Average Error: 0.0 → 0.1
Time: 1.5s
Precision: binary64
\[\frac{2}{e^{x} + e^{-x}} \]
↓
\[\sqrt[3]{{\left(\frac{1}{\cosh x}\right)}^{3}} \]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt[3]{{\left(\frac{1}{\cosh x}\right)}^{3}}

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))
↓
(FPCore (x) :precision binary64 (cbrt (pow (/ 1.0 (cosh x)) 3.0)))
double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}
↓
double code(double x) {
	return cbrt(pow((1.0 / cosh(x)), 3.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Applied cosh-undef_binary640.0

    \[\leadsto \frac{2}{\color{blue}{2 \cdot \cosh x}} \]

  3. Applied add-sqr-sqrt_binary640.5

    \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{2 \cdot \cosh x} \]

  4. Applied times-frac_binary640.5

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{\cosh x}} \]

  5. Applied add-cbrt-cube_binary640.1

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{\cosh x}\right) \cdot \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{\cosh x}\right)\right) \cdot \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{\cosh x}\right)}} \]

  6. Simplified0.1

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{\cosh x}\right)}^{3}}} \]

  7. Final simplification0.1

    \[\leadsto \sqrt[3]{{\left(\frac{1}{\cosh x}\right)}^{3}} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))