Average Error: 4.4 → 0.8
Time: 24.2s
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 r23530 = 2.0;
        double r23531 = x;
        double r23532 = r23530 * r23531;
        double r23533 = exp(r23532);
        double r23534 = 1.0;
        double r23535 = r23533 - r23534;
        double r23536 = exp(r23531);
        double r23537 = r23536 - r23534;
        double r23538 = r23535 / r23537;
        double r23539 = sqrt(r23538);
        return r23539;
}
↓
double f(double x) {
        double r23540 = x;
        double r23541 = -3.356135868947112e-05;
        bool r23542 = r23540 <= r23541;
        double r23543 = 2.0;
        double r23544 = r23543 * r23540;
        double r23545 = exp(r23544);
        double r23546 = sqrt(r23545);
        double r23547 = 1.0;
        double r23548 = -r23547;
        double r23549 = fma(r23546, r23546, r23548);
        double r23550 = exp(r23540);
        double r23551 = r23550 - r23547;
        double r23552 = r23549 / r23551;
        double r23553 = sqrt(r23552);
        double r23554 = 0.5;
        double r23555 = fma(r23554, r23540, r23547);
        double r23556 = fma(r23540, r23555, r23543);
        double r23557 = sqrt(r23556);
        double r23558 = r23542 ? r23553 : r23557;
        return r23558;
}

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))))