Average Error: 40.6 → 0.1
Time: 4.6s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
↓
\[\sqrt{\frac{1 + {\left(e^{x}\right)}^{3}}{{\left(e^{x}\right)}^{2} - \mathsf{expm1}\left(x\right)}} \]
(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))
↓
(FPCore (x)
 :precision binary64
 (sqrt (/ (+ 1.0 (pow (exp x) 3.0)) (- (pow (exp x) 2.0) (expm1 x)))))
double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}
↓
double code(double x) {
	return sqrt(((1.0 + pow(exp(x), 3.0)) / (pow(exp(x), 2.0) - expm1(x))));
}

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.sqrt(((1.0 + Math.pow(Math.exp(x), 3.0)) / (Math.pow(Math.exp(x), 2.0) - Math.expm1(x))));
}

def code(x):
	return math.sqrt(((math.exp((2.0 * x)) - 1.0) / (math.exp(x) - 1.0)))
↓
def code(x):
	return math.sqrt(((1.0 + math.pow(math.exp(x), 3.0)) / (math.pow(math.exp(x), 2.0) - math.expm1(x))))

function code(x)
	return sqrt(Float64(Float64(exp(Float64(2.0 * x)) - 1.0) / Float64(exp(x) - 1.0)))
end
↓
function code(x)
	return sqrt(Float64(Float64(1.0 + (exp(x) ^ 3.0)) / Float64((exp(x) ^ 2.0) - expm1(x))))
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[Sqrt[N[(N[(1.0 + N[Power[N[Exp[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Exp[x], $MachinePrecision], 2.0], $MachinePrecision] - N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{1 + {\left(e^{x}\right)}^{3}}{{\left(e^{x}\right)}^{2} - \mathsf{expm1}\left(x\right)}}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 40.6

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

  2. Simplified0.0

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

  3. Applied egg-rr0.1

    \[\leadsto \sqrt{\color{blue}{\left(1 + {\left(e^{x}\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(e^{x}, \mathsf{expm1}\left(x\right), 1\right)}}} \]

  4. Taylor expanded in x around inf 0.1

    \[\leadsto \sqrt{\color{blue}{\frac{1 + {\left(e^{x}\right)}^{3}}{\left(1 + {\left(e^{x}\right)}^{2}\right) - e^{x}}}} \]

  5. Simplified0.1

    \[\leadsto \sqrt{\color{blue}{\frac{1 + {\left(e^{x}\right)}^{3}}{{\left(e^{x}\right)}^{2} - \mathsf{expm1}\left(x\right)}}} \]

  6. Final simplification0.1

    \[\leadsto \sqrt{\frac{1 + {\left(e^{x}\right)}^{3}}{{\left(e^{x}\right)}^{2} - \mathsf{expm1}\left(x\right)}} \]

Reproduce

herbie shell --seed 2022146 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))