Hyperbolic secant

Average Error: 0.0 → 0.1

Time: 2.2s

Precision: binary64

Math

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied add-cbrt-cube 0.1

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

4. Applied add-cbrt-cube 0.1

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

5. Applied cbrt-undiv 0.1

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

6. Simplified 0.1

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020169 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (neg x)))))