Average Error: 4.6 → 0.8
Time: 23.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.40645755983236070131526737614358069095 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x \cdot 0.4999999999999997779553950749686919152737, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.40645755983236070131526737614358069095 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x \cdot 0.4999999999999997779553950749686919152737, \mathsf{fma}\left(1, x, 2\right)\right)}\\

\end{array}
double f(double x) {
        double r1092203 = 2.0;
        double r1092204 = x;
        double r1092205 = r1092203 * r1092204;
        double r1092206 = exp(r1092205);
        double r1092207 = 1.0;
        double r1092208 = r1092206 - r1092207;
        double r1092209 = exp(r1092204);
        double r1092210 = r1092209 - r1092207;
        double r1092211 = r1092208 / r1092210;
        double r1092212 = sqrt(r1092211);
        return r1092212;
}
↓
double f(double x) {
        double r1092213 = x;
        double r1092214 = -1.4064575598323607e-05;
        bool r1092215 = r1092213 <= r1092214;
        double r1092216 = 2.0;
        double r1092217 = r1092216 * r1092213;
        double r1092218 = exp(r1092217);
        double r1092219 = sqrt(r1092218);
        double r1092220 = log(r1092219);
        double r1092221 = exp(r1092220);
        double r1092222 = 1.0;
        double r1092223 = -r1092222;
        double r1092224 = fma(r1092219, r1092221, r1092223);
        double r1092225 = exp(r1092213);
        double r1092226 = r1092225 - r1092222;
        double r1092227 = r1092224 / r1092226;
        double r1092228 = sqrt(r1092227);
        double r1092229 = 0.4999999999999998;
        double r1092230 = r1092213 * r1092229;
        double r1092231 = fma(r1092222, r1092213, r1092216);
        double r1092232 = fma(r1092213, r1092230, r1092231);
        double r1092233 = sqrt(r1092232);
        double r1092234 = r1092215 ? r1092228 : r1092233;
        return r1092234;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -1.4064575598323607e-05

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied fma-neg0.0

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

    6. Applied add-exp-log0.0

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

    if -1.4064575598323607e-05 < x

    1. Initial program 34.3

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

    3. Applied add-sqr-sqrt32.2

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

    4. Applied fma-neg26.8

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

    5. Taylor expanded around 0 6.0

      \[\leadsto \sqrt{\color{blue}{1 \cdot x + \left(0.4999999999999997779553950749686919152737 \cdot {x}^{2} + 2\right)}}\]

    6. Simplified6.0

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.4999999999999997779553950749686919152737, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.40645755983236070131526737614358069095 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x \cdot 0.4999999999999997779553950749686919152737, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))