Average Error: 4.2 → 0.1
Time: 26.8s
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 r532690 = 2.0;
        double r532691 = x;
        double r532692 = r532690 * r532691;
        double r532693 = exp(r532692);
        double r532694 = 1.0;
        double r532695 = r532693 - r532694;
        double r532696 = exp(r532691);
        double r532697 = r532696 - r532694;
        double r532698 = r532695 / r532697;
        double r532699 = sqrt(r532698);
        return r532699;
}
↓
double f(double x) {
        double r532700 = x;
        double r532701 = exp(r532700);
        double r532702 = sqrt(r532701);
        double r532703 = 1.0;
        double r532704 = hypot(r532702, r532703);
        return r532704;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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))))