math.cos on complex, real part

Average Error: 0.0 → 0.1

Time: 5.4s

Precision: binary64

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]

↓

\[\left(0.5 \cdot \cos re\right) \cdot \frac{{\left(e^{im}\right)}^{-3} + {\left(e^{im}\right)}^{3}}{{\left(e^{im}\right)}^{-2} + \left({\left(e^{im}\right)}^{2} - 1\right)}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.0

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
2. Using strategy rm

3. Applied flip3-+ 0.1

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\frac{{\left(e^{-im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{-im} \cdot e^{-im} + \left(e^{im} \cdot e^{im} - e^{-im} \cdot e^{im}\right)}}\]

4. Simplified 0.1

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \frac{\color{blue}{{\left(e^{im}\right)}^{-3} + {\left(e^{im}\right)}^{3}}}{e^{-im} \cdot e^{-im} + \left(e^{im} \cdot e^{im} - e^{-im} \cdot e^{im}\right)}\]

5. Simplified 0.1

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \frac{{\left(e^{im}\right)}^{-3} + {\left(e^{im}\right)}^{3}}{\color{blue}{{\left(e^{im}\right)}^{-2} + \left({\left(e^{im}\right)}^{2} - 1\right)}}\]

6. Final simplification 0.1

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \frac{{\left(e^{im}\right)}^{-3} + {\left(e^{im}\right)}^{3}}{{\left(e^{im}\right)}^{-2} + \left({\left(e^{im}\right)}^{2} - 1\right)}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (neg im)) (exp im))))