Average Error: 4.5 → 0.1
Time: 18.7s
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 r22150 = 2.0;
        double r22151 = x;
        double r22152 = r22150 * r22151;
        double r22153 = exp(r22152);
        double r22154 = 1.0;
        double r22155 = r22153 - r22154;
        double r22156 = exp(r22151);
        double r22157 = r22156 - r22154;
        double r22158 = r22155 / r22157;
        double r22159 = sqrt(r22158);
        return r22159;
}
↓
double f(double x) {
        double r22160 = 1.0;
        double r22161 = x;
        double r22162 = exp(r22161);
        double r22163 = r22162 + r22160;
        double r22164 = r22160 * r22163;
        double r22165 = sqrt(r22164);
        return r22165;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.5

    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
  2. Using strategy rm

  3. Applied flip--4.1

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

  4. Applied associate-/r/4.1

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

  5. Simplified2.7

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

  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 2019347 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))