Average Error: 4.3 → 0.2
Time: 20.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{e^{x \cdot 3} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{e^{x \cdot 3} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}
double f(double x) {
        double r340861 = 2.0;
        double r340862 = x;
        double r340863 = r340861 * r340862;
        double r340864 = exp(r340863);
        double r340865 = 1.0;
        double r340866 = r340864 - r340865;
        double r340867 = exp(r340862);
        double r340868 = r340867 - r340865;
        double r340869 = r340866 / r340868;
        double r340870 = sqrt(r340869);
        return r340870;
}
↓
double f(double x) {
        double r340871 = x;
        double r340872 = 3.0;
        double r340873 = r340871 * r340872;
        double r340874 = exp(r340873);
        double r340875 = 1.0;
        double r340876 = r340874 + r340875;
        double r340877 = exp(r340871);
        double r340878 = r340877 * r340877;
        double r340879 = r340875 - r340877;
        double r340880 = r340878 + r340879;
        double r340881 = r340876 / r340880;
        double r340882 = sqrt(r340881);
        return r340882;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{e^{x} + 1}}\]
  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 pow-exp0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019121 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))