Average Error: 0.0 → 0.0
Time: 43.9s
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[\left(\left(0.5 \cdot \sin re\right) \cdot \sqrt{e^{im}}\right) \cdot \sqrt{e^{im}} + \frac{0.5}{\sqrt{e^{im}}} \cdot \frac{\sin re}{\sqrt{e^{im}}}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]

  2. Simplified0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-*l*0.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied times-frac0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019050 
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))