Average Error: 4.4 → 0.1
Time: 25.9s
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 r968771 = 2.0;
        double r968772 = x;
        double r968773 = r968771 * r968772;
        double r968774 = exp(r968773);
        double r968775 = 1.0;
        double r968776 = r968774 - r968775;
        double r968777 = exp(r968772);
        double r968778 = r968777 - r968775;
        double r968779 = r968776 / r968778;
        double r968780 = sqrt(r968779);
        return r968780;
}
↓
double f(double x) {
        double r968781 = 1.0;
        double r968782 = x;
        double r968783 = exp(r968782);
        double r968784 = sqrt(r968783);
        double r968785 = hypot(r968781, r968784);
        return r968785;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Initial program 4.4

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

  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 2019158 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))