Average Error: 4.4 → 0.1
Time: 34.4s
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 r903331 = 2.0;
        double r903332 = x;
        double r903333 = r903331 * r903332;
        double r903334 = exp(r903333);
        double r903335 = 1.0;
        double r903336 = r903334 - r903335;
        double r903337 = exp(r903332);
        double r903338 = r903337 - r903335;
        double r903339 = r903336 / r903338;
        double r903340 = sqrt(r903339);
        return r903340;
}
↓
double f(double x) {
        double r903341 = x;
        double r903342 = exp(r903341);
        double r903343 = 1.0;
        double r903344 = r903342 + r903343;
        double r903345 = sqrt(r903344);
        double r903346 = log(r903345);
        double r903347 = exp(r903346);
        return r903347;
}

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. Simplified0.1

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

  4. Applied add-exp-log0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))