Average Error: 0.0 → 0.0
Time: 4.7s
Precision: binary64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\cos re \cdot \left(e^{im} \cdot 0.5 + \frac{0.5}{e^{im}}\right)\]

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 distribute-lft-in0.0

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

  4. Simplified0.0

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

  5. Simplified0.0

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

  6. Taylor expanded around inf 0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020181 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (neg im)) (exp im))))