Average Error: 0.0 → 0.0
Time: 1.2min
Precision: binary64
Cost: 1984
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\left(\left|\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{0.5}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt{2}}{2}}}\right) \cdot \left(\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\sqrt{2}}{\cosh x}}\right)\]
\frac{2}{e^{x} + e^{-x}}
↓
\left(\left|\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{0.5}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt{2}}{2}}}\right) \cdot \left(\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\sqrt{2}}{\cosh x}}\right)
(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))
↓
(FPCore (x)
 :precision binary64
 (*
  (* (fabs (* (cbrt (sqrt 2.0)) (cbrt 0.5))) (sqrt (cbrt (/ (sqrt 2.0) 2.0))))
  (* (sqrt (/ 2.0 (+ (exp x) (exp (- x))))) (sqrt (/ (sqrt 2.0) (cosh x))))))
double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}
↓
double code(double x) {
	return (fabs(cbrt(sqrt(2.0)) * cbrt(0.5)) * sqrt(cbrt(sqrt(2.0) / 2.0))) * (sqrt(2.0 / (exp(x) + exp(-x))) * sqrt(sqrt(2.0) / cosh(x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs
Alternative 1
Accuracy0.0
Cost1344
\[\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{\cosh x}}\]
Alternative 2
Accuracy0.5
Cost1408
\[\left(\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{e^{x} + e^{-x}}}\]
Alternative 3
Accuracy31.8
Cost2816
\[\left(\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{{\left(e^{x}\right)}^{3} + {\left(e^{x}\right)}^{-3}}}\right) \cdot \sqrt{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}\]

Derivation

  1. Initial program 0.0

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

  3. Applied add-sqr-sqrt_binary64_11230.0

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

  5. Applied cosh-undef_binary64_12950.0

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

  6. Applied add-sqr-sqrt_binary64_11230.0

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

  7. Applied times-frac_binary64_11070.0

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

  8. Applied sqrt-prod_binary64_11170.0

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

  9. Applied associate-*l*_binary64_10420.0

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

  10. Simplified0.0

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

  12. Applied add-cube-cbrt_binary64_11360.0

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

  13. Applied sqrt-prod_binary64_11170.0

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

  14. Simplified0.0

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

  16. Applied div-inv_binary64_10980.0

    \[\leadsto \left(\left|\sqrt[3]{\color{blue}{\sqrt{2} \cdot \frac{1}{2}}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt{2}}{2}}}\right) \cdot \left(\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\sqrt{2}}{\cosh x}}\right)\]

  17. Applied cbrt-prod_binary64_11320.0

    \[\leadsto \left(\left|\color{blue}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\frac{1}{2}}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt{2}}{2}}}\right) \cdot \left(\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\sqrt{2}}{\cosh x}}\right)\]

  18. Simplified0.0

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

  19. Final simplification0.0

    \[\leadsto \left(\left|\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{0.5}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt{2}}{2}}}\right) \cdot \left(\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\sqrt{2}}{\cosh x}}\right)\]

Reproduce

herbie shell --seed 2020322 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))