Average Error: 4.1 → 0.1
Time: 18.7s
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 r504012 = 2.0;
        double r504013 = x;
        double r504014 = r504012 * r504013;
        double r504015 = exp(r504014);
        double r504016 = 1.0;
        double r504017 = r504015 - r504016;
        double r504018 = exp(r504013);
        double r504019 = r504018 - r504016;
        double r504020 = r504017 / r504019;
        double r504021 = sqrt(r504020);
        return r504021;
}
↓
double f(double x) {
        double r504022 = x;
        double r504023 = exp(r504022);
        double r504024 = log1p(r504023);
        double r504025 = exp(r504024);
        double r504026 = sqrt(r504025);
        return r504026;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\sqrt{e^{x} + 1}}\]
  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 2019141 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))