Average Error: 4.4 → 0.0
Time: 24.3s
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 r472972 = 2.0;
        double r472973 = x;
        double r472974 = r472972 * r472973;
        double r472975 = exp(r472974);
        double r472976 = 1.0;
        double r472977 = r472975 - r472976;
        double r472978 = exp(r472973);
        double r472979 = r472978 - r472976;
        double r472980 = r472977 / r472979;
        double r472981 = sqrt(r472980);
        return r472981;
}
↓
double f(double x) {
        double r472982 = x;
        double r472983 = exp(r472982);
        double r472984 = log1p(r472983);
        double r472985 = exp(r472984);
        double r472986 = sqrt(r472985);
        return r472986;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.4

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

  2. Simplified0.0

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

  4. Applied add-exp-log0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))