Average Error: 4.4 → 0.7
Time: 26.1s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.196894471233741566924913620091619481656 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \frac{x}{\frac{\sqrt{2}}{x}}, \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \sqrt{2}\right)\right) - \frac{x}{\frac{\sqrt{2}}{x}} \cdot \left(\frac{0.75}{3 \cdot 3} + \frac{0.125}{2}\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.196894471233741566924913620091619481656 \cdot 10^{-8}:\\
\;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \frac{x}{\frac{\sqrt{2}}{x}}, \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \sqrt{2}\right)\right) - \frac{x}{\frac{\sqrt{2}}{x}} \cdot \left(\frac{0.75}{3 \cdot 3} + \frac{0.125}{2}\right)\\

\end{array}
double f(double x) {
        double r1434960 = 2.0;
        double r1434961 = x;
        double r1434962 = r1434960 * r1434961;
        double r1434963 = exp(r1434962);
        double r1434964 = 1.0;
        double r1434965 = r1434963 - r1434964;
        double r1434966 = exp(r1434961);
        double r1434967 = r1434966 - r1434964;
        double r1434968 = r1434965 / r1434967;
        double r1434969 = sqrt(r1434968);
        return r1434969;
}
↓
double f(double x) {
        double r1434970 = x;
        double r1434971 = -2.1968944712337416e-08;
        bool r1434972 = r1434970 <= r1434971;
        double r1434973 = 2.0;
        double r1434974 = r1434973 * r1434970;
        double r1434975 = exp(r1434974);
        double r1434976 = sqrt(r1434975);
        double r1434977 = 1.0;
        double r1434978 = -r1434977;
        double r1434979 = fma(r1434976, r1434976, r1434978);
        double r1434980 = exp(r1434970);
        double r1434981 = r1434980 - r1434977;
        double r1434982 = r1434979 / r1434981;
        double r1434983 = sqrt(r1434982);
        double r1434984 = 0.3333333333333333;
        double r1434985 = sqrt(r1434973);
        double r1434986 = r1434985 / r1434970;
        double r1434987 = r1434970 / r1434986;
        double r1434988 = 0.5;
        double r1434989 = r1434970 / r1434985;
        double r1434990 = fma(r1434988, r1434989, r1434985);
        double r1434991 = fma(r1434984, r1434987, r1434990);
        double r1434992 = 0.75;
        double r1434993 = 3.0;
        double r1434994 = r1434993 * r1434993;
        double r1434995 = r1434992 / r1434994;
        double r1434996 = 0.125;
        double r1434997 = r1434996 / r1434973;
        double r1434998 = r1434995 + r1434997;
        double r1434999 = r1434987 * r1434998;
        double r1435000 = r1434991 - r1434999;
        double r1435001 = r1434972 ? r1434983 : r1435000;
        return r1435001;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -2.1968944712337416e-08

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied fma-neg0.0

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

    if -2.1968944712337416e-08 < x

    1. Initial program 35.6

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

    2. Taylor expanded around 0 6.0

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

    3. Simplified6.0

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, 1\right), 2\right)}}\]
    4. Using strategy rm

    5. Applied expm1-log1p-u6.4

      \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, 1\right), 2\right)\right)\right)}}\]

    6. Taylor expanded around 0 6.0

      \[\leadsto \color{blue}{\left(\sqrt{2} + \left(0.5 \cdot \frac{x}{\sqrt{2}} + 0.3333333333333333148296162562473909929395 \cdot \frac{{x}^{2}}{\sqrt{2}}\right)\right) - \left(0.75 \cdot \frac{{x}^{2}}{\sqrt{2} \cdot {3}^{2}} + 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}\right)}\]

    7. Simplified6.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \frac{x}{\frac{\sqrt{2}}{x}}, \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \sqrt{2}\right)\right) - \frac{x}{\frac{\sqrt{2}}{x}} \cdot \left(\frac{0.75}{3 \cdot 3} + \frac{0.125}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.196894471233741566924913620091619481656 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \frac{x}{\frac{\sqrt{2}}{x}}, \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \sqrt{2}\right)\right) - \frac{x}{\frac{\sqrt{2}}{x}} \cdot \left(\frac{0.75}{3 \cdot 3} + \frac{0.125}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))