Average Error: 4.4 → 0.8
Time: 24.1s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.356135868947112289167877818840679537971 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, 1\right), 2\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.356135868947112289167877818840679537971 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, 1\right), 2\right)}\\

\end{array}
double f(double x) {
        double r23492 = 2.0;
        double r23493 = x;
        double r23494 = r23492 * r23493;
        double r23495 = exp(r23494);
        double r23496 = 1.0;
        double r23497 = r23495 - r23496;
        double r23498 = exp(r23493);
        double r23499 = r23498 - r23496;
        double r23500 = r23497 / r23499;
        double r23501 = sqrt(r23500);
        return r23501;
}
↓
double f(double x) {
        double r23502 = x;
        double r23503 = -3.356135868947112e-05;
        bool r23504 = r23502 <= r23503;
        double r23505 = 2.0;
        double r23506 = r23505 * r23502;
        double r23507 = exp(r23506);
        double r23508 = sqrt(r23507);
        double r23509 = 1.0;
        double r23510 = -r23509;
        double r23511 = fma(r23508, r23508, r23510);
        double r23512 = exp(r23502);
        double r23513 = r23512 - r23509;
        double r23514 = r23511 / r23513;
        double r23515 = sqrt(r23514);
        double r23516 = 0.5;
        double r23517 = fma(r23516, r23502, r23509);
        double r23518 = fma(r23502, r23517, r23505);
        double r23519 = sqrt(r23518);
        double r23520 = r23504 ? r23515 : r23519;
        return r23520;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3.356135868947112e-05

    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

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

    if -3.356135868947112e-05 < x

    1. Initial program 34.2

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

    2. Taylor expanded around 0 6.2

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

    3. Simplified6.2

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.356135868947112289167877818840679537971 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, 1\right), 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))