Average Error: 4.7 → 0.1
Time: 15.5s
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 r187795 = 2.0;
        double r187796 = x;
        double r187797 = r187795 * r187796;
        double r187798 = exp(r187797);
        double r187799 = 1.0;
        double r187800 = r187798 - r187799;
        double r187801 = exp(r187796);
        double r187802 = r187801 - r187799;
        double r187803 = r187800 / r187802;
        double r187804 = sqrt(r187803);
        return r187804;
}
↓
double f(double x) {
        double r187805 = 1.0;
        double r187806 = x;
        double r187807 = exp(r187806);
        double r187808 = r187805 + r187807;
        double r187809 = sqrt(r187808);
        return r187809;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.7

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))