Average Error: 0.0 → 0.0
Time: 3.2s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{2} \cdot \frac{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{2} \cdot \frac{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}
double f(double x) {
        double r75855 = 2.0;
        double r75856 = x;
        double r75857 = exp(r75856);
        double r75858 = -r75856;
        double r75859 = exp(r75858);
        double r75860 = r75857 + r75859;
        double r75861 = r75855 / r75860;
        return r75861;
}
↓
double f(double x) {
        double r75862 = 2.0;
        double r75863 = sqrt(r75862);
        double r75864 = x;
        double r75865 = exp(r75864);
        double r75866 = -r75864;
        double r75867 = exp(r75866);
        double r75868 = r75865 + r75867;
        double r75869 = sqrt(r75868);
        double r75870 = r75863 / r75869;
        double r75871 = r75870 / r75869;
        double r75872 = r75863 * r75871;
        return r75872;
}

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 *-un-lft-identity0.0

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied times-frac0.5

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

  6. Simplified0.5

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

  8. Applied add-sqr-sqrt0.0

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

  9. Applied associate-/r*0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))