Average Error: 4.3 → 0.1
Time: 20.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[e^{\log \left(\sqrt{e^{x} + 1}\right)}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
e^{\log \left(\sqrt{e^{x} + 1}\right)}
double f(double x) {
        double r351809 = 2.0;
        double r351810 = x;
        double r351811 = r351809 * r351810;
        double r351812 = exp(r351811);
        double r351813 = 1.0;
        double r351814 = r351812 - r351813;
        double r351815 = exp(r351810);
        double r351816 = r351815 - r351813;
        double r351817 = r351814 / r351816;
        double r351818 = sqrt(r351817);
        return r351818;
}
↓
double f(double x) {
        double r351819 = x;
        double r351820 = exp(r351819);
        double r351821 = 1.0;
        double r351822 = r351820 + r351821;
        double r351823 = sqrt(r351822);
        double r351824 = log(r351823);
        double r351825 = exp(r351824);
        return r351825;
}

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. Simplified0.1

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied *-un-lft-identity0.1

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

  6. Applied hypot-def0.1

    \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)}\]
  7. Using strategy rm

  8. Applied add-exp-log0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))