Average Error: 4.3 → 0.9
Time: 6.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.3888208797845926 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{2 \cdot x} - 1\right)\right)}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.3888208797845926 \cdot 10^{-12}:\\
\;\;\;\;\sqrt{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{2 \cdot x} - 1\right)\right)}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\

\end{array}
double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}
↓
double code(double x) {
	double temp;
	if ((x <= -5.3888208797845926e-12)) {
		temp = sqrt((expm1(log1p((exp((2.0 * x)) - 1.0))) / (fma(-1.0, 1.0, exp((x + x))) / (exp(x) + 1.0))));
	} else {
		temp = sqrt(fma(0.5, pow(x, 2.0), fma(1.0, x, 2.0)));
	}
	return temp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -5.3888208797845926e-12

    1. Initial program 0.5

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    2. Using strategy rm

    3. Applied flip--0.3

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]

    4. Simplified0.0

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\color{blue}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{e^{x} + 1}}}\]
    5. Using strategy rm

    6. Applied expm1-log1p-u0.0

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{2 \cdot x} - 1\right)\right)}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\]

    if -5.3888208797845926e-12 < x

    1. Initial program 35.8

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

    2. Taylor expanded around 0 7.9

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]

    3. Simplified7.9

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.3888208797845926 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{2 \cdot x} - 1\right)\right)}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))