Average Error: 4.3 → 0.1
Time: 22.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}
double f(double x) {
        double r539319 = 2.0;
        double r539320 = x;
        double r539321 = r539319 * r539320;
        double r539322 = exp(r539321);
        double r539323 = 1.0;
        double r539324 = r539322 - r539323;
        double r539325 = exp(r539320);
        double r539326 = r539325 - r539323;
        double r539327 = r539324 / r539326;
        double r539328 = sqrt(r539327);
        return r539328;
}
↓
double f(double x) {
        double r539329 = x;
        double r539330 = r539329 + r539329;
        double r539331 = expm1(r539330);
        double r539332 = expm1(r539329);
        double r539333 = r539331 / r539332;
        double r539334 = sqrt(r539333);
        return r539334;
}

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 flip-+3.6

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

  5. Simplified4.7

    \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}}\]

  6. Simplified0.1

    \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}}\]

  7. Final simplification0.1

    \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}\]

Reproduce

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