Average Error: 4.3 → 0.1
Time: 5.9s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.95248520591072818947935714688592719751 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}}}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}\\ \mathbf{elif}\;x \le 5.644932717281176303190925487624055737173 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.95248520591072818947935714688592719751 \cdot 10^{-11}:\\
\;\;\;\;\sqrt{\frac{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}}}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}\\

\mathbf{elif}\;x \le 5.644932717281176303190925487624055737173 \cdot 10^{-17}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\

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

\end{array}
double f(double x) {
        double r13739 = 2.0;
        double r13740 = x;
        double r13741 = r13739 * r13740;
        double r13742 = exp(r13741);
        double r13743 = 1.0;
        double r13744 = r13742 - r13743;
        double r13745 = exp(r13740);
        double r13746 = r13745 - r13743;
        double r13747 = r13744 / r13746;
        double r13748 = sqrt(r13747);
        return r13748;
}
↓
double f(double x) {
        double r13749 = x;
        double r13750 = -3.952485205910728e-11;
        bool r13751 = r13749 <= r13750;
        double r13752 = 2.0;
        double r13753 = r13752 * r13749;
        double r13754 = exp(r13753);
        double r13755 = 1.0;
        double r13756 = r13754 - r13755;
        double r13757 = -r13755;
        double r13758 = r13749 + r13749;
        double r13759 = exp(r13758);
        double r13760 = fma(r13757, r13755, r13759);
        double r13761 = exp(r13749);
        double r13762 = 3.0;
        double r13763 = pow(r13761, r13762);
        double r13764 = pow(r13755, r13762);
        double r13765 = r13763 + r13764;
        double r13766 = r13760 / r13765;
        double r13767 = r13756 / r13766;
        double r13768 = r13761 * r13761;
        double r13769 = r13755 * r13755;
        double r13770 = r13761 * r13755;
        double r13771 = r13769 - r13770;
        double r13772 = r13768 + r13771;
        double r13773 = r13767 / r13772;
        double r13774 = sqrt(r13773);
        double r13775 = 5.644932717281176e-17;
        bool r13776 = r13749 <= r13775;
        double r13777 = 0.5;
        double r13778 = 2.0;
        double r13779 = pow(r13749, r13778);
        double r13780 = fma(r13755, r13749, r13752);
        double r13781 = fma(r13777, r13779, r13780);
        double r13782 = sqrt(r13781);
        double r13783 = r13761 + r13755;
        double r13784 = r13760 / r13783;
        double r13785 = r13756 / r13784;
        double r13786 = sqrt(r13785);
        double r13787 = r13776 ? r13782 : r13786;
        double r13788 = r13751 ? r13774 : r13787;
        return r13788;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -3.952485205910728e-11

    1. Initial program 0.4

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

    3. Applied flip--0.2

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

    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 flip3-+0.0

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{\color{blue}{\frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}}}}\]

    7. Applied associate-/r/0.0

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)}}}\]

    8. Applied associate-/r*0.0

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}}}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}}\]

    if -3.952485205910728e-11 < x < 5.644932717281176e-17

    1. Initial program 52.9

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]

    if 5.644932717281176e-17 < x

    1. Initial program 17.1

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

    3. Applied flip--13.0

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

    4. Simplified1.9

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.95248520591072818947935714688592719751 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}}}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}\\ \mathbf{elif}\;x \le 5.644932717281176303190925487624055737173 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))