), -0.5);
}

Python

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

↓

def code(x):
	return math.pow(math.pow(math.cosh(x), 2.0), -0.5)

Julia

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

↓

function code(x)
	return (cosh(x) ^ 2.0) ^ -0.5
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (cosh(x) ^ 2.0) ^ -0.5;
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], 2.0], $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \cosh x}} \]

3. Applied egg-rr 0.0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{{\cosh x}^{2}}}} \]

4. Applied egg-rr 0.0

   \[\leadsto \color{blue}{{\left({\cosh x}^{2}\right)}^{-0.5}} \]

5. Final simplification 0.0

   \[\leadsto {\left({\cosh x}^{2}\right)}^{-0.5} \]

Alternatives

Alternative 1
Error	0.0
Cost	6592

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

Alternative 2
Error	31.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))