Average Error: 0.0 → 0.0
Time: 24.1s
Precision: 64
Internal Precision: 384
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[(\left(e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) + \left(\frac{\cos re \cdot 0.5}{e^{im}}\right))_*\]

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 add-sqr-sqrt0.0

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

  4. Applied fma-def0.0

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

  5. Taylor expanded around -inf 0.0

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot (\left(\sqrt{e^{-im}}\right) \cdot \color{blue}{\left(\sqrt{e^{-1 \cdot im}}\right)} + \left(e^{im}\right))_*\]

  6. Applied simplify0.0

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

Runtime

Time bar (total: 24.1s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))