Hyperbolic secant

Average Error: 0.0 → 0.0

Time: 2.8s

Precision: binary64

Cost: 13248

Math

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

↓

\[\frac{2}{e^{x} + \frac{1}{e^{x}}} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (/ 1.0 (exp x)))))

C

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

↓

double code(double x) {
	return 2.0 / (exp(x) + (1.0 / 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) + (1.0d0 / 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) + (1.0 / 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) + (1.0 / 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) + Float64(1.0 / exp(x))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = 2.0 / (exp(x) + (1.0 / 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[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.5

   \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \cosh x\right)\right)}} \]

3. Taylor expanded in x around inf 0.0

   \[\leadsto \frac{2}{\color{blue}{\frac{1}{e^{x}} + e^{x}}} \]

4. Final simplification 0.0

   \[\leadsto \frac{2}{e^{x} + \frac{1}{e^{x}}} \]

Alternatives

Alternative 1
Error	0.0
Cost	6720

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

Alternative 2
Error	31.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))