Average Error: 0.0 → 0.0
Time: 12.0s
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 r697588 = 2.0;
        double r697589 = x;
        double r697590 = exp(r697589);
        double r697591 = -r697589;
        double r697592 = exp(r697591);
        double r697593 = r697590 + r697592;
        double r697594 = r697588 / r697593;
        return r697594;
}
↓
double f(double x) {
        double r697595 = 2.0;
        double r697596 = x;
        double r697597 = exp(r697596);
        double r697598 = -r697596;
        double r697599 = exp(r697598);
        double r697600 = r697597 + r697599;
        double r697601 = r697595 / r697600;
        double r697602 = sqrt(r697601);
        double r697603 = r697602 * r697602;
        return r697603;
}

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 2019128 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))