Average Error: 4.4 → 0.1
Time: 20.0s
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 r264877 = 2.0;
        double r264878 = x;
        double r264879 = r264877 * r264878;
        double r264880 = exp(r264879);
        double r264881 = 1.0;
        double r264882 = r264880 - r264881;
        double r264883 = exp(r264878);
        double r264884 = r264883 - r264881;
        double r264885 = r264882 / r264884;
        double r264886 = sqrt(r264885);
        return r264886;
}
↓
double f(double x) {
        double r264887 = 1.0;
        double r264888 = x;
        double r264889 = exp(r264888);
        double r264890 = r264887 + r264889;
        double r264891 = sqrt(r264890);
        return r264891;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 4.4

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))