Average Error: 4.6 → 0.7
Time: 27.6s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.52185052166229348137156751295945711862 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \mathsf{fma}\left(\frac{x}{\sqrt{2}}, 0.5, \sqrt{2}\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.52185052166229348137156751295945711862 \cdot 10^{-17}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \mathsf{fma}\left(\frac{x}{\sqrt{2}}, 0.5, \sqrt{2}\right)\\

\end{array}
double f(double x) {
        double r24831 = 2.0;
        double r24832 = x;
        double r24833 = r24831 * r24832;
        double r24834 = exp(r24833);
        double r24835 = 1.0;
        double r24836 = r24834 - r24835;
        double r24837 = exp(r24832);
        double r24838 = r24837 - r24835;
        double r24839 = r24836 / r24838;
        double r24840 = sqrt(r24839);
        return r24840;
}
↓
double f(double x) {
        double r24841 = x;
        double r24842 = -5.5218505216622935e-17;
        bool r24843 = r24841 <= r24842;
        double r24844 = 2.0;
        double r24845 = r24844 * r24841;
        double r24846 = exp(r24845);
        double r24847 = 1.0;
        double r24848 = r24846 - r24847;
        double r24849 = r24841 + r24841;
        double r24850 = exp(r24849);
        double r24851 = r24847 * r24847;
        double r24852 = r24850 - r24851;
        double r24853 = r24848 / r24852;
        double r24854 = exp(r24841);
        double r24855 = sqrt(r24854);
        double r24856 = fma(r24855, r24855, r24847);
        double r24857 = r24853 * r24856;
        double r24858 = sqrt(r24857);
        double r24859 = 2.0;
        double r24860 = pow(r24841, r24859);
        double r24861 = sqrt(r24844);
        double r24862 = r24860 / r24861;
        double r24863 = 0.25;
        double r24864 = 0.125;
        double r24865 = r24864 / r24844;
        double r24866 = r24863 - r24865;
        double r24867 = r24862 * r24866;
        double r24868 = r24841 / r24861;
        double r24869 = 0.5;
        double r24870 = fma(r24868, r24869, r24861);
        double r24871 = r24867 + r24870;
        double r24872 = r24843 ? r24858 : r24871;
        return r24872;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -5.5218505216622935e-17

    1. Initial program 0.9

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

    3. Applied flip--0.7

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

    4. Applied associate-/r/0.7

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

    5. Simplified0.0

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

    7. Applied add-sqr-sqrt0.0

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

    8. Applied fma-def0.0

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

    if -5.5218505216622935e-17 < x

    1. Initial program 37.9

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

    3. Applied flip--35.3

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

    4. Applied associate-/r/35.3

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

    5. Simplified28.3

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

    6. Taylor expanded around 0 7.3

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]

    7. Simplified7.3

      \[\leadsto \color{blue}{\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \mathsf{fma}\left(\frac{x}{\sqrt{2}}, 0.5, \sqrt{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.52185052166229348137156751295945711862 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \mathsf{fma}\left(\frac{x}{\sqrt{2}}, 0.5, \sqrt{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))