Average Error: 4.8 → 0.4
Time: 3.8s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.85360628086390432 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{elif}\;x \le 1.4337737389655462 \cdot 10^{-4}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\log \left(e^{e^{x} - 1}\right)}}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.85360628086390432 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\

\mathbf{elif}\;x \le 1.4337737389655462 \cdot 10^{-4}:\\
\;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\log \left(e^{e^{x} - 1}\right)}}\\

\end{array}
double code(double x) {
	return ((double) sqrt(((double) (((double) (((double) exp(((double) (2.0 * x)))) - 1.0)) / ((double) (((double) exp(x)) - 1.0))))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -2.8536062808639043e-05)) {
		VAR = ((double) sqrt(((double) (((double) (((double) (((double) sqrt(((double) exp(((double) (2.0 * x)))))) + ((double) sqrt(1.0)))) * ((double) (((double) sqrt(((double) exp(((double) (2.0 * x)))))) - ((double) sqrt(1.0)))))) / ((double) (((double) exp(x)) - 1.0))))));
	} else {
		double VAR_1;
		if ((x <= 0.00014337737389655462)) {
			VAR_1 = ((double) sqrt(((double) (((double) (x * ((double) (1.0 + ((double) (0.5 * x)))))) + 2.0))));
		} else {
			VAR_1 = ((double) sqrt(((double) (((double) (((double) exp(((double) (2.0 * x)))) - 1.0)) / ((double) log(((double) exp(((double) (((double) exp(x)) - 1.0))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -2.85360628086390432e-5

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    if -2.85360628086390432e-5 < x < 1.4337737389655462e-4

    1. Initial program 39.5

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 1.4337737389655462e-4 < x

    1. Initial program 6.3

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

    3. Applied add-log-exp6.3

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

    4. Applied add-log-exp20.0

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

    5. Applied diff-log20.2

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

    6. Simplified20.1

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\log \color{blue}{\left(e^{e^{x} - 1}\right)}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.85360628086390432 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{elif}\;x \le 1.4337737389655462 \cdot 10^{-4}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\log \left(e^{e^{x} - 1}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))