Average Error: 4.5 → 0.8
Time: 25.9s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.47195806647322485446235246220325620925 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right), 2\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9.47195806647322485446235246220325620925 \cdot 10^{-6}:\\
\;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right), 2\right)}\\

\end{array}
double f(double x) {
        double r1326818 = 2.0;
        double r1326819 = x;
        double r1326820 = r1326818 * r1326819;
        double r1326821 = exp(r1326820);
        double r1326822 = 1.0;
        double r1326823 = r1326821 - r1326822;
        double r1326824 = exp(r1326819);
        double r1326825 = r1326824 - r1326822;
        double r1326826 = r1326823 / r1326825;
        double r1326827 = sqrt(r1326826);
        return r1326827;
}
↓
double f(double x) {
        double r1326828 = x;
        double r1326829 = -9.471958066473225e-06;
        bool r1326830 = r1326828 <= r1326829;
        double r1326831 = 2.0;
        double r1326832 = r1326831 * r1326828;
        double r1326833 = exp(r1326832);
        double r1326834 = sqrt(r1326833);
        double r1326835 = log(r1326834);
        double r1326836 = exp(r1326835);
        double r1326837 = 1.0;
        double r1326838 = -r1326837;
        double r1326839 = fma(r1326834, r1326836, r1326838);
        double r1326840 = exp(r1326828);
        double r1326841 = r1326840 - r1326837;
        double r1326842 = r1326839 / r1326841;
        double r1326843 = sqrt(r1326842);
        double r1326844 = 0.4999999999999998;
        double r1326845 = fma(r1326844, r1326828, r1326837);
        double r1326846 = fma(r1326828, r1326845, r1326831);
        double r1326847 = sqrt(r1326846);
        double r1326848 = r1326830 ? r1326843 : r1326847;
        return r1326848;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -9.471958066473225e-06

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied fma-neg0.0

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

    6. Applied add-exp-log0.0

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

    if -9.471958066473225e-06 < x

    1. Initial program 34.5

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

    3. Applied add-sqr-sqrt31.9

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

    4. Applied fma-neg26.5

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

    6. Applied add-exp-log26.5

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

    7. Taylor expanded around 0 6.5

      \[\leadsto \sqrt{\color{blue}{1 \cdot x + \left(0.4999999999999997779553950749686919152737 \cdot {x}^{2} + 2\right)}}\]

    8. Simplified6.5

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right), 2\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.47195806647322485446235246220325620925 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right), 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))