Average Error: 4.3 → 0.1
Time: 18.7s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)
double f(double x) {
        double r346766 = 2.0;
        double r346767 = x;
        double r346768 = r346766 * r346767;
        double r346769 = exp(r346768);
        double r346770 = 1.0;
        double r346771 = r346769 - r346770;
        double r346772 = exp(r346767);
        double r346773 = r346772 - r346770;
        double r346774 = r346771 / r346773;
        double r346775 = sqrt(r346774);
        return r346775;
}
↓
double f(double x) {
        double r346776 = 1.0;
        double r346777 = x;
        double r346778 = exp(r346777);
        double r346779 = sqrt(r346778);
        double r346780 = hypot(r346776, r346779);
        return r346780;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Initial program 4.3

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

  2. Simplified0.1

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied *-un-lft-identity0.1

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

  6. Applied hypot-def0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))