Average Error: 4.6 → 0.0
Time: 31.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\mathsf{hypot}\left(\left(\sqrt{e^{x}}\right), 1\right)\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\mathsf{hypot}\left(\left(\sqrt{e^{x}}\right), 1\right)
double f(double x) {
        double r1127736 = 2.0;
        double r1127737 = x;
        double r1127738 = r1127736 * r1127737;
        double r1127739 = exp(r1127738);
        double r1127740 = 1.0;
        double r1127741 = r1127739 - r1127740;
        double r1127742 = exp(r1127737);
        double r1127743 = r1127742 - r1127740;
        double r1127744 = r1127741 / r1127743;
        double r1127745 = sqrt(r1127744);
        return r1127745;
}
↓
double f(double x) {
        double r1127746 = x;
        double r1127747 = exp(r1127746);
        double r1127748 = sqrt(r1127747);
        double r1127749 = 1.0;
        double r1127750 = hypot(r1127748, r1127749);
        return r1127750;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.6

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

  2. Simplified0.0

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied hypot-def0.0

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\left(\sqrt{e^{x}}\right), 1\right)}\]

  7. Final simplification0.0

    \[\leadsto \mathsf{hypot}\left(\left(\sqrt{e^{x}}\right), 1\right)\]

Reproduce

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