Average Error: 4.6 → 0.1
Time: 26.2s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{{\left(e^{x}\right)}^{3} + 1}{e^{x} \cdot e^{x} + \frac{1 - e^{x} \cdot e^{x}}{e^{x} + 1}}}\]
double f(double x) {
        double r501132 = 2.0;
        double r501133 = x;
        double r501134 = r501132 * r501133;
        double r501135 = exp(r501134);
        double r501136 = 1.0;
        double r501137 = r501135 - r501136;
        double r501138 = exp(r501133);
        double r501139 = r501138 - r501136;
        double r501140 = r501137 / r501139;
        double r501141 = sqrt(r501140);
        return r501141;
}
↓
double f(double x) {
        double r501142 = x;
        double r501143 = exp(r501142);
        double r501144 = 3.0;
        double r501145 = pow(r501143, r501144);
        double r501146 = 1.0;
        double r501147 = r501145 + r501146;
        double r501148 = r501143 * r501143;
        double r501149 = r501146 - r501148;
        double r501150 = r501143 + r501146;
        double r501151 = r501149 / r501150;
        double r501152 = r501148 + r501151;
        double r501153 = r501147 / r501152;
        double r501154 = sqrt(r501153);
        return r501154;
}


\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{{\left(e^{x}\right)}^{3} + 1}{e^{x} \cdot e^{x} + \frac{1 - e^{x} \cdot e^{x}}{e^{x} + 1}}}

Error

Bits error versus x

Derivation

  1. Initial program 4.6

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

  2. Simplified0.1

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

  4. Applied 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 flip--0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019101 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))