Average Error: 4.4 → 0.8
Time: 5.4s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.38762274541548188 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9.38762274541548188 \cdot 10^{-6}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\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 f(double x) {
        double r11846 = 2.0;
        double r11847 = x;
        double r11848 = r11846 * r11847;
        double r11849 = exp(r11848);
        double r11850 = 1.0;
        double r11851 = r11849 - r11850;
        double r11852 = exp(r11847);
        double r11853 = r11852 - r11850;
        double r11854 = r11851 / r11853;
        double r11855 = sqrt(r11854);
        return r11855;
}
↓
double f(double x) {
        double r11856 = x;
        double r11857 = -9.387622745415482e-06;
        bool r11858 = r11856 <= r11857;
        double r11859 = 2.0;
        double r11860 = r11859 * r11856;
        double r11861 = exp(r11860);
        double r11862 = 1.0;
        double r11863 = r11861 - r11862;
        double r11864 = -r11862;
        double r11865 = r11856 + r11856;
        double r11866 = exp(r11865);
        double r11867 = fma(r11864, r11862, r11866);
        double r11868 = exp(r11856);
        double r11869 = r11868 + r11862;
        double r11870 = r11867 / r11869;
        double r11871 = r11863 / r11870;
        double r11872 = sqrt(r11871);
        double r11873 = 0.5;
        double r11874 = 2.0;
        double r11875 = pow(r11856, r11874);
        double r11876 = fma(r11862, r11856, r11859);
        double r11877 = fma(r11873, r11875, r11876);
        double r11878 = sqrt(r11877);
        double r11879 = r11858 ? r11872 : r11878;
        return r11879;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -9.387622745415482e-06

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    4. Simplified0.0

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

    if -9.387622745415482e-06 < x

    1. Initial program 33.3

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

    2. Taylor expanded around 0 6.3

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

    3. Simplified6.3

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.38762274541548188 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))