Average Error: 4.4 → 0.1
Time: 36.2s
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 r1037470 = 2.0;
        double r1037471 = x;
        double r1037472 = r1037470 * r1037471;
        double r1037473 = exp(r1037472);
        double r1037474 = 1.0;
        double r1037475 = r1037473 - r1037474;
        double r1037476 = exp(r1037471);
        double r1037477 = r1037476 - r1037474;
        double r1037478 = r1037475 / r1037477;
        double r1037479 = sqrt(r1037478);
        return r1037479;
}
↓
double f(double x) {
        double r1037480 = x;
        double r1037481 = exp(r1037480);
        double r1037482 = 1.0;
        double r1037483 = r1037481 + r1037482;
        double r1037484 = sqrt(r1037483);
        double r1037485 = log(r1037484);
        double r1037486 = exp(r1037485);
        return r1037486;
}

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 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))