Average Error: 4.3 → 0.1
Time: 16.0s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\left(\sqrt{e^{x}}\right)^2 + 1^2}^*\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\left(\sqrt{e^{x}}\right)^2 + 1^2}^*
double f(double x) {
        double r259375 = 2.0;
        double r259376 = x;
        double r259377 = r259375 * r259376;
        double r259378 = exp(r259377);
        double r259379 = 1.0;
        double r259380 = r259378 - r259379;
        double r259381 = exp(r259376);
        double r259382 = r259381 - r259379;
        double r259383 = r259380 / r259382;
        double r259384 = sqrt(r259383);
        return r259384;
}
↓
double f(double x) {
        double r259385 = x;
        double r259386 = exp(r259385);
        double r259387 = sqrt(r259386);
        double r259388 = 1.0;
        double r259389 = hypot(r259387, r259388);
        return r259389;
}

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 *-un-lft-identity0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied hypot-def0.1

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

  7. Final simplification0.1

    \[\leadsto \sqrt{\left(\sqrt{e^{x}}\right)^2 + 1^2}^*\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))