Average Error: 0.0 → 0.0

Time: 29.6s

Precision: 64

Internal Precision: 128

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `re`

Derivation

Initial program 0.0
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
Simplified0.0
\[\leadsto \color{blue}{(\left(e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) + \left(\frac{0.5 \cdot \cos re}{e^{im}}\right))_*}\]
Using strategy rm
Applied fma-udef0.0
\[\leadsto \color{blue}{e^{im} \cdot \left(0.5 \cdot \cos re\right) + \frac{0.5 \cdot \cos re}{e^{im}}}\]
Final simplification0.0
\[\leadsto \frac{\cos re \cdot 0.5}{e^{im}} + \left(\cos re \cdot 0.5\right) \cdot e^{im}\]

Reproduce

herbie shell --seed 2019091 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))