Hyperbolic secant

Average Error: 0.0 → 0.0

Time: 17.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r4507588 = 2.0;
        double r4507589 = x;
        double r4507590 = exp(r4507589);
        double r4507591 = -r4507589;
        double r4507592 = exp(r4507591);
        double r4507593 = r4507590 + r4507592;
        double r4507594 = r4507588 / r4507593;
        return r4507594;
}

↓

double f(double x) {
        double r4507595 = 2.0;
        double r4507596 = x;
        double r4507597 = exp(r4507596);
        double r4507598 = -r4507596;
        double r4507599 = exp(r4507598);
        double r4507600 = r4507597 + r4507599;
        double r4507601 = r4507595 / r4507600;
        double r4507602 = cbrt(r4507601);
        double r4507603 = sqrt(r4507602);
        double r4507604 = r4507602 * r4507603;
        double r4507605 = r4507604 * r4507603;
        double r4507606 = sqrt(r4507595);
        double r4507607 = r4507606 / r4507600;
        double r4507608 = r4507606 * r4507607;
        double r4507609 = cbrt(r4507608);
        double r4507610 = r4507605 * r4507609;
        return r4507610;
}

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

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

6. Applied associate-*r* 0.0

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

8. Applied *-un-lft-identity 0.0

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

9. Applied add-sqr-sqrt 0.0

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

10. Applied times-frac 0.0

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

11. Simplified 0.0

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

12. Final simplification 0.0

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

Reproduce

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