Euler formula real part (p55)

Average Error: 0.0 → 0.1

Time: 4.5s

Precision: 64

Math

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

↓

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

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 flip3-+ 0.1

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

4. Applied associate-/l/ 0.1

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

5. Applied associate-*l/ 0.1

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

7. Applied unpow3 0.1

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

8. Simplified 0.1

   \[\leadsto \Re(\left(\frac{\left(\color{blue}{e^{x + x}} \cdot e^{x} + {\left(e^{-x}\right)}^{3}\right) \cdot \cos y}{2 \cdot \left(e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)\right)} + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
9. Using strategy rm

10. Applied prod-exp 0.1

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

11. Final simplification 0.1

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

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)))))