Average Error: 0.0 → 0.0
Time: 12.0s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\frac{\cos y}{e^{x}}\right))_*}{2}\]

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

    \[\leadsto \frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\frac{\cos y}{e^{x}}\right))_*}{2}\]

  3. Final simplification0.0

    \[\leadsto \frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\frac{\cos y}{e^{x}}\right))_*}{2}\]

Runtime

Time bar (total: 12.0s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(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)))))