Average Error: 4.6 → 0.2
Time: 20.2s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{1 + \sqrt[3]{{\left(e^{x}\right)}^{3} \cdot \left({\left(e^{x}\right)}^{3} \cdot {\left(e^{x}\right)}^{3}\right)}}{\left(1 - e^{x}\right) + e^{x} \cdot e^{x}}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{1 + \sqrt[3]{{\left(e^{x}\right)}^{3} \cdot \left({\left(e^{x}\right)}^{3} \cdot {\left(e^{x}\right)}^{3}\right)}}{\left(1 - e^{x}\right) + e^{x} \cdot e^{x}}}
double f(double x) {
        double r386284 = 2.0;
        double r386285 = x;
        double r386286 = r386284 * r386285;
        double r386287 = exp(r386286);
        double r386288 = 1.0;
        double r386289 = r386287 - r386288;
        double r386290 = exp(r386285);
        double r386291 = r386290 - r386288;
        double r386292 = r386289 / r386291;
        double r386293 = sqrt(r386292);
        return r386293;
}
↓
double f(double x) {
        double r386294 = 1.0;
        double r386295 = x;
        double r386296 = exp(r386295);
        double r386297 = 3.0;
        double r386298 = pow(r386296, r386297);
        double r386299 = r386298 * r386298;
        double r386300 = r386298 * r386299;
        double r386301 = cbrt(r386300);
        double r386302 = r386294 + r386301;
        double r386303 = r386294 - r386296;
        double r386304 = r386296 * r386296;
        double r386305 = r386303 + r386304;
        double r386306 = r386302 / r386305;
        double r386307 = sqrt(r386306);
        return r386307;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.6

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

  2. Simplified0.0

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

  4. Applied flip3-+0.1

    \[\leadsto \sqrt{\color{blue}{\frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube0.2

    \[\leadsto \sqrt{\frac{\color{blue}{\sqrt[3]{\left({\left(e^{x}\right)}^{3} \cdot {\left(e^{x}\right)}^{3}\right) \cdot {\left(e^{x}\right)}^{3}}} + {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}\]

  7. Final simplification0.2

    \[\leadsto \sqrt{\frac{1 + \sqrt[3]{{\left(e^{x}\right)}^{3} \cdot \left({\left(e^{x}\right)}^{3} \cdot {\left(e^{x}\right)}^{3}\right)}}{\left(1 - e^{x}\right) + e^{x} \cdot e^{x}}}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))