Average Error: 4.7 → 0.8
Time: 13.0s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.2779266924479184 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \left({\left(e^{x}\right)}^{3} + {1}^{3}\right)}}{\sqrt{1 \cdot \left(1 - e^{x}\right) + e^{2 \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.2779266924479184 \cdot 10^{-5}:\\
\;\;\;\;\frac{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \left({\left(e^{x}\right)}^{3} + {1}^{3}\right)}}{\sqrt{1 \cdot \left(1 - e^{x}\right) + e^{2 \cdot x}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\

\end{array}
double f(double x) {
        double r12009 = 2.0;
        double r12010 = x;
        double r12011 = r12009 * r12010;
        double r12012 = exp(r12011);
        double r12013 = 1.0;
        double r12014 = r12012 - r12013;
        double r12015 = exp(r12010);
        double r12016 = r12015 - r12013;
        double r12017 = r12014 / r12016;
        double r12018 = sqrt(r12017);
        return r12018;
}
↓
double f(double x) {
        double r12019 = x;
        double r12020 = -1.2779266924479184e-05;
        bool r12021 = r12019 <= r12020;
        double r12022 = 2.0;
        double r12023 = r12022 * r12019;
        double r12024 = exp(r12023);
        double r12025 = 1.0;
        double r12026 = r12024 - r12025;
        double r12027 = r12019 + r12019;
        double r12028 = exp(r12027);
        double r12029 = r12025 * r12025;
        double r12030 = r12028 - r12029;
        double r12031 = r12026 / r12030;
        double r12032 = exp(r12019);
        double r12033 = 3.0;
        double r12034 = pow(r12032, r12033);
        double r12035 = pow(r12025, r12033);
        double r12036 = r12034 + r12035;
        double r12037 = r12031 * r12036;
        double r12038 = sqrt(r12037);
        double r12039 = r12025 - r12032;
        double r12040 = r12025 * r12039;
        double r12041 = 2.0;
        double r12042 = r12041 * r12019;
        double r12043 = exp(r12042);
        double r12044 = r12040 + r12043;
        double r12045 = sqrt(r12044);
        double r12046 = r12038 / r12045;
        double r12047 = 0.5;
        double r12048 = r12047 * r12019;
        double r12049 = r12048 + r12025;
        double r12050 = r12019 * r12049;
        double r12051 = r12022 + r12050;
        double r12052 = sqrt(r12051);
        double r12053 = r12021 ? r12046 : r12052;
        return r12053;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -1.2779266924479184e-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

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

    7. Applied *-un-lft-identity0.0

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

    9. Applied flip3-+0.0

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

    10. Applied associate-*r/0.0

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

    11. Applied sqrt-div0.0

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

    12. Simplified0.0

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

    if -1.2779266924479184e-05 < x

    1. Initial program 34.3

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

    2. Taylor expanded around 0 5.9

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

    3. Simplified5.9

      \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(0.5 \cdot x + 1\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.2779266924479184 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \left({\left(e^{x}\right)}^{3} + {1}^{3}\right)}}{\sqrt{1 \cdot \left(1 - e^{x}\right) + e^{2 \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020044 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))