Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \le 2.0000000483992446:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{12}, {x}^{4}, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}\right) \cdot \sqrt[3]{e^{x} + e^{-x}}}\\ \end{array}\]
\frac{2}{e^{x} + e^{-x}}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} + e^{-x} \le 2.0000000483992446:\\
\;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{12}, {x}^{4}, 2\right)\right)}\\

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

\end{array}
double code(double x) {
	return (2.0 / (exp(x) + exp(-x)));
}
↓
double code(double x) {
	double VAR;
	if (((exp(x) + exp(-x)) <= 2.0000000483992446)) {
		VAR = (2.0 / fma(x, x, fma(0.08333333333333333, pow(x, 4.0), 2.0)));
	} else {
		VAR = (2.0 / ((cbrt((exp(x) + exp(-x))) * cbrt((exp(x) + exp(-x)))) * cbrt((exp(x) + exp(-x)))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (+ (exp x) (exp (- x))) < 2.0000000483992446

    1. Initial program 0.0

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \frac{2}{\color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + 2\right)}}\]

    3. Simplified0.0

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{12}, {x}^{4}, 2\right)\right)}}\]

    if 2.0000000483992446 < (+ (exp x) (exp (- x)))

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

      \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}\right) \cdot \sqrt[3]{e^{x} + e^{-x}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \le 2.0000000483992446:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{12}, {x}^{4}, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}\right) \cdot \sqrt[3]{e^{x} + e^{-x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))