Average Error: 4.3 → 0.1
Time: 20.2s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\mathsf{hypot}\left(\sqrt{e^{x}}, 1\right)\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\mathsf{hypot}\left(\sqrt{e^{x}}, 1\right)
double f(double x) {
        double r580088 = 2.0;
        double r580089 = x;
        double r580090 = r580088 * r580089;
        double r580091 = exp(r580090);
        double r580092 = 1.0;
        double r580093 = r580091 - r580092;
        double r580094 = exp(r580089);
        double r580095 = r580094 - r580092;
        double r580096 = r580093 / r580095;
        double r580097 = sqrt(r580096);
        return r580097;
}
↓
double f(double x) {
        double r580098 = x;
        double r580099 = exp(r580098);
        double r580100 = sqrt(r580099);
        double r580101 = 1.0;
        double r580102 = hypot(r580100, r580101);
        return r580102;
}

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

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

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))