Average Error: 0.0 → 0.0
Time: 13.8s
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 r68472 = 2.0;
        double r68473 = x;
        double r68474 = exp(r68473);
        double r68475 = -r68473;
        double r68476 = exp(r68475);
        double r68477 = r68474 + r68476;
        double r68478 = r68472 / r68477;
        return r68478;
}
↓
double f(double x) {
        double r68479 = 2.0;
        double r68480 = x;
        double r68481 = exp(r68480);
        double r68482 = -r68480;
        double r68483 = exp(r68482);
        double r68484 = r68481 + r68483;
        double r68485 = r68479 / r68484;
        double r68486 = sqrt(r68485);
        double r68487 = sqrt(r68479);
        double r68488 = sqrt(r68487);
        double r68489 = r68486 * r68488;
        double r68490 = r68487 / r68484;
        double r68491 = sqrt(r68490);
        double r68492 = r68489 * r68491;
        return r68492;
}

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)))))