Average Error: 40.8 → 0.0
Time: 4.4s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
↓
\[{\left({\left(\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}\right)}^{1.5}\right)}^{0.3333333333333333} \]
(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))
↓
(FPCore (x)
 :precision binary64
 (pow (pow (/ (expm1 (+ x x)) (expm1 x)) 1.5) 0.3333333333333333))
double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}
↓
double code(double x) {
	return pow(pow((expm1((x + x)) / expm1(x)), 1.5), 0.3333333333333333);
}

public static double code(double x) {
	return Math.sqrt(((Math.exp((2.0 * x)) - 1.0) / (Math.exp(x) - 1.0)));
}
↓
public static double code(double x) {
	return Math.pow(Math.pow((Math.expm1((x + x)) / Math.expm1(x)), 1.5), 0.3333333333333333);
}

def code(x):
	return math.sqrt(((math.exp((2.0 * x)) - 1.0) / (math.exp(x) - 1.0)))
↓
def code(x):
	return math.pow(math.pow((math.expm1((x + x)) / math.expm1(x)), 1.5), 0.3333333333333333)

function code(x)
	return sqrt(Float64(Float64(exp(Float64(2.0 * x)) - 1.0) / Float64(exp(x) - 1.0)))
end
↓
function code(x)
	return (Float64(expm1(Float64(x + x)) / expm1(x)) ^ 1.5) ^ 0.3333333333333333
end

code[x_] := N[Sqrt[N[(N[(N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := N[Power[N[Power[N[(N[(Exp[N[(x + x), $MachinePrecision]] - 1), $MachinePrecision] / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
{\left({\left(\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}\right)}^{1.5}\right)}^{0.3333333333333333}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 40.8

    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

  3. Applied egg-rr0.0

    \[\leadsto \color{blue}{{\left({\left(\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

  4. Final simplification0.0

    \[\leadsto {\left({\left(\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}\right)}^{1.5}\right)}^{0.3333333333333333} \]

Reproduce

herbie shell --seed 2022203 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))