Average Error: 0.0 → 0.0
Time: 15.9s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}} \cdot \frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}\right)\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}} \cdot \frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}\right)
double f(double x) {
        double r1200440 = 2.0;
        double r1200441 = x;
        double r1200442 = exp(r1200441);
        double r1200443 = -r1200441;
        double r1200444 = exp(r1200443);
        double r1200445 = r1200442 + r1200444;
        double r1200446 = r1200440 / r1200445;
        return r1200446;
}
↓
double f(double x) {
        double r1200447 = 2.0;
        double r1200448 = sqrt(r1200447);
        double r1200449 = x;
        double r1200450 = exp(r1200449);
        double r1200451 = -r1200449;
        double r1200452 = exp(r1200451);
        double r1200453 = r1200450 + r1200452;
        double r1200454 = sqrt(r1200453);
        double r1200455 = r1200448 / r1200454;
        double r1200456 = sqrt(r1200455);
        double r1200457 = r1200456 * r1200455;
        double r1200458 = r1200456 * r1200457;
        return r1200458;
}

Error

Bits error versus x

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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

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

  8. Applied associate-*r*0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))