Average Error: 40.7 → 0.1
Time: 4.0s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
↓
\[\left|\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}\right| \]
(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))
↓
(FPCore (x) :precision binary64 (fabs (sqrt (/ (expm1 (+ x x)) (expm1 x)))))
double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}
↓
double code(double x) {
	return fabs(sqrt((expm1((x + x)) / 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.abs(Math.sqrt((Math.expm1((x + x)) / 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.fabs(math.sqrt((math.expm1((x + x)) / 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 abs(sqrt(Float64(expm1(Float64(x + x)) / 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[Abs[N[Sqrt[N[(N[(Exp[N[(x + x), $MachinePrecision]] - 1), $MachinePrecision] / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 40.7

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

  2. Simplified0.0

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

  3. Applied egg-rr39.6

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

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022166 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))