Average Error: 4.2 → 0.1
Time: 19.6s
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 r527137 = 2.0;
        double r527138 = x;
        double r527139 = r527137 * r527138;
        double r527140 = exp(r527139);
        double r527141 = 1.0;
        double r527142 = r527140 - r527141;
        double r527143 = exp(r527138);
        double r527144 = r527143 - r527141;
        double r527145 = r527142 / r527144;
        double r527146 = sqrt(r527145);
        return r527146;
}
↓
double f(double x) {
        double r527147 = x;
        double r527148 = exp(r527147);
        double r527149 = sqrt(r527148);
        double r527150 = 1.0;
        double r527151 = hypot(r527149, r527150);
        return r527151;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 4.2

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

  2. Simplified0.1

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

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

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied hypot-def0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))