Average Error: 4.7 → 0.1
Time: 5.4s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.029879697390661017842097835206516265316 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{elif}\;x \le 5.549336637458991264453317788129656623006 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.029879697390661017842097835206516265316 \cdot 10^{-5}:\\
\;\;\;\;\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^{x}}, 1\right)}\\

\mathbf{elif}\;x \le 5.549336637458991264453317788129656623006 \cdot 10^{-17}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 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 r11204 = 2.0;
        double r11205 = x;
        double r11206 = r11204 * r11205;
        double r11207 = exp(r11206);
        double r11208 = 1.0;
        double r11209 = r11207 - r11208;
        double r11210 = exp(r11205);
        double r11211 = r11210 - r11208;
        double r11212 = r11209 / r11211;
        double r11213 = sqrt(r11212);
        return r11213;
}
↓
double f(double x) {
        double r11214 = x;
        double r11215 = -1.029879697390661e-05;
        bool r11216 = r11214 <= r11215;
        double r11217 = 2.0;
        double r11218 = r11217 * r11214;
        double r11219 = exp(r11218);
        double r11220 = 1.0;
        double r11221 = r11219 - r11220;
        double r11222 = -r11220;
        double r11223 = r11214 + r11214;
        double r11224 = exp(r11223);
        double r11225 = fma(r11222, r11220, r11224);
        double r11226 = r11221 / r11225;
        double r11227 = exp(r11214);
        double r11228 = sqrt(r11227);
        double r11229 = fma(r11228, r11228, r11220);
        double r11230 = r11226 * r11229;
        double r11231 = sqrt(r11230);
        double r11232 = 5.549336637458991e-17;
        bool r11233 = r11214 <= r11232;
        double r11234 = 0.5;
        double r11235 = 2.0;
        double r11236 = pow(r11214, r11235);
        double r11237 = fma(r11220, r11214, r11217);
        double r11238 = fma(r11234, r11236, r11237);
        double r11239 = sqrt(r11238);
        double r11240 = r11227 + r11220;
        double r11241 = r11226 * r11240;
        double r11242 = sqrt(r11241);
        double r11243 = r11233 ? r11239 : r11242;
        double r11244 = r11216 ? r11231 : r11243;
        return r11244;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -1.029879697390661e-05

    1. Initial program 0.1

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

    3. Applied flip--0.0

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

    4. Applied associate-/r/0.0

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

    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

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

    if -1.029879697390661e-05 < x < 5.549336637458991e-17

    1. Initial program 48.4

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]

    3. Simplified0.1

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]

    if 5.549336637458991e-17 < x

    1. Initial program 18.0

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

    3. Applied flip--14.2

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

    4. Applied associate-/r/14.2

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

    5. Simplified1.9

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.029879697390661017842097835206516265316 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{elif}\;x \le 5.549336637458991264453317788129656623006 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))