Average Error: 4.3 → 0.1
Time: 14.2s
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 r268539 = 2.0;
        double r268540 = x;
        double r268541 = r268539 * r268540;
        double r268542 = exp(r268541);
        double r268543 = 1.0;
        double r268544 = r268542 - r268543;
        double r268545 = exp(r268540);
        double r268546 = r268545 - r268543;
        double r268547 = r268544 / r268546;
        double r268548 = sqrt(r268547);
        return r268548;
}
↓
double f(double x) {
        double r268549 = x;
        double r268550 = exp(r268549);
        double r268551 = 1.0;
        double r268552 = r268550 + r268551;
        double r268553 = sqrt(r268552);
        return r268553;
}

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. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))