Hyperbolic secant

Average Accuracy: 100.0% → 100.0%

Time: 1.7s

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 100.0%

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

2. Taylor expanded in x around inf 100.0%

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

3. Simplified 100.0%

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

   [Start] 100.0	\[ \frac{2}{e^{-x} + e^{x}} \]
   metadata-eval [<=] 100.0	\[ \frac{\color{blue}{1 \cdot 2}}{e^{-x} + e^{x}} \]
   associate-*l/ [<=] 100.0	\[ \color{blue}{\frac{1}{e^{-x} + e^{x}} \cdot 2} \]
   associate-/r/ [<=] 100.0	\[ \color{blue}{\frac{1}{\frac{e^{-x} + e^{x}}{2}}} \]
   +-commutative [=>] 100.0	\[ \frac{1}{\frac{\color{blue}{e^{x} + e^{-x}}}{2}} \]
   cosh-def [<=] 100.0	\[ \frac{1}{\color{blue}{\cosh x}} \]

4. Taylor expanded in x around inf 100.0%

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

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6592

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

Alternative 2
Accuracy	49.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023125 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))