Average Error: 0.0 → 0.1
Time: 2.8s
Precision: binary64
Cost: 19328
\[\frac{2}{e^{x} + e^{-x}} \]
↓
\[\sqrt{{\cosh x}^{-2}} \]
(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))
↓
(FPCore (x) :precision binary64 (sqrt (pow (cosh x) -2.0)))
double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}
↓
double code(double x) {
	return sqrt(pow(cosh(x), -2.0));
}

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

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}
↓
public static double code(double x) {
	return Math.sqrt(Math.pow(Math.cosh(x), -2.0));
}

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))
↓
def code(x):
	return math.sqrt(math.pow(math.cosh(x), -2.0))

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end
↓
function code(x)
	return sqrt((cosh(x) ^ -2.0))
end

function tmp = code(x)
	tmp = 2.0 / (exp(x) + exp(-x));
end
↓
function tmp = code(x)
	tmp = sqrt((cosh(x) ^ -2.0));
end

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

\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{{\cosh x}^{-2}}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded in x around inf 0.0

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

  3. Simplified0.0

    \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]
    Proof
    (/.f64 1 (cosh.f64 x)): 0 points increase in error, 0 points decrease in error
    (/.f64 1 (Rewrite=> cosh-def_binary64 (/.f64 (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) 2))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) 2)): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 2) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite=> metadata-eval 2) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
    (/.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 (neg.f64 x)) (exp.f64 x)))): 0 points increase in error, 0 points decrease in error

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

    \[\leadsto \sqrt{{\cosh x}^{-2}} \]

Alternatives

Alternative 1
Error0.0
Cost6592
\[\frac{1}{\cosh x} \]
Alternative 2
Error0.9
Cost840
\[\begin{array}{l} t_0 := \left(1 + \frac{2}{x \cdot x}\right) + -1\\ \mathbf{if}\;x \leq -1450000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1750000:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error15.0
Cost712
\[\begin{array}{l} t_0 := \frac{2}{x \cdot x}\\ \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error15.1
Cost584
\[\begin{array}{l} t_0 := \frac{2}{x \cdot x}\\ \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error14.9
Cost448
\[\frac{2}{2 + x \cdot x} \]
Alternative 6
Error30.8
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022335 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))