Hyperbolic secant

Average Error: 0.0 → 0.1

Time: 1.7s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r60558 = 2.0;
        double r60559 = x;
        double r60560 = exp(r60559);
        double r60561 = -r60559;
        double r60562 = exp(r60561);
        double r60563 = r60560 + r60562;
        double r60564 = r60558 / r60563;
        return r60564;
}

↓

double f(double x) {
        double r60565 = 2.0;
        double r60566 = -1.0;
        double r60567 = x;
        double r60568 = r60566 * r60567;
        double r60569 = exp(r60568);
        double r60570 = exp(r60567);
        double r60571 = r60569 + r60570;
        double r60572 = r60565 / r60571;
        double r60573 = 3.0;
        double r60574 = pow(r60572, r60573);
        double r60575 = cbrt(r60574);
        return r60575;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied add-cbrt-cube 0.1

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

4. Applied add-cbrt-cube 0.1

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

5. Applied cbrt-undiv 0.1

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

6. Simplified 0.1

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))