Hyperbolic secant

Average Error: 0.0 → 0.0

Time: 2.5s

Precision: binary64

Cost: 6592

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ 1.0 (cosh x)))

C

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

↓

double code(double x) {
	return 1.0 / cosh(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 = 1.0d0 / cosh(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 1.0 / Math.cosh(x);
}

Python

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

↓

def code(x):
	return 1.0 / math.cosh(x)

Julia

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

↓

function code(x)
	return Float64(1.0 / cosh(x))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = 1.0 / cosh(x);
end

Wolfram

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]

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. Simplified 0.0

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]
   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. Final simplification 0.0

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

Alternatives

Alternative 1
Error	1.0
Cost	840

\[\begin{array}{l} t_0 := \left(1 + \frac{2}{x \cdot x}\right) + -1\\ \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	15.8
Cost	712

\[\begin{array}{l} t_0 := \frac{2}{x \cdot x}\\ \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	16.0
Cost	584

\[\begin{array}{l} t_0 := \frac{2}{x \cdot x}\\ \mathbf{if}\;x \leq -1.42:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.8
Cost	448

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

Alternative 5
Error	31.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))