Hyperbolic secant

Average Error: 0.0 → 0.2

Time: 2.9s

Precision: binary64

Cost: 6976

Math

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

↓

\[2 \cdot \left(\left(1 + \frac{0.5}{\cosh x}\right) + -1\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (* 2.0 (+ (+ 1.0 (/ 0.5 (cosh x))) -1.0)))

C

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

↓

double code(double x) {
	return 2.0 * ((1.0 + (0.5 / cosh(x))) + -1.0);
}

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 = 2.0d0 * ((1.0d0 + (0.5d0 / cosh(x))) + (-1.0d0))
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 2.0 * ((1.0 + (0.5 / Math.cosh(x))) + -1.0);
}

Python

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

↓

def code(x):
	return 2.0 * ((1.0 + (0.5 / math.cosh(x))) + -1.0)

Julia

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

↓

function code(x)
	return Float64(2.0 * Float64(Float64(1.0 + Float64(0.5 / cosh(x))) + -1.0))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = 2.0 * ((1.0 + (0.5 / cosh(x))) + -1.0);
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[(1.0 + N[(0.5 / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Applied egg-rr 0.2

   \[\leadsto 2 \cdot \color{blue}{\left(\left(1 + \frac{0.5}{\cosh x}\right) - 1\right)} \]

4. Final simplification 0.2

   \[\leadsto 2 \cdot \left(\left(1 + \frac{0.5}{\cosh x}\right) + -1\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	6720

\[2 \cdot \frac{0.5}{\cosh x} \]

Alternative 2
Error	31.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022295 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))