Average Error: 4.3 → 0.1
Time: 25.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{\log \left(1 + e^{x}\right)}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{e^{\log \left(1 + e^{x}\right)}}
double f(double x) {
        double r962890 = 2.0;
        double r962891 = x;
        double r962892 = r962890 * r962891;
        double r962893 = exp(r962892);
        double r962894 = 1.0;
        double r962895 = r962893 - r962894;
        double r962896 = exp(r962891);
        double r962897 = r962896 - r962894;
        double r962898 = r962895 / r962897;
        double r962899 = sqrt(r962898);
        return r962899;
}
↓
double f(double x) {
        double r962900 = 1.0;
        double r962901 = x;
        double r962902 = exp(r962901);
        double r962903 = r962900 + r962902;
        double r962904 = log(r962903);
        double r962905 = exp(r962904);
        double r962906 = sqrt(r962905);
        return r962906;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.3

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\sqrt{1 + e^{x}}}\]
  3. Using strategy rm

  4. Applied add-exp-log0.1

    \[\leadsto \sqrt{\color{blue}{e^{\log \left(1 + e^{x}\right)}}}\]

  5. Final simplification0.1

    \[\leadsto \sqrt{e^{\log \left(1 + e^{x}\right)}}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))