Average Error: 0.0 → 0.0
Time: 3.9s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\sqrt{2}}{\frac{e^{x} + e^{-x}}{\sqrt{2}}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\sqrt{2}}{\frac{e^{x} + e^{-x}}{\sqrt{2}}}}
double f(double x) {
        double r43603 = 2.0;
        double r43604 = x;
        double r43605 = exp(r43604);
        double r43606 = -r43604;
        double r43607 = exp(r43606);
        double r43608 = r43605 + r43607;
        double r43609 = r43603 / r43608;
        return r43609;
}
↓
double f(double x) {
        double r43610 = 2.0;
        double r43611 = x;
        double r43612 = exp(r43611);
        double r43613 = -r43611;
        double r43614 = exp(r43613);
        double r43615 = r43612 + r43614;
        double r43616 = r43610 / r43615;
        double r43617 = sqrt(r43616);
        double r43618 = sqrt(r43610);
        double r43619 = r43615 / r43618;
        double r43620 = r43618 / r43619;
        double r43621 = sqrt(r43620);
        double r43622 = r43617 * r43621;
        return r43622;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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-sqr-sqrt0.0

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

  6. Applied associate-/l*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))