Average Error: 4.2 → 0.1
Time: 5.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.105334618224627253660437971286073308481 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, {\left(e^{1}\right)}^{\left(\frac{1}{2} \cdot x\right)}, 1\right)}\\ \mathbf{elif}\;x \le 5.520088747928307920653874590734725838643 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.105334618224627253660437971286073308481 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, {\left(e^{1}\right)}^{\left(\frac{1}{2} \cdot x\right)}, 1\right)}\\

\mathbf{elif}\;x \le 5.520088747928307920653874590734725838643 \cdot 10^{-17}:\\
\;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\

\end{array}
double f(double x) {
        double r12322 = 2.0;
        double r12323 = x;
        double r12324 = r12322 * r12323;
        double r12325 = exp(r12324);
        double r12326 = 1.0;
        double r12327 = r12325 - r12326;
        double r12328 = exp(r12323);
        double r12329 = r12328 - r12326;
        double r12330 = r12327 / r12329;
        double r12331 = sqrt(r12330);
        return r12331;
}
↓
double f(double x) {
        double r12332 = x;
        double r12333 = -1.1053346182246273e-16;
        bool r12334 = r12332 <= r12333;
        double r12335 = 2.0;
        double r12336 = r12335 * r12332;
        double r12337 = exp(r12336);
        double r12338 = 1.0;
        double r12339 = r12337 - r12338;
        double r12340 = -r12338;
        double r12341 = r12332 + r12332;
        double r12342 = exp(r12341);
        double r12343 = fma(r12340, r12338, r12342);
        double r12344 = r12339 / r12343;
        double r12345 = exp(r12332);
        double r12346 = sqrt(r12345);
        double r12347 = 1.0;
        double r12348 = exp(r12347);
        double r12349 = 0.5;
        double r12350 = r12349 * r12332;
        double r12351 = pow(r12348, r12350);
        double r12352 = fma(r12346, r12351, r12338);
        double r12353 = r12344 * r12352;
        double r12354 = sqrt(r12353);
        double r12355 = 5.520088747928308e-17;
        bool r12356 = r12332 <= r12355;
        double r12357 = 0.5;
        double r12358 = sqrt(r12335);
        double r12359 = r12332 / r12358;
        double r12360 = r12357 * r12359;
        double r12361 = 2.0;
        double r12362 = pow(r12332, r12361);
        double r12363 = r12362 / r12358;
        double r12364 = 0.25;
        double r12365 = 0.125;
        double r12366 = r12365 / r12335;
        double r12367 = r12364 - r12366;
        double r12368 = r12363 * r12367;
        double r12369 = r12358 + r12368;
        double r12370 = r12360 + r12369;
        double r12371 = r12345 + r12338;
        double r12372 = r12344 * r12371;
        double r12373 = sqrt(r12372);
        double r12374 = r12356 ? r12370 : r12373;
        double r12375 = r12334 ? r12354 : r12374;
        return r12375;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -1.1053346182246273e-16

    1. Initial program 0.8

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

    3. Applied flip--0.6

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

    4. Applied associate-/r/0.6

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

    5. Simplified0.0

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

    7. Applied add-sqr-sqrt0.0

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

    8. Applied fma-def0.0

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

    10. Applied *-un-lft-identity0.0

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

    11. Applied exp-prod0.0

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

    12. Applied sqrt-pow10.0

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

    13. Simplified0.0

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

    if -1.1053346182246273e-16 < x < 5.520088747928308e-17

    1. Initial program 63.4

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 5.520088747928308e-17 < x

    1. Initial program 16.6

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

    3. Applied flip--12.6

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

    4. Applied associate-/r/12.6

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

    5. Simplified1.7

      \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}} \cdot \left(e^{x} + 1\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.105334618224627253660437971286073308481 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, {\left(e^{1}\right)}^{\left(\frac{1}{2} \cdot x\right)}, 1\right)}\\ \mathbf{elif}\;x \le 5.520088747928307920653874590734725838643 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))