Average Error: 0.0 → 0.0
Time: 1.7s
Precision: binary64
Cost: 6592
\[\frac{2}{e^{x} + e^{-x}} \]
↓
\[\frac{1}{\cosh x} \]
(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))
↓
(FPCore (x) :precision binary64 (/ 1.0 (cosh x)))
double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}
↓
double code(double x) {
	return 1.0 / cosh(x);
}

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 = 1.0d0 / cosh(x)
end function

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}
↓
public static double code(double x) {
	return 1.0 / Math.cosh(x);
}

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))
↓
def code(x):
	return 1.0 / math.cosh(x)

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end
↓
function code(x)
	return Float64(1.0 / cosh(x))
end

function tmp = code(x)
	tmp = 2.0 / (exp(x) + exp(-x));
end
↓
function tmp = code(x)
	tmp = 1.0 / cosh(x);
end

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(1.0 / N[Cosh[x], $MachinePrecision]), $MachinePrecision]

\frac{2}{e^{x} + e^{-x}}
↓
\frac{1}{\cosh x}

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
    (/.f64 1 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 (neg.f64 x)) (exp.f64 x))) 2)): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 (+.f64 (exp.f64 (neg.f64 x)) (exp.f64 x))) 2)): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 2) (+.f64 (exp.f64 (neg.f64 x)) (exp.f64 x)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite=> metadata-eval 2) (+.f64 (exp.f64 (neg.f64 x)) (exp.f64 x))): 0 points increase in error, 0 points decrease in error

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error31.1
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022340 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))