Average Error: 4.8 → 0.8
Time: 42.4s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.998472655500366441157916730375632141659 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 2\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.998472655500366441157916730375632141659 \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(x, 0.5, 1\right), 2\right)}\\

\end{array}
double f(double x) {
        double r1658458 = 2.0;
        double r1658459 = x;
        double r1658460 = r1658458 * r1658459;
        double r1658461 = exp(r1658460);
        double r1658462 = 1.0;
        double r1658463 = r1658461 - r1658462;
        double r1658464 = exp(r1658459);
        double r1658465 = r1658464 - r1658462;
        double r1658466 = r1658463 / r1658465;
        double r1658467 = sqrt(r1658466);
        return r1658467;
}
↓
double f(double x) {
        double r1658468 = x;
        double r1658469 = -1.9984726555003664e-05;
        bool r1658470 = r1658468 <= r1658469;
        double r1658471 = 2.0;
        double r1658472 = r1658471 * r1658468;
        double r1658473 = exp(r1658472);
        double r1658474 = sqrt(r1658473);
        double r1658475 = 1.0;
        double r1658476 = -r1658475;
        double r1658477 = fma(r1658474, r1658474, r1658476);
        double r1658478 = exp(r1658468);
        double r1658479 = r1658478 - r1658475;
        double r1658480 = r1658477 / r1658479;
        double r1658481 = sqrt(r1658480);
        double r1658482 = 0.5;
        double r1658483 = fma(r1658468, r1658482, r1658475);
        double r1658484 = fma(r1658468, r1658483, r1658471);
        double r1658485 = sqrt(r1658484);
        double r1658486 = r1658470 ? r1658481 : r1658485;
        return r1658486;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -1.9984726555003664e-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 -1.9984726555003664e-05 < x

    1. Initial program 34.0

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

    2. Taylor expanded around 0 6.0

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

    3. Simplified6.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.998472655500366441157916730375632141659 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))