Hyperbolic secant

Average Error: 0.01% → 0.02%

Time: 3.7s

Precision: binary64

Cost: 19328

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = sqrt(cosh(x)) ** (-2.0d0)
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.sqrt(Math.cosh(x)), -2.0);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x)
	tmp = sqrt(cosh(x)) ^ -2.0;
end

Wolfram

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

↓

code[x_] := N[Power[N[Sqrt[N[Cosh[x], $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 0.01

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

2. Taylor expanded in x around inf 0.01

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

3. Simplified 0.01

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]
   Proof

   [Start] 0.01	\[ \frac{2}{e^{-x} + e^{x}} \]
   metadata-eval [<=] 0.01	\[ \frac{\color{blue}{1 \cdot 2}}{e^{-x} + e^{x}} \]
   associate-*l/ [<=] 0.01	\[ \color{blue}{\frac{1}{e^{-x} + e^{x}} \cdot 2} \]
   associate-/r/ [<=] 0.01	\[ \color{blue}{\frac{1}{\frac{e^{-x} + e^{x}}{2}}} \]
   +-commutative [=>] 0.01	\[ \frac{1}{\frac{\color{blue}{e^{x} + e^{-x}}}{2}} \]
   cosh-def [<=] 0.01	\[ \frac{1}{\color{blue}{\cosh x}} \]

4. Applied egg-rr 0.03

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

5. Simplified 0.02

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

   [Start] 0.03	\[ {\left(\sqrt{\cosh x}\right)}^{-1} \cdot {\left(\sqrt{\cosh x}\right)}^{-1} \]
   pow-sqr [=>] 0.02	\[ \color{blue}{{\left(\sqrt{\cosh x}\right)}^{\left(2 \cdot -1\right)}} \]
   metadata-eval [=>] 0.02	\[ {\left(\sqrt{\cosh x}\right)}^{\color{blue}{-2}} \]

6. Final simplification 0.02

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

Alternatives

Alternative 1
Error	0.01%
Cost	6592

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

Alternative 2
Error	1.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -135000000 \lor \neg \left(x \leq 36000\right):\\ \;\;\;\;\left(1 + \frac{2}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \end{array} \]

Alternative 3
Error	23.11%
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	23.42%
Cost	585

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

Alternative 5
Error	23.1%
Cost	448

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

Alternative 6
Error	49.22%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))