Average Error: 4.4 → 0.0
Time: 18.5s
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 r300572 = 2.0;
        double r300573 = x;
        double r300574 = r300572 * r300573;
        double r300575 = exp(r300574);
        double r300576 = 1.0;
        double r300577 = r300575 - r300576;
        double r300578 = exp(r300573);
        double r300579 = r300578 - r300576;
        double r300580 = r300577 / r300579;
        double r300581 = sqrt(r300580);
        return r300581;
}
↓
double f(double x) {
        double r300582 = x;
        double r300583 = exp(r300582);
        double r300584 = log1p(r300583);
        double r300585 = exp(r300584);
        double r300586 = sqrt(r300585);
        return r300586;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.4

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

  2. Simplified0.0

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

  4. Applied add-exp-log0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))