Average Error: 4.4 → 0.2
Time: 42.7s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{\log \left(e^{\left(x + x\right) + x} + 1\right) - \log \left(1 - e^{x} \cdot \left(1 - e^{x}\right)\right)}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{e^{\log \left(e^{\left(x + x\right) + x} + 1\right) - \log \left(1 - e^{x} \cdot \left(1 - e^{x}\right)\right)}}
double f(double x) {
        double r1157739 = 2.0;
        double r1157740 = x;
        double r1157741 = r1157739 * r1157740;
        double r1157742 = exp(r1157741);
        double r1157743 = 1.0;
        double r1157744 = r1157742 - r1157743;
        double r1157745 = exp(r1157740);
        double r1157746 = r1157745 - r1157743;
        double r1157747 = r1157744 / r1157746;
        double r1157748 = sqrt(r1157747);
        return r1157748;
}
↓
double f(double x) {
        double r1157749 = x;
        double r1157750 = r1157749 + r1157749;
        double r1157751 = r1157750 + r1157749;
        double r1157752 = exp(r1157751);
        double r1157753 = 1.0;
        double r1157754 = r1157752 + r1157753;
        double r1157755 = log(r1157754);
        double r1157756 = exp(r1157749);
        double r1157757 = r1157753 - r1157756;
        double r1157758 = r1157756 * r1157757;
        double r1157759 = r1157753 - r1157758;
        double r1157760 = log(r1157759);
        double r1157761 = r1157755 - r1157760;
        double r1157762 = exp(r1157761);
        double r1157763 = sqrt(r1157762);
        return r1157763;
}

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. Simplified0.0

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

  4. Applied flip3-+0.1

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

  6. Applied add-exp-log0.1

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

  7. Applied add-exp-log0.2

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

  8. Applied div-exp0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

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