Average Error: 4.4 → 0.7
Time: 6.4s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.5283208540140548 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.5283208540140548 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;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)\\

\end{array}
double f(double x) {
        double r18427 = 2.0;
        double r18428 = x;
        double r18429 = r18427 * r18428;
        double r18430 = exp(r18429);
        double r18431 = 1.0;
        double r18432 = r18430 - r18431;
        double r18433 = exp(r18428);
        double r18434 = r18433 - r18431;
        double r18435 = r18432 / r18434;
        double r18436 = sqrt(r18435);
        return r18436;
}
↓
double f(double x) {
        double r18437 = x;
        double r18438 = -8.528320854014055e-16;
        bool r18439 = r18437 <= r18438;
        double r18440 = 2.0;
        double r18441 = r18440 * r18437;
        double r18442 = exp(r18441);
        double r18443 = 1.0;
        double r18444 = r18442 - r18443;
        double r18445 = -r18443;
        double r18446 = r18437 + r18437;
        double r18447 = exp(r18446);
        double r18448 = fma(r18445, r18443, r18447);
        double r18449 = exp(r18437);
        double r18450 = 3.0;
        double r18451 = pow(r18449, r18450);
        double r18452 = pow(r18443, r18450);
        double r18453 = r18451 + r18452;
        double r18454 = r18448 / r18453;
        double r18455 = r18449 * r18449;
        double r18456 = r18443 * r18443;
        double r18457 = r18449 * r18443;
        double r18458 = r18456 - r18457;
        double r18459 = r18455 + r18458;
        double r18460 = r18454 * r18459;
        double r18461 = r18444 / r18460;
        double r18462 = sqrt(r18461);
        double r18463 = 0.5;
        double r18464 = sqrt(r18440);
        double r18465 = r18437 / r18464;
        double r18466 = r18463 * r18465;
        double r18467 = 2.0;
        double r18468 = pow(r18437, r18467);
        double r18469 = r18468 / r18464;
        double r18470 = 0.25;
        double r18471 = 0.125;
        double r18472 = r18471 / r18440;
        double r18473 = r18470 - r18472;
        double r18474 = r18469 * r18473;
        double r18475 = r18464 + r18474;
        double r18476 = r18466 + r18475;
        double r18477 = r18439 ? r18462 : r18476;
        return r18477;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -8.528320854014055e-16

    1. Initial program 0.7

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

    3. Applied flip--0.5

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

    4. Simplified0.0

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

    6. Applied flip3-+0.0

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

    7. Applied associate-/r/0.0

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

    if -8.528320854014055e-16 < x

    1. Initial program 37.2

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

    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

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.5283208540140548 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))