Hyperbolic secant

Average Error: 0.0 → 0.1

Time: 13.6s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r37514 = 2.0;
        double r37515 = x;
        double r37516 = exp(r37515);
        double r37517 = -r37515;
        double r37518 = exp(r37517);
        double r37519 = r37516 + r37518;
        double r37520 = r37514 / r37519;
        return r37520;
}

↓

double f(double x) {
        double r37521 = 2.0;
        double r37522 = x;
        double r37523 = -r37522;
        double r37524 = exp(r37523);
        double r37525 = exp(r37522);
        double r37526 = r37524 + r37525;
        double r37527 = r37521 / r37526;
        double r37528 = 3.0;
        double r37529 = pow(r37527, r37528);
        double r37530 = cbrt(r37529);
        return r37530;
}

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

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))