math.cos on complex, real part

Average Error: 0.0 → 0.0

Time: 8.8s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

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-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

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

5. Simplified 0.0

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

6. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020182 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (neg im)) (exp im))))