Average Error: 4.5 → 0.7
Time: 19.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.109419166854957547087282512125511104578 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.109419166854957547087282512125511104578 \cdot 10^{-10}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{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 r20631 = 2.0;
        double r20632 = x;
        double r20633 = r20631 * r20632;
        double r20634 = exp(r20633);
        double r20635 = 1.0;
        double r20636 = r20634 - r20635;
        double r20637 = exp(r20632);
        double r20638 = r20637 - r20635;
        double r20639 = r20636 / r20638;
        double r20640 = sqrt(r20639);
        return r20640;
}
↓
double f(double x) {
        double r20641 = x;
        double r20642 = -1.1094191668549575e-10;
        bool r20643 = r20641 <= r20642;
        double r20644 = 2.0;
        double r20645 = r20644 * r20641;
        double r20646 = exp(r20645);
        double r20647 = 1.0;
        double r20648 = r20646 - r20647;
        double r20649 = r20641 + r20641;
        double r20650 = exp(r20649);
        double r20651 = r20647 * r20647;
        double r20652 = r20650 - r20651;
        double r20653 = r20648 / r20652;
        double r20654 = sqrt(r20653);
        double r20655 = exp(r20641);
        double r20656 = r20655 + r20647;
        double r20657 = sqrt(r20656);
        double r20658 = r20654 * r20657;
        double r20659 = 0.5;
        double r20660 = 2.0;
        double r20661 = pow(r20641, r20660);
        double r20662 = fma(r20647, r20641, r20644);
        double r20663 = fma(r20659, r20661, r20662);
        double r20664 = sqrt(r20663);
        double r20665 = r20643 ? r20658 : r20664;
        return r20665;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -1.1094191668549575e-10

    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. Applied associate-/r/0.2

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

    5. Applied sqrt-prod0.2

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

    6. Simplified0.0

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

    if -1.1094191668549575e-10 < x

    1. Initial program 36.5

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

    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 simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.109419166854957547087282512125511104578 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))