Average Error: 0.0 → 0.2
Time: 8.2s
Precision: binary64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\frac{\left(0.5 \cdot \cos re\right) \cdot \left({\left(e^{-im}\right)}^{3} + {\left(e^{im}\right)}^{3}\right)}{e^{-im} \cdot e^{-im} + \left(e^{im} \cdot e^{im} - e^{-im} \cdot e^{im}\right)}\]

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-+0.2

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

  4. Applied associate-*r/0.2

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

  5. Final simplification0.2

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

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))))