Average Error: 4.6 → 0.1
Time: 21.7s
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 r694458 = 2.0;
        double r694459 = x;
        double r694460 = r694458 * r694459;
        double r694461 = exp(r694460);
        double r694462 = 1.0;
        double r694463 = r694461 - r694462;
        double r694464 = exp(r694459);
        double r694465 = r694464 - r694462;
        double r694466 = r694463 / r694465;
        double r694467 = sqrt(r694466);
        return r694467;
}
↓
double f(double x) {
        double r694468 = x;
        double r694469 = exp(r694468);
        double r694470 = 1.0;
        double r694471 = r694469 + r694470;
        double r694472 = sqrt(r694471);
        return r694472;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.6

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

  2. Simplified0.1

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

  3. Taylor expanded around -inf 0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))