Average Error: 4.2 → 0.1
Time: 20.7s
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 r755805 = 2.0;
        double r755806 = x;
        double r755807 = r755805 * r755806;
        double r755808 = exp(r755807);
        double r755809 = 1.0;
        double r755810 = r755808 - r755809;
        double r755811 = exp(r755806);
        double r755812 = r755811 - r755809;
        double r755813 = r755810 / r755812;
        double r755814 = sqrt(r755813);
        return r755814;
}
↓
double f(double x) {
        double r755815 = 1.0;
        double r755816 = x;
        double r755817 = exp(r755816);
        double r755818 = r755815 + r755817;
        double r755819 = sqrt(r755818);
        return r755819;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.2

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))