Average Error: 4.3 → 0.1
Time: 23.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[e^{\log \left(\sqrt{e^{x} + 1}\right)}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
e^{\log \left(\sqrt{e^{x} + 1}\right)}
double f(double x) {
        double r830513 = 2.0;
        double r830514 = x;
        double r830515 = r830513 * r830514;
        double r830516 = exp(r830515);
        double r830517 = 1.0;
        double r830518 = r830516 - r830517;
        double r830519 = exp(r830514);
        double r830520 = r830519 - r830517;
        double r830521 = r830518 / r830520;
        double r830522 = sqrt(r830521);
        return r830522;
}
↓
double f(double x) {
        double r830523 = x;
        double r830524 = exp(r830523);
        double r830525 = 1.0;
        double r830526 = r830524 + r830525;
        double r830527 = sqrt(r830526);
        double r830528 = log(r830527);
        double r830529 = exp(r830528);
        return r830529;
}

Error

Bits error versus x

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Results

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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 add-exp-log0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019133 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))