Hyperbolic secant

Average Error: 0.0 → 0.1

Time: 13.3s

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 r54563 = 2.0;
        double r54564 = x;
        double r54565 = exp(r54564);
        double r54566 = -r54564;
        double r54567 = exp(r54566);
        double r54568 = r54565 + r54567;
        double r54569 = r54563 / r54568;
        return r54569;
}

↓

double f(double x) {
        double r54570 = 2.0;
        double r54571 = x;
        double r54572 = -r54571;
        double r54573 = exp(r54572);
        double r54574 = exp(r54571);
        double r54575 = r54573 + r54574;
        double r54576 = r54570 / r54575;
        double r54577 = 3.0;
        double r54578 = pow(r54576, r54577);
        double r54579 = cbrt(r54578);
        return r54579;
}

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)))))