Euler formula real part (p55)

Average Error: 0.0 → 0.0

Time: 6.0s

Precision: 64

Math

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

↓

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

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. Using strategy rm

3. Applied *-un-lft-identity 0.0

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

4. Applied *-un-lft-identity 0.0

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

5. Applied times-frac 0.0

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

6. Applied associate-*l* 0.0

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

7. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020066 
(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)))))