Average Error: 4.1 → 0.1
Time: 22.1s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{1 + e^{x}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{1 + e^{x}}
double f(double x) {
        double r623598 = 2.0;
        double r623599 = x;
        double r623600 = r623598 * r623599;
        double r623601 = exp(r623600);
        double r623602 = 1.0;
        double r623603 = r623601 - r623602;
        double r623604 = exp(r623599);
        double r623605 = r623604 - r623602;
        double r623606 = r623603 / r623605;
        double r623607 = sqrt(r623606);
        return r623607;
}
↓
double f(double x) {
        double r623608 = 1.0;
        double r623609 = x;
        double r623610 = exp(r623609);
        double r623611 = r623608 + r623610;
        double r623612 = sqrt(r623611);
        return r623612;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\sqrt{e^{x} + 1}}\]
  3. Using strategy rm

  4. Applied +-commutative0.1

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

  5. Final simplification0.1

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

Reproduce

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