Average Error: 0.0 → 0.0
Time: 5.3s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{\sqrt{1}}{\sqrt{\sqrt{e^{x} + e^{-x}}}} \cdot \frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{\sqrt{e^{x} + e^{-x}}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\frac{\sqrt{1}}{\sqrt{\sqrt{e^{x} + e^{-x}}}} \cdot \frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{\sqrt{e^{x} + e^{-x}}}}
double f(double x) {
        double r75432 = 2.0;
        double r75433 = x;
        double r75434 = exp(r75433);
        double r75435 = -r75433;
        double r75436 = exp(r75435);
        double r75437 = r75434 + r75436;
        double r75438 = r75432 / r75437;
        return r75438;
}
↓
double f(double x) {
        double r75439 = 1.0;
        double r75440 = sqrt(r75439);
        double r75441 = x;
        double r75442 = exp(r75441);
        double r75443 = -r75441;
        double r75444 = exp(r75443);
        double r75445 = r75442 + r75444;
        double r75446 = sqrt(r75445);
        double r75447 = sqrt(r75446);
        double r75448 = r75440 / r75447;
        double r75449 = 2.0;
        double r75450 = r75449 / r75446;
        double r75451 = r75450 / r75447;
        double r75452 = r75448 * r75451;
        return r75452;
}

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-sqrt0.8

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied times-frac0.0

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}} \cdot \frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.0

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

  8. Applied sqrt-prod0.5

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

  9. Applied *-un-lft-identity0.5

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

  10. Applied sqrt-prod0.5

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

  11. Applied times-frac0.5

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

  12. Applied associate-*l*0.5

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

  13. Simplified0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))