Average Error: 4.3 → 0.0
Time: 18.8s
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 r446544 = 2.0;
        double r446545 = x;
        double r446546 = r446544 * r446545;
        double r446547 = exp(r446546);
        double r446548 = 1.0;
        double r446549 = r446547 - r446548;
        double r446550 = exp(r446545);
        double r446551 = r446550 - r446548;
        double r446552 = r446549 / r446551;
        double r446553 = sqrt(r446552);
        return r446553;
}
↓
double f(double x) {
        double r446554 = x;
        double r446555 = exp(r446554);
        double r446556 = 1.0;
        double r446557 = r446555 + r446556;
        double r446558 = sqrt(r446557);
        return r446558;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.3

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))