Average Error: 0.0 → 0.0
Time: 14.9s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \left(\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \left(\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}
double f(double x) {
        double r1808406 = 2.0;
        double r1808407 = x;
        double r1808408 = exp(r1808407);
        double r1808409 = -r1808407;
        double r1808410 = exp(r1808409);
        double r1808411 = r1808408 + r1808410;
        double r1808412 = r1808406 / r1808411;
        return r1808412;
}
↓
double f(double x) {
        double r1808413 = 2.0;
        double r1808414 = x;
        double r1808415 = exp(r1808414);
        double r1808416 = -r1808414;
        double r1808417 = exp(r1808416);
        double r1808418 = r1808415 + r1808417;
        double r1808419 = r1808413 / r1808418;
        double r1808420 = cbrt(r1808419);
        double r1808421 = r1808420 * r1808420;
        double r1808422 = r1808420 * r1808421;
        double r1808423 = sqrt(r1808422);
        double r1808424 = sqrt(r1808419);
        double r1808425 = r1808423 * r1808424;
        return r1808425;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Initial program 0.0

    \[\frac{2}{e^{x} + e^{-x}}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.0

    \[\leadsto \color{blue}{\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.0

    \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right) \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]

  6. Final simplification0.0

    \[\leadsto \sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \left(\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))