Average Error: 4.3 → 0.1
Time: 26.2s
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 r1202315 = 2.0;
        double r1202316 = x;
        double r1202317 = r1202315 * r1202316;
        double r1202318 = exp(r1202317);
        double r1202319 = 1.0;
        double r1202320 = r1202318 - r1202319;
        double r1202321 = exp(r1202316);
        double r1202322 = r1202321 - r1202319;
        double r1202323 = r1202320 / r1202322;
        double r1202324 = sqrt(r1202323);
        return r1202324;
}
↓
double f(double x) {
        double r1202325 = x;
        double r1202326 = exp(r1202325);
        double r1202327 = 1.0;
        double r1202328 = r1202326 + r1202327;
        double r1202329 = sqrt(r1202328);
        double r1202330 = log(r1202329);
        double r1202331 = exp(r1202330);
        return r1202331;
}

Error

Bits error versus x

Try it out

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. Simplified0.1

    \[\leadsto \color{blue}{\sqrt{e^{x} + 1}}\]
  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 2019163 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))