Average Error: 4.3 → 0.1
Time: 25.2s
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 r840658 = 2.0;
        double r840659 = x;
        double r840660 = r840658 * r840659;
        double r840661 = exp(r840660);
        double r840662 = 1.0;
        double r840663 = r840661 - r840662;
        double r840664 = exp(r840659);
        double r840665 = r840664 - r840662;
        double r840666 = r840663 / r840665;
        double r840667 = sqrt(r840666);
        return r840667;
}
↓
double f(double x) {
        double r840668 = x;
        double r840669 = exp(r840668);
        double r840670 = 1.0;
        double r840671 = r840669 + r840670;
        double r840672 = sqrt(r840671);
        return r840672;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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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 2019138 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))