Euler formula real part (p55)

Average Error: 0.0 → 0.1

Time: 5.1s

Precision: 64

Math

\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

↓

\[\sqrt{\frac{\sqrt[3]{{\left(e^{-1 \cdot x} + e^{x}\right)}^{3}}}{2}} \cdot \left(\sqrt{\frac{\sqrt[3]{{\left(e^{-1 \cdot x} + e^{x}\right)}^{3}}}{2}} \cdot \cos y\right)\]

Error

Bits error versus x

Bits error versus y

Derivation

1. Initial program 0.0

   \[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

2. Simplified 0.0

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

4. Applied add-cbrt-cube 0.1

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

5. Simplified 0.1

   \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(e^{-1 \cdot x} + e^{x}\right)}^{3}}}}{2} \cdot \cos y\]
6. Using strategy rm

7. Applied add-sqr-sqrt 0.1

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

8. Applied associate-*l* 0.1

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

9. Final simplification 0.1

   \[\leadsto \sqrt{\frac{\sqrt[3]{{\left(e^{-1 \cdot x} + e^{x}\right)}^{3}}}{2}} \cdot \left(\sqrt{\frac{\sqrt[3]{{\left(e^{-1 \cdot x} + e^{x}\right)}^{3}}}{2}} \cdot \cos y\right)\]

Reproduce

herbie shell --seed 2020078 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  :precision binary64
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))