Average Error: 4.4 → 0.1
Time: 17.6s
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 r428537 = 2.0;
        double r428538 = x;
        double r428539 = r428537 * r428538;
        double r428540 = exp(r428539);
        double r428541 = 1.0;
        double r428542 = r428540 - r428541;
        double r428543 = exp(r428538);
        double r428544 = r428543 - r428541;
        double r428545 = r428542 / r428544;
        double r428546 = sqrt(r428545);
        return r428546;
}
↓
double f(double x) {
        double r428547 = x;
        double r428548 = exp(r428547);
        double r428549 = sqrt(r428548);
        double r428550 = 1.0;
        double r428551 = hypot(r428549, r428550);
        return r428551;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.4

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

  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 2019132 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))