Average Error: 4.3 → 0.8
Time: 28.0s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.639623849017884427727010081959901910409 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{1 \cdot \left(1 - e^{x}\right) + e^{2 \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.639623849017884427727010081959901910409 \cdot 10^{-7}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{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 r20058 = 2.0;
        double r20059 = x;
        double r20060 = r20058 * r20059;
        double r20061 = exp(r20060);
        double r20062 = 1.0;
        double r20063 = r20061 - r20062;
        double r20064 = exp(r20059);
        double r20065 = r20064 - r20062;
        double r20066 = r20063 / r20065;
        double r20067 = sqrt(r20066);
        return r20067;
}
↓
double f(double x) {
        double r20068 = x;
        double r20069 = -3.6396238490178844e-07;
        bool r20070 = r20068 <= r20069;
        double r20071 = 2.0;
        double r20072 = r20071 * r20068;
        double r20073 = exp(r20072);
        double r20074 = 1.0;
        double r20075 = r20073 - r20074;
        double r20076 = r20068 + r20068;
        double r20077 = exp(r20076);
        double r20078 = r20074 * r20074;
        double r20079 = r20077 - r20078;
        double r20080 = r20075 / r20079;
        double r20081 = exp(r20068);
        double r20082 = 3.0;
        double r20083 = pow(r20081, r20082);
        double r20084 = pow(r20074, r20082);
        double r20085 = r20083 + r20084;
        double r20086 = r20074 - r20081;
        double r20087 = r20074 * r20086;
        double r20088 = 2.0;
        double r20089 = r20088 * r20068;
        double r20090 = exp(r20089);
        double r20091 = r20087 + r20090;
        double r20092 = r20085 / r20091;
        double r20093 = r20080 * r20092;
        double r20094 = sqrt(r20093);
        double r20095 = 0.5;
        double r20096 = r20095 * r20068;
        double r20097 = r20096 + r20074;
        double r20098 = r20068 * r20097;
        double r20099 = r20071 + r20098;
        double r20100 = sqrt(r20099);
        double r20101 = r20070 ? r20094 : r20100;
        return r20101;
}

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 < -3.6396238490178844e-07

    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

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

    7. Applied flip3-+0.0

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

    8. Simplified0.0

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

    if -3.6396238490178844e-07 < x

    1. Initial program 34.4

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

    2. Taylor expanded around 0 6.6

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

    3. Simplified6.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.639623849017884427727010081959901910409 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{1 \cdot \left(1 - e^{x}\right) + e^{2 \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))