Average Error: 4.0 → 0.0
Time: 2.0m
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{1} \cdot \sqrt{e^{x} + 1}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{1} \cdot \sqrt{e^{x} + 1}
double f(double x) {
        double r19858 = 2.0;
        double r19859 = x;
        double r19860 = r19858 * r19859;
        double r19861 = exp(r19860);
        double r19862 = 1.0;
        double r19863 = r19861 - r19862;
        double r19864 = exp(r19859);
        double r19865 = r19864 - r19862;
        double r19866 = r19863 / r19865;
        double r19867 = sqrt(r19866);
        return r19867;
}
↓
double f(double x) {
        double r19868 = 1.0;
        double r19869 = sqrt(r19868);
        double r19870 = x;
        double r19871 = exp(r19870);
        double r19872 = r19871 + r19868;
        double r19873 = sqrt(r19872);
        double r19874 = r19869 * r19873;
        return r19874;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.0

    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
  2. Using strategy rm

  3. Applied flip--3.6

    \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]

  4. Applied associate-/r/3.6

    \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]

  5. Applied sqrt-prod3.6

    \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}}\]

  6. Simplified2.5

    \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}}} \cdot \sqrt{e^{x} + 1}\]

  7. Taylor expanded around 0 0.0

    \[\leadsto \sqrt{\color{blue}{1}} \cdot \sqrt{e^{x} + 1}\]

  8. Final simplification0.0

    \[\leadsto \sqrt{1} \cdot \sqrt{e^{x} + 1}\]

Reproduce

herbie shell --seed 2019202 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))