math.cos on complex, real part

Average Error: 0.0 → 0.0

Time: 3.9s

Precision: binary64

Math

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

↓

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

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 distribute-lft-in 0.0

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

4. Simplified 0.0

   \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{e^{im}}} + \left(0.5 \cdot \cos re\right) \cdot e^{im}\]
5. Using strategy rm

6. Applied add-sqr-sqrt 0.0

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

7. Applied associate-*r* 0.0

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

8. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020155 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (neg im)) (exp im))))