Average Error: 4.4 → 0.0
Time: 16.8s
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 r367293 = 2.0;
        double r367294 = x;
        double r367295 = r367293 * r367294;
        double r367296 = exp(r367295);
        double r367297 = 1.0;
        double r367298 = r367296 - r367297;
        double r367299 = exp(r367294);
        double r367300 = r367299 - r367297;
        double r367301 = r367298 / r367300;
        double r367302 = sqrt(r367301);
        return r367302;
}
↓
double f(double x) {
        double r367303 = x;
        double r367304 = exp(r367303);
        double r367305 = 1.0;
        double r367306 = r367304 + r367305;
        double r367307 = sqrt(r367306);
        return r367307;
}

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. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))