Average Error: 0.0 → 0.0
Time: 18.1s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{2}{e^{x} + \frac{1}{e^{x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\frac{2}{e^{x} + \frac{1}{e^{x}}}
double f(double x) {
        double r1425529 = 2.0;
        double r1425530 = x;
        double r1425531 = exp(r1425530);
        double r1425532 = -r1425530;
        double r1425533 = exp(r1425532);
        double r1425534 = r1425531 + r1425533;
        double r1425535 = r1425529 / r1425534;
        return r1425535;
}
↓
double f(double x) {
        double r1425536 = 2.0;
        double r1425537 = x;
        double r1425538 = exp(r1425537);
        double r1425539 = 1.0;
        double r1425540 = r1425539 / r1425538;
        double r1425541 = r1425538 + r1425540;
        double r1425542 = r1425536 / r1425541;
        return r1425542;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 0.0

    \[\frac{2}{e^{x} + e^{-x}}\]

  2. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\frac{2}{e^{x} + e^{-x}}}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\frac{2}{e^{x} + \frac{1}{e^{x}}}}\]

  4. Final simplification0.0

    \[\leadsto \frac{2}{e^{x} + \frac{1}{e^{x}}}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))