Average Error: 4.3 → 0.1
Time: 19.9s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{\left(e^{x} \cdot e^{x}\right) \cdot e^{x} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{\left(e^{x} \cdot e^{x}\right) \cdot e^{x} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}
double f(double x) {
        double r566016 = 2.0;
        double r566017 = x;
        double r566018 = r566016 * r566017;
        double r566019 = exp(r566018);
        double r566020 = 1.0;
        double r566021 = r566019 - r566020;
        double r566022 = exp(r566017);
        double r566023 = r566022 - r566020;
        double r566024 = r566021 / r566023;
        double r566025 = sqrt(r566024);
        return r566025;
}
↓
double f(double x) {
        double r566026 = x;
        double r566027 = exp(r566026);
        double r566028 = r566027 * r566027;
        double r566029 = r566028 * r566027;
        double r566030 = 1.0;
        double r566031 = r566029 + r566030;
        double r566032 = r566030 - r566027;
        double r566033 = r566028 + r566032;
        double r566034 = r566031 / r566033;
        double r566035 = sqrt(r566034);
        return r566035;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.3

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

  2. Simplified0.1

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

  4. Applied add-cbrt-cube0.1

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

  6. Applied flip3-+0.1

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

  7. Applied flip3-+0.1

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

  8. Applied flip3-+0.1

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

  9. Applied frac-times0.2

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

  10. Applied frac-times0.2

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

  11. Applied cbrt-div0.2

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

  12. Simplified0.2

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019139 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))