Average Error: 4.4 → 0.1
Time: 20.4s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{e^{x} \cdot \left(\sqrt{e^{x}} \cdot \left(\sqrt{e^{x}} \cdot e^{x}\right)\right) + 1}{\left(1 - e^{x}\right) + e^{x} \cdot e^{x}}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{e^{x} \cdot \left(\sqrt{e^{x}} \cdot \left(\sqrt{e^{x}} \cdot e^{x}\right)\right) + 1}{\left(1 - e^{x}\right) + e^{x} \cdot e^{x}}}
double f(double x) {
        double r581747 = 2.0;
        double r581748 = x;
        double r581749 = r581747 * r581748;
        double r581750 = exp(r581749);
        double r581751 = 1.0;
        double r581752 = r581750 - r581751;
        double r581753 = exp(r581748);
        double r581754 = r581753 - r581751;
        double r581755 = r581752 / r581754;
        double r581756 = sqrt(r581755);
        return r581756;
}
↓
double f(double x) {
        double r581757 = x;
        double r581758 = exp(r581757);
        double r581759 = sqrt(r581758);
        double r581760 = r581759 * r581758;
        double r581761 = r581759 * r581760;
        double r581762 = r581758 * r581761;
        double r581763 = 1.0;
        double r581764 = r581762 + r581763;
        double r581765 = r581763 - r581758;
        double r581766 = r581758 * r581758;
        double r581767 = r581765 + r581766;
        double r581768 = r581764 / r581767;
        double r581769 = sqrt(r581768);
        return r581769;
}

Error

Bits error versus x

Try it out

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. Taylor expanded around inf 0.1

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

  5. Applied flip3-+0.1

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

  6. Simplified0.1

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

  7. Simplified0.1

    \[\leadsto \sqrt{\frac{e^{x} \cdot \left(e^{x} \cdot e^{x}\right) + 1}{\color{blue}{\left(1 - e^{x}\right) + e^{x} \cdot e^{x}}}}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt0.1

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

  10. Applied associate-*r*0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019132 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))