Hyperbolic secant

Average Error: 0.0 → 0.1

Time: 15.0s

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 r61666 = 2.0;
        double r61667 = x;
        double r61668 = exp(r61667);
        double r61669 = -r61667;
        double r61670 = exp(r61669);
        double r61671 = r61668 + r61670;
        double r61672 = r61666 / r61671;
        return r61672;
}

↓

double f(double x) {
        double r61673 = 2.0;
        double r61674 = x;
        double r61675 = exp(r61674);
        double r61676 = -r61674;
        double r61677 = exp(r61676);
        double r61678 = r61675 + r61677;
        double r61679 = r61673 / r61678;
        double r61680 = 3.0;
        double r61681 = pow(r61679, r61680);
        double r61682 = cbrt(r61681);
        return r61682;
}

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 2019199 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2.0 (+ (exp x) (exp (- x)))))