Average Error: 0.0 → 0.0
Time: 15.3s
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 r65179 = 2.0;
        double r65180 = x;
        double r65181 = exp(r65180);
        double r65182 = -r65180;
        double r65183 = exp(r65182);
        double r65184 = r65181 + r65183;
        double r65185 = r65179 / r65184;
        return r65185;
}
↓
double f(double x) {
        double r65186 = 2.0;
        double r65187 = x;
        double r65188 = exp(r65187);
        double r65189 = -r65187;
        double r65190 = exp(r65189);
        double r65191 = r65188 + r65190;
        double r65192 = r65186 / r65191;
        double r65193 = sqrt(r65192);
        double r65194 = sqrt(r65186);
        double r65195 = sqrt(r65194);
        double r65196 = r65193 * r65195;
        double r65197 = r65194 / r65191;
        double r65198 = sqrt(r65197);
        double r65199 = r65196 * r65198;
        return r65199;
}

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)))))