Average Error: 4.4 → 0.2
Time: 1.3m
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt[3]{\sqrt{e^{x} + 1} \cdot \left(\sqrt{e^{x} + 1} \cdot \sqrt{e^{x} + 1}\right)}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt[3]{\sqrt{e^{x} + 1} \cdot \left(\sqrt{e^{x} + 1} \cdot \sqrt{e^{x} + 1}\right)}
double f(double x) {
        double r2584137 = 2.0;
        double r2584138 = x;
        double r2584139 = r2584137 * r2584138;
        double r2584140 = exp(r2584139);
        double r2584141 = 1.0;
        double r2584142 = r2584140 - r2584141;
        double r2584143 = exp(r2584138);
        double r2584144 = r2584143 - r2584141;
        double r2584145 = r2584142 / r2584144;
        double r2584146 = sqrt(r2584145);
        return r2584146;
}
↓
double f(double x) {
        double r2584147 = x;
        double r2584148 = exp(r2584147);
        double r2584149 = 1.0;
        double r2584150 = r2584148 + r2584149;
        double r2584151 = sqrt(r2584150);
        double r2584152 = r2584151 * r2584151;
        double r2584153 = r2584151 * r2584152;
        double r2584154 = cbrt(r2584153);
        return r2584154;
}

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. Simplified0.1

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

  4. Applied add-cbrt-cube0.2

    \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{e^{x} + 1} \cdot \sqrt{e^{x} + 1}\right) \cdot \sqrt{e^{x} + 1}}}\]

  5. Final simplification0.2

    \[\leadsto \sqrt[3]{\sqrt{e^{x} + 1} \cdot \left(\sqrt{e^{x} + 1} \cdot \sqrt{e^{x} + 1}\right)}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))