Average Error: 4.4 → 0.1
Time: 19.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{1} \cdot \sqrt{e^{x} + 1}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{1} \cdot \sqrt{e^{x} + 1}
double f(double x) {
        double r20072 = 2.0;
        double r20073 = x;
        double r20074 = r20072 * r20073;
        double r20075 = exp(r20074);
        double r20076 = 1.0;
        double r20077 = r20075 - r20076;
        double r20078 = exp(r20073);
        double r20079 = r20078 - r20076;
        double r20080 = r20077 / r20079;
        double r20081 = sqrt(r20080);
        return r20081;
}
↓
double f(double x) {
        double r20082 = 1.0;
        double r20083 = sqrt(r20082);
        double r20084 = x;
        double r20085 = exp(r20084);
        double r20086 = r20085 + r20082;
        double r20087 = sqrt(r20086);
        double r20088 = r20083 * r20087;
        return r20088;
}

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--3.9

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

  4. Applied associate-/r/3.9

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

  5. Applied sqrt-prod3.9

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

  6. Simplified2.8

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

  7. Taylor expanded around 0 0.1

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

  8. Final simplification0.1

    \[\leadsto \sqrt{1} \cdot \sqrt{e^{x} + 1}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))