Average Error: 4.3 → 0.2
Time: 20.4s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{\left(e^{x} \cdot e^{x}\right) \cdot e^{x} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{\left(e^{x} \cdot e^{x}\right) \cdot e^{x} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}
double f(double x) {
        double r666849 = 2.0;
        double r666850 = x;
        double r666851 = r666849 * r666850;
        double r666852 = exp(r666851);
        double r666853 = 1.0;
        double r666854 = r666852 - r666853;
        double r666855 = exp(r666850);
        double r666856 = r666855 - r666853;
        double r666857 = r666854 / r666856;
        double r666858 = sqrt(r666857);
        return r666858;
}
↓
double f(double x) {
        double r666859 = x;
        double r666860 = exp(r666859);
        double r666861 = r666860 * r666860;
        double r666862 = r666861 * r666860;
        double r666863 = 1.0;
        double r666864 = r666862 + r666863;
        double r666865 = r666863 - r666860;
        double r666866 = r666861 + r666865;
        double r666867 = r666864 / r666866;
        double r666868 = sqrt(r666867);
        return r666868;
}

Error

Bits error versus x

Try it out

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. Simplified0.0

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

  4. Applied flip3-+0.2

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

  5. Simplified0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019129 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))