Hyperbolic secant

Average Error: 0.0 → 0.0

Time: 16.9s

Precision: 64

Math

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

↓

\[\left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)\right) \cdot \sqrt[3]{\sqrt{2} \cdot \frac{\sqrt{2}}{e^{x} + e^{-x}}}\]

C

double f(double x) {
        double r5389972 = 2.0;
        double r5389973 = x;
        double r5389974 = exp(r5389973);
        double r5389975 = -r5389973;
        double r5389976 = exp(r5389975);
        double r5389977 = r5389974 + r5389976;
        double r5389978 = r5389972 / r5389977;
        return r5389978;
}

↓

double f(double x) {
        double r5389979 = 2.0;
        double r5389980 = x;
        double r5389981 = exp(r5389980);
        double r5389982 = -r5389980;
        double r5389983 = exp(r5389982);
        double r5389984 = r5389981 + r5389983;
        double r5389985 = r5389979 / r5389984;
        double r5389986 = cbrt(r5389985);
        double r5389987 = sqrt(r5389986);
        double r5389988 = r5389987 * r5389986;
        double r5389989 = r5389987 * r5389988;
        double r5389990 = sqrt(r5389979);
        double r5389991 = r5389990 / r5389984;
        double r5389992 = r5389990 * r5389991;
        double r5389993 = cbrt(r5389992);
        double r5389994 = r5389989 * r5389993;
        return r5389994;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

   \[\frac{2}{e^{x} + e^{-x}}\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.0

   \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right) \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}\]
4. Using strategy rm

5. Applied add-sqr-sqrt 0.0

   \[\leadsto \left(\color{blue}{\left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}\right)} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right) \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\]

6. Applied associate-*l* 0.0

   \[\leadsto \color{blue}{\left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)\right)} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\]
7. Using strategy rm

8. Applied *-un-lft-identity 0.0

   \[\leadsto \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)\right) \cdot \sqrt[3]{\frac{2}{\color{blue}{1 \cdot \left(e^{x} + e^{-x}\right)}}}\]

9. Applied add-sqr-sqrt 0.0

   \[\leadsto \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)\right) \cdot \sqrt[3]{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{1 \cdot \left(e^{x} + e^{-x}\right)}}\]

10. Applied times-frac 0.0

    \[\leadsto \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{e^{x} + e^{-x}}}}\]

11. Simplified 0.0

    \[\leadsto \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)\right) \cdot \sqrt[3]{\color{blue}{\sqrt{2}} \cdot \frac{\sqrt{2}}{e^{x} + e^{-x}}}\]

12. Final simplification 0.0

    \[\leadsto \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)\right) \cdot \sqrt[3]{\sqrt{2} \cdot \frac{\sqrt{2}}{e^{x} + e^{-x}}}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2.0 (+ (exp x) (exp (- x)))))