Average Error: 4.4 → 0.1
Time: 18.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{x} + 1}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{e^{x} + 1}
double f(double x) {
        double r471407 = 2.0;
        double r471408 = x;
        double r471409 = r471407 * r471408;
        double r471410 = exp(r471409);
        double r471411 = 1.0;
        double r471412 = r471410 - r471411;
        double r471413 = exp(r471408);
        double r471414 = r471413 - r471411;
        double r471415 = r471412 / r471414;
        double r471416 = sqrt(r471415);
        return r471416;
}
↓
double f(double x) {
        double r471417 = x;
        double r471418 = exp(r471417);
        double r471419 = 1.0;
        double r471420 = r471418 + r471419;
        double r471421 = sqrt(r471420);
        return r471421;
}

Error

Bits error versus x

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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. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))