Average Error: 4.3 → 0.8
Time: 25.2s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.639623849017884427727010081959901910409 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{\frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{e^{x + x} + 1 \cdot \left(1 - e^{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]
\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 \sqrt{\frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{e^{x + x} + 1 \cdot \left(1 - e^{x}\right)}}\\

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

\end{array}
double f(double x) {
        double r19084 = 2.0;
        double r19085 = x;
        double r19086 = r19084 * r19085;
        double r19087 = exp(r19086);
        double r19088 = 1.0;
        double r19089 = r19087 - r19088;
        double r19090 = exp(r19085);
        double r19091 = r19090 - r19088;
        double r19092 = r19089 / r19091;
        double r19093 = sqrt(r19092);
        return r19093;
}
↓
double f(double x) {
        double r19094 = x;
        double r19095 = -3.6396238490178844e-07;
        bool r19096 = r19094 <= r19095;
        double r19097 = 2.0;
        double r19098 = r19097 * r19094;
        double r19099 = exp(r19098);
        double r19100 = 1.0;
        double r19101 = r19099 - r19100;
        double r19102 = r19094 + r19094;
        double r19103 = exp(r19102);
        double r19104 = r19100 * r19100;
        double r19105 = r19103 - r19104;
        double r19106 = r19101 / r19105;
        double r19107 = sqrt(r19106);
        double r19108 = exp(r19094);
        double r19109 = 3.0;
        double r19110 = pow(r19108, r19109);
        double r19111 = pow(r19100, r19109);
        double r19112 = r19110 + r19111;
        double r19113 = r19100 - r19108;
        double r19114 = r19100 * r19113;
        double r19115 = r19103 + r19114;
        double r19116 = r19112 / r19115;
        double r19117 = sqrt(r19116);
        double r19118 = r19107 * r19117;
        double r19119 = 0.5;
        double r19120 = r19119 * r19094;
        double r19121 = r19120 + r19100;
        double r19122 = r19094 * r19121;
        double r19123 = r19097 + r19122;
        double r19124 = sqrt(r19123);
        double r19125 = r19096 ? r19118 : r19124;
        return r19125;
}

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. Applied sqrt-prod0.1

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

    6. Simplified0.0

      \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}}} \cdot \sqrt{e^{x} + 1}\]
    7. Using strategy rm

    8. Applied flip3-+0.0

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

    9. Simplified0.0

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

    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

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

Reproduce

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