Average Error: 4.3 → 0.8
Time: 6.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.898814212803163338280231127452868378214 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left({\left(e^{x}\right)}^{3} + {1}^{3}\right)}}{\sqrt{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.898814212803163338280231127452868378214 \cdot 10^{-7}:\\
\;\;\;\;\frac{\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left({\left(e^{x}\right)}^{3} + {1}^{3}\right)}}{\sqrt{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\

\end{array}
double f(double x) {
        double r19915 = 2.0;
        double r19916 = x;
        double r19917 = r19915 * r19916;
        double r19918 = exp(r19917);
        double r19919 = 1.0;
        double r19920 = r19918 - r19919;
        double r19921 = exp(r19916);
        double r19922 = r19921 - r19919;
        double r19923 = r19920 / r19922;
        double r19924 = sqrt(r19923);
        return r19924;
}
↓
double f(double x) {
        double r19925 = x;
        double r19926 = -8.898814212803163e-07;
        bool r19927 = r19925 <= r19926;
        double r19928 = 2.0;
        double r19929 = r19928 * r19925;
        double r19930 = exp(r19929);
        double r19931 = 1.0;
        double r19932 = r19930 - r19931;
        double r19933 = -r19931;
        double r19934 = r19925 + r19925;
        double r19935 = exp(r19934);
        double r19936 = fma(r19933, r19931, r19935);
        double r19937 = r19932 / r19936;
        double r19938 = exp(r19925);
        double r19939 = 3.0;
        double r19940 = pow(r19938, r19939);
        double r19941 = pow(r19931, r19939);
        double r19942 = r19940 + r19941;
        double r19943 = r19937 * r19942;
        double r19944 = sqrt(r19943);
        double r19945 = r19938 * r19938;
        double r19946 = r19931 * r19931;
        double r19947 = r19938 * r19931;
        double r19948 = r19946 - r19947;
        double r19949 = r19945 + r19948;
        double r19950 = sqrt(r19949);
        double r19951 = r19944 / r19950;
        double r19952 = 0.5;
        double r19953 = sqrt(r19928);
        double r19954 = r19925 / r19953;
        double r19955 = r19952 * r19954;
        double r19956 = 2.0;
        double r19957 = pow(r19925, r19956);
        double r19958 = r19957 / r19953;
        double r19959 = 0.25;
        double r19960 = 0.125;
        double r19961 = r19960 / r19928;
        double r19962 = r19959 - r19961;
        double r19963 = r19958 * r19962;
        double r19964 = r19953 + r19963;
        double r19965 = r19955 + r19964;
        double r19966 = r19927 ? r19951 : r19965;
        return r19966;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -8.898814212803163e-07

    1. Initial program 0.1

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    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 flip3-+0.0

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

    8. Applied associate-*r/0.0

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

    9. Applied sqrt-div0.0

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

    if -8.898814212803163e-07 < x

    1. Initial program 35.2

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

    2. Taylor expanded around 0 6.9

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

    3. Simplified6.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.898814212803163338280231127452868378214 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left({\left(e^{x}\right)}^{3} + {1}^{3}\right)}}{\sqrt{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))