Average Error: 0.0 → 0.0
Time: 21.5s
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 r2333790 = 2.0;
        double r2333791 = x;
        double r2333792 = exp(r2333791);
        double r2333793 = -r2333791;
        double r2333794 = exp(r2333793);
        double r2333795 = r2333792 + r2333794;
        double r2333796 = r2333790 / r2333795;
        return r2333796;
}
↓
double f(double x) {
        double r2333797 = 2.0;
        double r2333798 = sqrt(r2333797);
        double r2333799 = x;
        double r2333800 = exp(r2333799);
        double r2333801 = -r2333799;
        double r2333802 = exp(r2333801);
        double r2333803 = r2333800 + r2333802;
        double r2333804 = sqrt(r2333803);
        double r2333805 = r2333798 / r2333804;
        double r2333806 = sqrt(r2333805);
        double r2333807 = r2333806 * r2333805;
        double r2333808 = r2333806 * r2333807;
        return r2333808;
}

Error

Bits error versus x

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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

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

  8. Applied associate-*l*0.0

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

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