Average Error: 4.6 → 0.3
Time: 26.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{\sqrt[3]{\left(\left(1 + e^{x}\right) \cdot \left(1 + e^{x}\right)\right) \cdot \left(1 + e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right)}}{\sqrt[3]{1 + \left(e^{x} \cdot e^{x} - e^{x}\right)}}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{\sqrt[3]{\left(\left(1 + e^{x}\right) \cdot \left(1 + e^{x}\right)\right) \cdot \left(1 + e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right)}}{\sqrt[3]{1 + \left(e^{x} \cdot e^{x} - e^{x}\right)}}}
double f(double x) {
        double r682931 = 2.0;
        double r682932 = x;
        double r682933 = r682931 * r682932;
        double r682934 = exp(r682933);
        double r682935 = 1.0;
        double r682936 = r682934 - r682935;
        double r682937 = exp(r682932);
        double r682938 = r682937 - r682935;
        double r682939 = r682936 / r682938;
        double r682940 = sqrt(r682939);
        return r682940;
}
↓
double f(double x) {
        double r682941 = 1.0;
        double r682942 = x;
        double r682943 = exp(r682942);
        double r682944 = r682941 + r682943;
        double r682945 = r682944 * r682944;
        double r682946 = r682943 * r682943;
        double r682947 = r682943 * r682946;
        double r682948 = r682941 + r682947;
        double r682949 = r682945 * r682948;
        double r682950 = cbrt(r682949);
        double r682951 = r682946 - r682943;
        double r682952 = r682941 + r682951;
        double r682953 = cbrt(r682952);
        double r682954 = r682950 / r682953;
        double r682955 = sqrt(r682954);
        return r682955;
}

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. Simplified0.1

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

  4. Applied add-cbrt-cube0.2

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

  6. Applied flip3-+0.2

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

  7. Applied associate-*r/0.2

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

  8. Applied associate-*l/0.2

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

  9. Applied cbrt-div0.3

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

  10. Simplified0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019134 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))