Average Error: 0.0 → 0.0
Time: 1.4m
Precision: 64
Internal Precision: 320
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} \cdot \cos y + \frac{\frac{\cos y}{\sqrt{e^{x}}}}{\sqrt{e^{x}}}}{2}\]

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. Initial simplification0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-/r*0.0

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

  6. Final simplification0.0

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018230 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))