Average Error: 4.3 → 0.1
Time: 23.7s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{\mathsf{log1p}\left(\left(e^{x}\right)\right)}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{e^{\mathsf{log1p}\left(\left(e^{x}\right)\right)}}
double f(double x) {
        double r740891 = 2.0;
        double r740892 = x;
        double r740893 = r740891 * r740892;
        double r740894 = exp(r740893);
        double r740895 = 1.0;
        double r740896 = r740894 - r740895;
        double r740897 = exp(r740892);
        double r740898 = r740897 - r740895;
        double r740899 = r740896 / r740898;
        double r740900 = sqrt(r740899);
        return r740900;
}
↓
double f(double x) {
        double r740901 = x;
        double r740902 = exp(r740901);
        double r740903 = log1p(r740902);
        double r740904 = exp(r740903);
        double r740905 = sqrt(r740904);
        return r740905;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.3

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

  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

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))