Average Error: 0.0 → 0.0
Time: 18.0s
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 r56439 = 2.0;
        double r56440 = x;
        double r56441 = exp(r56440);
        double r56442 = -r56440;
        double r56443 = exp(r56442);
        double r56444 = r56441 + r56443;
        double r56445 = r56439 / r56444;
        return r56445;
}
↓
double f(double x) {
        double r56446 = 2.0;
        double r56447 = x;
        double r56448 = exp(r56447);
        double r56449 = -r56447;
        double r56450 = exp(r56449);
        double r56451 = r56448 + r56450;
        double r56452 = r56446 / r56451;
        double r56453 = sqrt(r56452);
        double r56454 = sqrt(r56446);
        double r56455 = sqrt(r56454);
        double r56456 = r56453 * r56455;
        double r56457 = r56454 / r56451;
        double r56458 = sqrt(r56457);
        double r56459 = r56456 * r56458;
        return r56459;
}

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