Average Error: 4.4 → 0.9
Time: 6.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.5012786943274603 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}, \mathsf{fma}\left(\frac{{x}^{2}}{\sqrt{2}}, 0.25, \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \sqrt{2}\right)\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.5012786943274603 \cdot 10^{-8}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\right)\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 r21998 = 2.0;
        double r21999 = x;
        double r22000 = r21998 * r21999;
        double r22001 = exp(r22000);
        double r22002 = 1.0;
        double r22003 = r22001 - r22002;
        double r22004 = exp(r21999);
        double r22005 = r22004 - r22002;
        double r22006 = r22003 / r22005;
        double r22007 = sqrt(r22006);
        return r22007;
}
↓
double f(double x) {
        double r22008 = x;
        double r22009 = -2.5012786943274603e-08;
        bool r22010 = r22008 <= r22009;
        double r22011 = 2.0;
        double r22012 = r22011 * r22008;
        double r22013 = exp(r22012);
        double r22014 = 1.0;
        double r22015 = r22013 - r22014;
        double r22016 = -r22014;
        double r22017 = r22008 + r22008;
        double r22018 = exp(r22017);
        double r22019 = fma(r22016, r22014, r22018);
        double r22020 = r22015 / r22019;
        double r22021 = exp(r22008);
        double r22022 = r22021 + r22014;
        double r22023 = r22020 * r22022;
        double r22024 = sqrt(r22023);
        double r22025 = log1p(r22024);
        double r22026 = expm1(r22025);
        double r22027 = 0.125;
        double r22028 = -r22027;
        double r22029 = 2.0;
        double r22030 = pow(r22008, r22029);
        double r22031 = sqrt(r22011);
        double r22032 = 3.0;
        double r22033 = pow(r22031, r22032);
        double r22034 = r22030 / r22033;
        double r22035 = r22030 / r22031;
        double r22036 = 0.25;
        double r22037 = 0.5;
        double r22038 = r22008 / r22031;
        double r22039 = fma(r22037, r22038, r22031);
        double r22040 = fma(r22035, r22036, r22039);
        double r22041 = fma(r22028, r22034, r22040);
        double r22042 = r22010 ? r22026 : r22041;
        return r22042;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -2.5012786943274603e-08

    1. Initial program 0.2

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

    3. Applied flip--0.1

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

    4. Applied associate-/r/0.1

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

    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 expm1-log1p-u0.0

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\right)\right)}\]

    if -2.5012786943274603e-08 < x

    1. Initial program 35.3

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

    3. Applied flip--32.2

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

    4. Applied associate-/r/32.2

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

    5. Simplified23.7

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

    6. Taylor expanded around 0 7.4

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

    7. Simplified7.4

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.5012786943274603 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}, \mathsf{fma}\left(\frac{{x}^{2}}{\sqrt{2}}, 0.25, \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \sqrt{2}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))