Average Error: 4.3 → 0.8
Time: 6.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.6734196112405429 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}, \mathsf{fma}\left(\frac{{x}^{2}}{\sqrt{2}}, 0.25, \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \sqrt{2}\right)\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.6734196112405429 \cdot 10^{-12}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-0.125, \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}, \mathsf{fma}\left(\frac{{x}^{2}}{\sqrt{2}}, 0.25, \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \sqrt{2}\right)\right)\right)\\

\end{array}
double f(double x) {
        double r18477 = 2.0;
        double r18478 = x;
        double r18479 = r18477 * r18478;
        double r18480 = exp(r18479);
        double r18481 = 1.0;
        double r18482 = r18480 - r18481;
        double r18483 = exp(r18478);
        double r18484 = r18483 - r18481;
        double r18485 = r18482 / r18484;
        double r18486 = sqrt(r18485);
        return r18486;
}
↓
double f(double x) {
        double r18487 = x;
        double r18488 = -1.673419611240543e-12;
        bool r18489 = r18487 <= r18488;
        double r18490 = 2.0;
        double r18491 = r18490 * r18487;
        double r18492 = exp(r18491);
        double r18493 = 1.0;
        double r18494 = r18492 - r18493;
        double r18495 = -r18493;
        double r18496 = r18487 + r18487;
        double r18497 = exp(r18496);
        double r18498 = fma(r18495, r18493, r18497);
        double r18499 = r18494 / r18498;
        double r18500 = exp(r18487);
        double r18501 = r18500 + r18493;
        double r18502 = r18499 * r18501;
        double r18503 = sqrt(r18502);
        double r18504 = 0.125;
        double r18505 = -r18504;
        double r18506 = 2.0;
        double r18507 = pow(r18487, r18506);
        double r18508 = sqrt(r18490);
        double r18509 = 3.0;
        double r18510 = pow(r18508, r18509);
        double r18511 = r18507 / r18510;
        double r18512 = r18507 / r18508;
        double r18513 = 0.25;
        double r18514 = 0.5;
        double r18515 = r18487 / r18508;
        double r18516 = fma(r18514, r18515, r18508);
        double r18517 = fma(r18512, r18513, r18516);
        double r18518 = fma(r18505, r18511, r18517);
        double r18519 = r18489 ? r18503 : r18518;
        return r18519;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -1.673419611240543e-12

    1. Initial program 0.5

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

    3. Applied flip--0.3

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

    4. Applied associate-/r/0.3

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

    5. Simplified0.0

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

    if -1.673419611240543e-12 < x

    1. Initial program 35.6

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

    3. Applied flip--33.2

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

    4. Applied associate-/r/33.2

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

    5. Simplified25.2

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

    6. Taylor expanded around 0 7.6

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

    7. Simplified7.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6734196112405429 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}, \mathsf{fma}\left(\frac{{x}^{2}}{\sqrt{2}}, 0.25, \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \sqrt{2}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))