Average Error: 4.4 → 0.0
Time: 16.4s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\left(\sqrt{e^{x}}\right)^2 + 1^2}^*\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\left(\sqrt{e^{x}}\right)^2 + 1^2}^*
double f(double x) {
        double r252015 = 2.0;
        double r252016 = x;
        double r252017 = r252015 * r252016;
        double r252018 = exp(r252017);
        double r252019 = 1.0;
        double r252020 = r252018 - r252019;
        double r252021 = exp(r252016);
        double r252022 = r252021 - r252019;
        double r252023 = r252020 / r252022;
        double r252024 = sqrt(r252023);
        return r252024;
}
↓
double f(double x) {
        double r252025 = x;
        double r252026 = exp(r252025);
        double r252027 = sqrt(r252026);
        double r252028 = 1.0;
        double r252029 = hypot(r252027, r252028);
        return r252029;
}

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

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied hypot-def0.0

    \[\leadsto \color{blue}{\sqrt{\left(\sqrt{e^{x}}\right)^2 + 1^2}^*}\]

  7. Final simplification0.0

    \[\leadsto \sqrt{\left(\sqrt{e^{x}}\right)^2 + 1^2}^*\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))