Average Error: 4.2 → 0.1
Time: 14.6s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\mathsf{hypot}\left(\left(\sqrt{e^{x}}\right), 1\right)\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\mathsf{hypot}\left(\left(\sqrt{e^{x}}\right), 1\right)
double f(double x) {
        double r154087 = 2.0;
        double r154088 = x;
        double r154089 = r154087 * r154088;
        double r154090 = exp(r154089);
        double r154091 = 1.0;
        double r154092 = r154090 - r154091;
        double r154093 = exp(r154088);
        double r154094 = r154093 - r154091;
        double r154095 = r154092 / r154094;
        double r154096 = sqrt(r154095);
        return r154096;
}
↓
double f(double x) {
        double r154097 = x;
        double r154098 = exp(r154097);
        double r154099 = sqrt(r154098);
        double r154100 = 1.0;
        double r154101 = hypot(r154099, r154100);
        return r154101;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.2

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

  2. Simplified0.1

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

  4. Applied *-un-lft-identity0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied hypot-def0.1

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\left(\sqrt{e^{x}}\right), 1\right)}\]

  7. Final simplification0.1

    \[\leadsto \mathsf{hypot}\left(\left(\sqrt{e^{x}}\right), 1\right)\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))