Hyperbolic secant

Average Accuracy: 100.0% → 100.0%

Time: 3.8s

Precision: binary64

Cost: 13184

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = 2.0d0 / (exp(x) + exp(-x))
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 2.0 / (Math.exp(x) + Math.exp(-x));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	98.8%
Cost	6976

\[\left(1 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right) + -1 \]

Alternative 2
Accuracy	99.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Accuracy	58.9%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Accuracy	99.2%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Accuracy	51.0%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))