Average Error: 4.2 → 0.1
Time: 20.9s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{1}{\frac{1 + e^{x} \cdot \left(e^{x} - 1\right)}{1 + \left(e^{x} \cdot e^{x}\right) \cdot e^{x}}}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{1}{\frac{1 + e^{x} \cdot \left(e^{x} - 1\right)}{1 + \left(e^{x} \cdot e^{x}\right) \cdot e^{x}}}}
double f(double x) {
        double r1150296 = 2.0;
        double r1150297 = x;
        double r1150298 = r1150296 * r1150297;
        double r1150299 = exp(r1150298);
        double r1150300 = 1.0;
        double r1150301 = r1150299 - r1150300;
        double r1150302 = exp(r1150297);
        double r1150303 = r1150302 - r1150300;
        double r1150304 = r1150301 / r1150303;
        double r1150305 = sqrt(r1150304);
        return r1150305;
}
↓
double f(double x) {
        double r1150306 = 1.0;
        double r1150307 = x;
        double r1150308 = exp(r1150307);
        double r1150309 = r1150308 - r1150306;
        double r1150310 = r1150308 * r1150309;
        double r1150311 = r1150306 + r1150310;
        double r1150312 = r1150308 * r1150308;
        double r1150313 = r1150312 * r1150308;
        double r1150314 = r1150306 + r1150313;
        double r1150315 = r1150311 / r1150314;
        double r1150316 = r1150306 / r1150315;
        double r1150317 = sqrt(r1150316);
        return r1150317;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.2

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

  2. Simplified0.1

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

  4. Applied flip3-+0.1

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

  5. Simplified0.1

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

  6. Simplified0.1

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

  8. Applied clear-num0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))