Average Error: 0.0 → 0.0
Time: 18.3s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}
double f(double x) {
        double r1775089 = 2.0;
        double r1775090 = x;
        double r1775091 = exp(r1775090);
        double r1775092 = -r1775090;
        double r1775093 = exp(r1775092);
        double r1775094 = r1775091 + r1775093;
        double r1775095 = r1775089 / r1775094;
        return r1775095;
}
↓
double f(double x) {
        double r1775096 = 2.0;
        double r1775097 = x;
        double r1775098 = exp(r1775097);
        double r1775099 = -r1775097;
        double r1775100 = exp(r1775099);
        double r1775101 = r1775098 + r1775100;
        double r1775102 = r1775096 / r1775101;
        double r1775103 = sqrt(r1775102);
        double r1775104 = r1775103 * r1775103;
        return r1775104;
}

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. 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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))