Average Error: 4.3 → 0.9
Time: 6.7s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.403797957856243383586276017582772368542 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + e^{\log \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.403797957856243383586276017582772368542 \cdot 10^{-7}:\\
\;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + e^{\log \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\right)\\

\end{array}
double f(double x) {
        double r18478 = 2.0;
        double r18479 = x;
        double r18480 = r18478 * r18479;
        double r18481 = exp(r18480);
        double r18482 = 1.0;
        double r18483 = r18481 - r18482;
        double r18484 = exp(r18479);
        double r18485 = r18484 - r18482;
        double r18486 = r18483 / r18485;
        double r18487 = sqrt(r18486);
        return r18487;
}
↓
double f(double x) {
        double r18488 = x;
        double r18489 = -8.403797957856243e-07;
        bool r18490 = r18488 <= r18489;
        double r18491 = 2.0;
        double r18492 = r18491 * r18488;
        double r18493 = exp(r18492);
        double r18494 = sqrt(r18493);
        double r18495 = 1.0;
        double r18496 = sqrt(r18495);
        double r18497 = r18494 + r18496;
        double r18498 = 1.0;
        double r18499 = r18497 / r18498;
        double r18500 = sqrt(r18499);
        double r18501 = r18494 - r18496;
        double r18502 = exp(r18488);
        double r18503 = r18502 - r18495;
        double r18504 = r18501 / r18503;
        double r18505 = sqrt(r18504);
        double r18506 = r18500 * r18505;
        double r18507 = 0.5;
        double r18508 = sqrt(r18491);
        double r18509 = r18488 / r18508;
        double r18510 = r18507 * r18509;
        double r18511 = 2.0;
        double r18512 = pow(r18488, r18511);
        double r18513 = r18512 / r18508;
        double r18514 = 0.25;
        double r18515 = 0.125;
        double r18516 = r18515 / r18491;
        double r18517 = r18514 - r18516;
        double r18518 = r18513 * r18517;
        double r18519 = log(r18518);
        double r18520 = exp(r18519);
        double r18521 = r18508 + r18520;
        double r18522 = r18510 + r18521;
        double r18523 = r18490 ? r18506 : r18522;
        return r18523;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -8.403797957856243e-07

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied difference-of-squares0.0

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

    7. Applied times-frac0.0

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

    8. Applied sqrt-prod0.0

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

    if -8.403797957856243e-07 < x

    1. Initial program 35.2

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

    2. Taylor expanded around 0 6.9

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

    3. Simplified6.9

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

    5. Applied add-exp-log6.9

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

    6. Applied add-exp-log6.9

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

    7. Applied add-exp-log31.5

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

    8. Applied pow-exp31.5

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

    9. Applied div-exp31.5

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

    10. Applied prod-exp31.5

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

    11. Simplified6.9

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.403797957856243383586276017582772368542 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + e^{\log \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))