Average Error: 0.0 → 0.0
Time: 3.9s
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 r1786717 = 2.0;
        double r1786718 = x;
        double r1786719 = exp(r1786718);
        double r1786720 = -r1786718;
        double r1786721 = exp(r1786720);
        double r1786722 = r1786719 + r1786721;
        double r1786723 = r1786717 / r1786722;
        return r1786723;
}
↓
double f(double x) {
        double r1786724 = 2.0;
        double r1786725 = x;
        double r1786726 = exp(r1786725);
        double r1786727 = 1.0;
        double r1786728 = r1786727 / r1786726;
        double r1786729 = r1786726 + r1786728;
        double r1786730 = r1786724 / r1786729;
        return r1786730;
}

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

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2.0 (+ (exp x) (exp (- x)))))