Average Error: 0.0 → 0.1
Time: 1.3s
Precision: binary64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt[3]{0.5 \cdot \frac{2}{{\cosh x}^{3}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt[3]{0.5 \cdot \frac{2}{{\cosh x}^{3}}}
(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))
↓
(FPCore (x) :precision binary64 (cbrt (* 0.5 (/ 2.0 (pow (cosh x) 3.0)))))
double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}
↓
double code(double x) {
	return cbrt(0.5 * (2.0 / pow(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. Using strategy rm

  3. Applied add-cbrt-cube_binary640.1

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

  4. Simplified0.1

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

  6. Applied cosh-undef_binary640.1

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

  7. Applied add-cube-cbrt_binary641.0

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

  8. Applied times-frac_binary641.0

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

  9. Applied unpow-prod-down_binary640.8

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

  10. Simplified0.1

    \[\leadsto \sqrt[3]{\color{blue}{0.5} \cdot {\left(\frac{\sqrt[3]{2}}{\cosh x}\right)}^{3}}\]

  11. Simplified0.1

    \[\leadsto \sqrt[3]{0.5 \cdot \color{blue}{\frac{2}{{\cosh x}^{3}}}}\]

  12. Final simplification0.1

    \[\leadsto \sqrt[3]{0.5 \cdot \frac{2}{{\cosh x}^{3}}}\]

Reproduce

herbie shell --seed 2021175 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))