Euler formula real part (p55)

Average Error: 0.0 → 0.0

Time: 6.9s

Precision: 64

Math

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

↓

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

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 clear-num 0.0

   \[\leadsto \color{blue}{\frac{1}{\frac{2}{e^{x} + e^{-x}}}} \cdot \cos y\]

5. Applied associate-*l/ 0.0

   \[\leadsto \color{blue}{\frac{1 \cdot \cos y}{\frac{2}{e^{x} + e^{-x}}}}\]

6. Simplified 0.0

   \[\leadsto \frac{\color{blue}{\cos y}}{\frac{2}{e^{x} + e^{-x}}}\]

7. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020114 
(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)))))