Average Error: 4.0 → 0.1
Time: 37.7s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{\mathsf{log1p}\left(e^{x}\right)}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{e^{\mathsf{log1p}\left(e^{x}\right)}}
double f(double x) {
        double r601350 = 2.0;
        double r601351 = x;
        double r601352 = r601350 * r601351;
        double r601353 = exp(r601352);
        double r601354 = 1.0;
        double r601355 = r601353 - r601354;
        double r601356 = exp(r601351);
        double r601357 = r601356 - r601354;
        double r601358 = r601355 / r601357;
        double r601359 = sqrt(r601358);
        return r601359;
}
↓
double f(double x) {
        double r601360 = x;
        double r601361 = exp(r601360);
        double r601362 = log1p(r601361);
        double r601363 = exp(r601362);
        double r601364 = sqrt(r601363);
        return r601364;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{e^{x} + 1}}\]
  3. Using strategy rm

  4. Applied add-exp-log0.1

    \[\leadsto \sqrt{\color{blue}{e^{\log \left(e^{x} + 1\right)}}}\]

  5. Simplified0.1

    \[\leadsto \sqrt{e^{\color{blue}{\mathsf{log1p}\left(e^{x}\right)}}}\]

  6. Final simplification0.1

    \[\leadsto \sqrt{e^{\mathsf{log1p}\left(e^{x}\right)}}\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))