Average Error: 4.3 → 0.1
Time: 18.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{\mathsf{log1p}\left(e^{x}\right)}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{e^{\mathsf{log1p}\left(e^{x}\right)}}
double f(double x) {
        double r284795 = 2.0;
        double r284796 = x;
        double r284797 = r284795 * r284796;
        double r284798 = exp(r284797);
        double r284799 = 1.0;
        double r284800 = r284798 - r284799;
        double r284801 = exp(r284796);
        double r284802 = r284801 - r284799;
        double r284803 = r284800 / r284802;
        double r284804 = sqrt(r284803);
        return r284804;
}
↓
double f(double x) {
        double r284805 = x;
        double r284806 = exp(r284805);
        double r284807 = log1p(r284806);
        double r284808 = exp(r284807);
        double r284809 = sqrt(r284808);
        return r284809;
}

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. Simplified0.0

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

  4. Applied add-exp-log0.1

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

  5. Simplified0.1

    \[\leadsto \sqrt{e^{\color{blue}{\mathsf{log1p}\left(e^{x}\right)}}}\]

  6. Final simplification0.1

    \[\leadsto \sqrt{e^{\mathsf{log1p}\left(e^{x}\right)}}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))