Hyperbolic secant

Average Error: 0.0 → 0.1

Time: 3.7s

Precision: binary64

Cost: 25920

Math

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

↓

\[\sqrt[3]{{\left({\left(\frac{1}{\cosh x}\right)}^{1.5}\right)}^{2}} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (cbrt (pow (pow (/ 1.0 (cosh x)) 1.5) 2.0)))

C

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

↓

double code(double x) {
	return cbrt(pow(pow((1.0 / cosh(x)), 1.5), 2.0));
}

Java

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

↓

public static double code(double x) {
	return Math.cbrt(Math.pow(Math.pow((1.0 / Math.cosh(x)), 1.5), 2.0));
}

Julia

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

↓

function code(x)
	return cbrt(((Float64(1.0 / cosh(x)) ^ 1.5) ^ 2.0))
end

Wolfram

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

↓

code[x_] := N[Power[N[Power[N[Power[N[(1.0 / N[Cosh[x], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{2}{2 \cdot \cosh x} \cdot \frac{2}{2 \cdot \cosh x}\right) \cdot \frac{2}{2 \cdot \cosh x}}} \]

3. Applied egg-rr 0.1

   \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(\frac{1}{\cosh x}\right)}^{1.5}\right)}^{2}}} \]

4. Final simplification 0.1

   \[\leadsto \sqrt[3]{{\left({\left(\frac{1}{\cosh x}\right)}^{1.5}\right)}^{2}} \]

Alternatives

Alternative 1
Error	0.0
Cost	19392

\[{\left({\cosh x}^{-0.5}\right)}^{2} \]

Alternative 2
Error	0.1
Cost	19328

\[\sqrt[3]{{\cosh x}^{-3}} \]

Alternative 3
Error	0.0
Cost	6592

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

Alternative 4
Error	31.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023016 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))