Average Error: 4.3 → 0.9
Time: 56.1s
Precision: 64
\[\sqrt{\frac{e^{2.0 \cdot x} - 1.0}{e^{x} - 1.0}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.6879468583117964 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1.0} + \sqrt{e^{2.0 \cdot x}}}{\left(\left(\frac{e^{x} - 1.0}{\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2.0} + \frac{x \cdot 0.5}{\sqrt{2.0}}\right) + \left(0.25 - \frac{0.125}{2.0}\right) \cdot \frac{x}{\frac{\sqrt{2.0}}{x}}\\ \end{array}\]
\sqrt{\frac{e^{2.0 \cdot x} - 1.0}{e^{x} - 1.0}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.6879468583117964 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{\sqrt{1.0} + \sqrt{e^{2.0 \cdot x}}}{\left(\left(\frac{e^{x} - 1.0}{\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}}\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt{2.0} + \frac{x \cdot 0.5}{\sqrt{2.0}}\right) + \left(0.25 - \frac{0.125}{2.0}\right) \cdot \frac{x}{\frac{\sqrt{2.0}}{x}}\\

\end{array}
double f(double x) {
        double r2089555 = 2.0;
        double r2089556 = x;
        double r2089557 = r2089555 * r2089556;
        double r2089558 = exp(r2089557);
        double r2089559 = 1.0;
        double r2089560 = r2089558 - r2089559;
        double r2089561 = exp(r2089556);
        double r2089562 = r2089561 - r2089559;
        double r2089563 = r2089560 / r2089562;
        double r2089564 = sqrt(r2089563);
        return r2089564;
}
↓
double f(double x) {
        double r2089565 = x;
        double r2089566 = -1.6879468583117964e-16;
        bool r2089567 = r2089565 <= r2089566;
        double r2089568 = 1.0;
        double r2089569 = sqrt(r2089568);
        double r2089570 = 2.0;
        double r2089571 = r2089570 * r2089565;
        double r2089572 = exp(r2089571);
        double r2089573 = sqrt(r2089572);
        double r2089574 = r2089569 + r2089573;
        double r2089575 = exp(r2089565);
        double r2089576 = r2089575 - r2089568;
        double r2089577 = r2089573 - r2089569;
        double r2089578 = r2089576 / r2089577;
        double r2089579 = /* ERROR: no posit support in C */;
        double r2089580 = /* ERROR: no posit support in C */;
        double r2089581 = r2089574 / r2089580;
        double r2089582 = sqrt(r2089581);
        double r2089583 = sqrt(r2089570);
        double r2089584 = 0.5;
        double r2089585 = r2089565 * r2089584;
        double r2089586 = r2089585 / r2089583;
        double r2089587 = r2089583 + r2089586;
        double r2089588 = 0.25;
        double r2089589 = 0.125;
        double r2089590 = r2089589 / r2089570;
        double r2089591 = r2089588 - r2089590;
        double r2089592 = r2089583 / r2089565;
        double r2089593 = r2089565 / r2089592;
        double r2089594 = r2089591 * r2089593;
        double r2089595 = r2089587 + r2089594;
        double r2089596 = r2089567 ? r2089582 : r2089595;
        return r2089596;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -1.6879468583117964e-16

    1. Initial program 0.8

      \[\sqrt{\frac{e^{2.0 \cdot x} - 1.0}{e^{x} - 1.0}}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.8

      \[\leadsto \sqrt{\frac{e^{2.0 \cdot x} - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}{e^{x} - 1.0}}\]

    4. Applied add-sqr-sqrt0.7

      \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2.0 \cdot x}} \cdot \sqrt{e^{2.0 \cdot x}}} - \sqrt{1.0} \cdot \sqrt{1.0}}{e^{x} - 1.0}}\]

    5. Applied difference-of-squares0.3

      \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2.0 \cdot x}} + \sqrt{1.0}\right) \cdot \left(\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}\right)}}{e^{x} - 1.0}}\]

    6. Applied associate-/l*0.3

      \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{e^{2.0 \cdot x}} + \sqrt{1.0}}{\frac{e^{x} - 1.0}{\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}}}}}\]
    7. Using strategy rm

    8. Applied insert-posit160.1

      \[\leadsto \sqrt{\frac{\sqrt{e^{2.0 \cdot x}} + \sqrt{1.0}}{\color{blue}{\left(\left(\frac{e^{x} - 1.0}{\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}}\right)\right)}}}\]

    if -1.6879468583117964e-16 < x

    1. Initial program 37.6

      \[\sqrt{\frac{e^{2.0 \cdot x} - 1.0}{e^{x} - 1.0}}\]

    2. Taylor expanded around 0 8.4

      \[\leadsto \color{blue}{\left(\sqrt{2.0} + \left(0.5 \cdot \frac{x}{\sqrt{2.0}} + 0.25 \cdot \frac{{x}^{2}}{\sqrt{2.0}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2.0}\right)}^{3}}}\]

    3. Simplified8.4

      \[\leadsto \color{blue}{\left(\frac{0.5 \cdot x}{\sqrt{2.0}} + \sqrt{2.0}\right) + \frac{x}{\frac{\sqrt{2.0}}{x}} \cdot \left(0.25 - \frac{0.125}{2.0}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6879468583117964 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1.0} + \sqrt{e^{2.0 \cdot x}}}{\left(\left(\frac{e^{x} - 1.0}{\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2.0} + \frac{x \cdot 0.5}{\sqrt{2.0}}\right) + \left(0.25 - \frac{0.125}{2.0}\right) \cdot \frac{x}{\frac{\sqrt{2.0}}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))