Average Error: 0.0 → 0.0
Time: 16.9s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\left(\sqrt{\frac{2}{e^{-x} + e^{x}}} \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\frac{\sqrt{2}}{e^{x} + e^{-x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\left(\sqrt{\frac{2}{e^{-x} + e^{x}}} \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\frac{\sqrt{2}}{e^{x} + e^{-x}}}
double f(double x) {
        double r55991 = 2.0;
        double r55992 = x;
        double r55993 = exp(r55992);
        double r55994 = -r55992;
        double r55995 = exp(r55994);
        double r55996 = r55993 + r55995;
        double r55997 = r55991 / r55996;
        return r55997;
}
↓
double f(double x) {
        double r55998 = 2.0;
        double r55999 = x;
        double r56000 = -r55999;
        double r56001 = exp(r56000);
        double r56002 = exp(r55999);
        double r56003 = r56001 + r56002;
        double r56004 = r55998 / r56003;
        double r56005 = sqrt(r56004);
        double r56006 = sqrt(r55998);
        double r56007 = sqrt(r56006);
        double r56008 = r56005 * r56007;
        double r56009 = r56002 + r56001;
        double r56010 = r56006 / r56009;
        double r56011 = sqrt(r56010);
        double r56012 = r56008 * r56011;
        return r56012;
}

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 *-un-lft-identity0.0

    \[\leadsto \sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{\color{blue}{1 \cdot \left(e^{x} + e^{-x}\right)}}}\]

  6. Applied add-sqr-sqrt0.0

    \[\leadsto \sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{1 \cdot \left(e^{x} + e^{-x}\right)}}\]

  7. Applied times-frac0.0

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

  8. Applied sqrt-prod0.0

    \[\leadsto \sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \color{blue}{\left(\sqrt{\frac{\sqrt{2}}{1}} \cdot \sqrt{\frac{\sqrt{2}}{e^{x} + e^{-x}}}\right)}\]

  9. Applied associate-*r*0.0

    \[\leadsto \color{blue}{\left(\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\sqrt{2}}{1}}\right) \cdot \sqrt{\frac{\sqrt{2}}{e^{x} + e^{-x}}}}\]

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))