Average Error: 4.4 → 0.0
Time: 15.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)
double f(double x) {
        double r312275 = 2.0;
        double r312276 = x;
        double r312277 = r312275 * r312276;
        double r312278 = exp(r312277);
        double r312279 = 1.0;
        double r312280 = r312278 - r312279;
        double r312281 = exp(r312276);
        double r312282 = r312281 - r312279;
        double r312283 = r312280 / r312282;
        double r312284 = sqrt(r312283);
        return r312284;
}
↓
double f(double x) {
        double r312285 = 1.0;
        double r312286 = x;
        double r312287 = exp(r312286);
        double r312288 = sqrt(r312287);
        double r312289 = hypot(r312285, r312288);
        return r312289;
}

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

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied hypot-def0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))