Average Error: 0.0 → 0.1
Time: 23.7s
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\frac{\cos re \cdot \left(\frac{0.5}{e^{im}} \cdot \left(\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}}\right) + \left(\left(0.5 \cdot e^{im}\right) \cdot \left(0.5 \cdot e^{im}\right)\right) \cdot \left(0.5 \cdot e^{im}\right)\right)}{\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}} + \left(0.5 \cdot e^{im} - \frac{0.5}{e^{im}}\right) \cdot \left(0.5 \cdot e^{im}\right)}\]

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. Simplified0.0

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

  4. Applied flip3-+0.1

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

  5. Applied associate-*l/0.1

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

  6. Simplified0.1

    \[\leadsto \frac{\color{blue}{\left(\left(\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}}\right) \cdot \frac{0.5}{e^{im}} + \left(\left(0.5 \cdot e^{im}\right) \cdot \left(0.5 \cdot e^{im}\right)\right) \cdot \left(0.5 \cdot e^{im}\right)\right) \cdot \cos re}}{\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}} + \left(\left(e^{im} \cdot 0.5\right) \cdot \left(e^{im} \cdot 0.5\right) - \frac{0.5}{e^{im}} \cdot \left(e^{im} \cdot 0.5\right)\right)}\]
  7. Using strategy rm

  8. Applied distribute-rgt-out--0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019053 
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))