Average Error: 4.4 → 0.1
Time: 12.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{1 \cdot \left(e^{x} + 1\right)}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{1 \cdot \left(e^{x} + 1\right)}
double f(double x) {
        double r12180 = 2.0;
        double r12181 = x;
        double r12182 = r12180 * r12181;
        double r12183 = exp(r12182);
        double r12184 = 1.0;
        double r12185 = r12183 - r12184;
        double r12186 = exp(r12181);
        double r12187 = r12186 - r12184;
        double r12188 = r12185 / r12187;
        double r12189 = sqrt(r12188);
        return r12189;
}
↓
double f(double x) {
        double r12190 = 1.0;
        double r12191 = x;
        double r12192 = exp(r12191);
        double r12193 = r12192 + r12190;
        double r12194 = r12190 * r12193;
        double r12195 = sqrt(r12194);
        return r12195;
}

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. Using strategy rm

  3. Applied flip--4.0

    \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]

  4. Applied associate-/r/4.0

    \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]

  5. Simplified2.9

    \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \left(e^{x} + 1\right)}\]

  6. Taylor expanded around 0 0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019350 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))