Average Error: 0.0 → 0.0
Time: 8.2s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \left(\left|\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right| \cdot \sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}\right)\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \left(\left|\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right| \cdot \sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}\right)
double f(double x) {
        double r53758 = 2.0;
        double r53759 = x;
        double r53760 = exp(r53759);
        double r53761 = -r53759;
        double r53762 = exp(r53761);
        double r53763 = r53760 + r53762;
        double r53764 = r53758 / r53763;
        return r53764;
}
↓
double f(double x) {
        double r53765 = 2.0;
        double r53766 = x;
        double r53767 = exp(r53766);
        double r53768 = -r53766;
        double r53769 = exp(r53768);
        double r53770 = r53767 + r53769;
        double r53771 = r53765 / r53770;
        double r53772 = sqrt(r53771);
        double r53773 = cbrt(r53771);
        double r53774 = fabs(r53773);
        double r53775 = sqrt(r53773);
        double r53776 = r53774 * r53775;
        double r53777 = r53772 * r53776;
        return r53777;
}

Error

Bits error versus x

Try it out

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-cube-cbrt0.0

    \[\leadsto \sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right) \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}}\]

  6. Applied sqrt-prod0.0

    \[\leadsto \sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}\right)}\]

  7. Simplified0.0

    \[\leadsto \sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \left(\color{blue}{\left|\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right|} \cdot \sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}\right)\]

  8. Final simplification0.0

    \[\leadsto \sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \left(\left|\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right| \cdot \sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}\right)\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))