Average Error: 4.6 → 0.7
Time: 24.7s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.52185052166229348137156751295945711862 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \mathsf{hypot}\left(\sqrt{e^{x}}, \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{2}}{\sqrt{2}}, 0.25 - \frac{0.125}{2}, \mathsf{fma}\left(\frac{x}{\sqrt{2}}, 0.5, \sqrt{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.52185052166229348137156751295945711862 \cdot 10^{-17}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \mathsf{hypot}\left(\sqrt{e^{x}}, \sqrt{1}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{{x}^{2}}{\sqrt{2}}, 0.25 - \frac{0.125}{2}, \mathsf{fma}\left(\frac{x}{\sqrt{2}}, 0.5, \sqrt{2}\right)\right)\\

\end{array}
double f(double x) {
        double r24220 = 2.0;
        double r24221 = x;
        double r24222 = r24220 * r24221;
        double r24223 = exp(r24222);
        double r24224 = 1.0;
        double r24225 = r24223 - r24224;
        double r24226 = exp(r24221);
        double r24227 = r24226 - r24224;
        double r24228 = r24225 / r24227;
        double r24229 = sqrt(r24228);
        return r24229;
}
↓
double f(double x) {
        double r24230 = x;
        double r24231 = -5.5218505216622935e-17;
        bool r24232 = r24230 <= r24231;
        double r24233 = 2.0;
        double r24234 = r24233 * r24230;
        double r24235 = exp(r24234);
        double r24236 = 1.0;
        double r24237 = r24235 - r24236;
        double r24238 = r24230 + r24230;
        double r24239 = exp(r24238);
        double r24240 = r24236 * r24236;
        double r24241 = r24239 - r24240;
        double r24242 = r24237 / r24241;
        double r24243 = sqrt(r24242);
        double r24244 = exp(r24230);
        double r24245 = sqrt(r24244);
        double r24246 = sqrt(r24236);
        double r24247 = hypot(r24245, r24246);
        double r24248 = r24243 * r24247;
        double r24249 = 2.0;
        double r24250 = pow(r24230, r24249);
        double r24251 = sqrt(r24233);
        double r24252 = r24250 / r24251;
        double r24253 = 0.25;
        double r24254 = 0.125;
        double r24255 = r24254 / r24233;
        double r24256 = r24253 - r24255;
        double r24257 = r24230 / r24251;
        double r24258 = 0.5;
        double r24259 = fma(r24257, r24258, r24251);
        double r24260 = fma(r24252, r24256, r24259);
        double r24261 = r24232 ? r24248 : r24260;
        return r24261;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -5.5218505216622935e-17

    1. Initial program 0.9

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

    3. Applied flip--0.7

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

    4. Applied associate-/r/0.7

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

    5. Applied sqrt-prod0.7

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

    6. Simplified0.0

      \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}}} \cdot \sqrt{e^{x} + 1}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt0.0

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

    9. Applied add-sqr-sqrt0.0

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

    10. Applied hypot-def0.0

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

    if -5.5218505216622935e-17 < x

    1. Initial program 37.9

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

    2. Taylor expanded around 0 7.3

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]

    3. Simplified7.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{\sqrt{2}}, 0.25 - \frac{0.125}{2}, \mathsf{fma}\left(\frac{x}{\sqrt{2}}, 0.5, \sqrt{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.52185052166229348137156751295945711862 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \mathsf{hypot}\left(\sqrt{e^{x}}, \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{2}}{\sqrt{2}}, 0.25 - \frac{0.125}{2}, \mathsf{fma}\left(\frac{x}{\sqrt{2}}, 0.5, \sqrt{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))