Average Error: 4.3 → 0.1
Time: 22.5s
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 r500039 = 2.0;
        double r500040 = x;
        double r500041 = r500039 * r500040;
        double r500042 = exp(r500041);
        double r500043 = 1.0;
        double r500044 = r500042 - r500043;
        double r500045 = exp(r500040);
        double r500046 = r500045 - r500043;
        double r500047 = r500044 / r500046;
        double r500048 = sqrt(r500047);
        return r500048;
}
↓
double f(double x) {
        double r500049 = 1.0;
        double r500050 = x;
        double r500051 = exp(r500050);
        double r500052 = r500049 + r500051;
        double r500053 = sqrt(r500052);
        return r500053;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 4.3

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019149 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))