Average Error: 4.1 → 0.1
Time: 25.2s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{x} + 1}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{e^{x} + 1}
double f(double x) {
        double r648683 = 2.0;
        double r648684 = x;
        double r648685 = r648683 * r648684;
        double r648686 = exp(r648685);
        double r648687 = 1.0;
        double r648688 = r648686 - r648687;
        double r648689 = exp(r648684);
        double r648690 = r648689 - r648687;
        double r648691 = r648688 / r648690;
        double r648692 = sqrt(r648691);
        return r648692;
}
↓
double f(double x) {
        double r648693 = x;
        double r648694 = exp(r648693);
        double r648695 = 1.0;
        double r648696 = r648694 + r648695;
        double r648697 = sqrt(r648696);
        return r648697;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.1

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))