Average Error: 4.4 → 0.1
Time: 20.6s
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 r364706 = 2.0;
        double r364707 = x;
        double r364708 = r364706 * r364707;
        double r364709 = exp(r364708);
        double r364710 = 1.0;
        double r364711 = r364709 - r364710;
        double r364712 = exp(r364707);
        double r364713 = r364712 - r364710;
        double r364714 = r364711 / r364713;
        double r364715 = sqrt(r364714);
        return r364715;
}
↓
double f(double x) {
        double r364716 = x;
        double r364717 = exp(r364716);
        double r364718 = sqrt(r364717);
        double r364719 = 1.0;
        double r364720 = hypot(r364718, r364719);
        return r364720;
}

Error

Bits error versus x

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Results

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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. 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 2019104 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))