Average Error: 0.0 → 0.0
Time: 4.6s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{\sqrt{2}}{\sqrt{\sqrt{e^{x} + e^{-x}}} \cdot \left|\sqrt[3]{e^{x} + e^{-x}}\right|} \cdot \frac{\frac{\sqrt{2}}{\sqrt{\sqrt{e^{x} + e^{-x}}}}}{\sqrt{\sqrt[3]{e^{x} + e^{-x}}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\frac{\sqrt{2}}{\sqrt{\sqrt{e^{x} + e^{-x}}} \cdot \left|\sqrt[3]{e^{x} + e^{-x}}\right|} \cdot \frac{\frac{\sqrt{2}}{\sqrt{\sqrt{e^{x} + e^{-x}}}}}{\sqrt{\sqrt[3]{e^{x} + e^{-x}}}}
double code(double x) {
	return (2.0 / (exp(x) + exp(-x)));
}
↓
double code(double x) {
	return ((sqrt(2.0) / (sqrt(sqrt((exp(x) + exp(-x)))) * fabs(cbrt((exp(x) + exp(-x)))))) * ((sqrt(2.0) / sqrt(sqrt((exp(x) + exp(-x))))) / sqrt(cbrt((exp(x) + exp(-x))))));
}

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-sqr-sqrt0.8

    \[\leadsto \frac{2}{\color{blue}{\sqrt{e^{x} + e^{-x}} \cdot \sqrt{e^{x} + e^{-x}}}}\]

  4. Applied associate-/r*0.5

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

  6. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{\color{blue}{\sqrt{e^{x} + e^{-x}} \cdot \sqrt{e^{x} + e^{-x}}}}}\]

  7. Applied sqrt-prod0.0

    \[\leadsto \frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\color{blue}{\sqrt{\sqrt{e^{x} + e^{-x}}} \cdot \sqrt{\sqrt{e^{x} + e^{-x}}}}}\]

  8. Applied add-cube-cbrt0.5

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

  9. Applied sqrt-prod0.5

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

  10. Applied add-sqr-sqrt0.5

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

  11. Applied times-frac0.0

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

  12. Applied times-frac0.0

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

  13. Simplified0.0

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

  14. Simplified0.0

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

  15. Final simplification0.0

    \[\leadsto \frac{\sqrt{2}}{\sqrt{\sqrt{e^{x} + e^{-x}}} \cdot \left|\sqrt[3]{e^{x} + e^{-x}}\right|} \cdot \frac{\frac{\sqrt{2}}{\sqrt{\sqrt{e^{x} + e^{-x}}}}}{\sqrt{\sqrt[3]{e^{x} + e^{-x}}}}\]

Reproduce

herbie shell --seed 2020058 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))