\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
Test:
math.sin on complex, imaginary part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 14.9 s
Input Error: 24.4
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\left(\cos re \cdot 0.5\right) \cdot \left(\left(-(\left({im}^{5}\right) * \frac{1}{1920} + \left((\frac{1}{24} * \left({im}^3\right) + im)_*\right))_*\right) \cdot \left(\sqrt{e^{im}} + \sqrt{e^{-im}}\right)\right)\)
  1. Started with
    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
    24.4
  2. Applied simplify to get
    \[\color{red}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)} \leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(\cos re \cdot 0.5\right)}\]
    24.4
  3. Using strategy rm
    24.4
  4. Applied add-sqr-sqrt to get
    \[\left(e^{-im} - \color{red}{e^{im}}\right) \cdot \left(\cos re \cdot 0.5\right) \leadsto \left(e^{-im} - \color{blue}{{\left(\sqrt{e^{im}}\right)}^2}\right) \cdot \left(\cos re \cdot 0.5\right)\]
    24.6
  5. Applied add-sqr-sqrt to get
    \[\left(\color{red}{e^{-im}} - {\left(\sqrt{e^{im}}\right)}^2\right) \cdot \left(\cos re \cdot 0.5\right) \leadsto \left(\color{blue}{{\left(\sqrt{e^{-im}}\right)}^2} - {\left(\sqrt{e^{im}}\right)}^2\right) \cdot \left(\cos re \cdot 0.5\right)\]
    24.6
  6. Applied difference-of-squares to get
    \[\color{red}{\left({\left(\sqrt{e^{-im}}\right)}^2 - {\left(\sqrt{e^{im}}\right)}^2\right)} \cdot \left(\cos re \cdot 0.5\right) \leadsto \color{blue}{\left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \left(\sqrt{e^{-im}} - \sqrt{e^{im}}\right)\right)} \cdot \left(\cos re \cdot 0.5\right)\]
    24.6
  7. Applied taylor to get
    \[\left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \left(\sqrt{e^{-im}} - \sqrt{e^{im}}\right)\right) \cdot \left(\cos re \cdot 0.5\right) \leadsto \left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \left(-\left(\frac{1}{1920} \cdot {im}^{5} + \left(im + \frac{1}{24} \cdot {im}^{3}\right)\right)\right)\right) \cdot \left(\cos re \cdot 0.5\right)\]
    0.3
  8. Taylor expanded around 0 to get
    \[\left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \color{red}{\left(-\left(\frac{1}{1920} \cdot {im}^{5} + \left(im + \frac{1}{24} \cdot {im}^{3}\right)\right)\right)}\right) \cdot \left(\cos re \cdot 0.5\right) \leadsto \left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \color{blue}{\left(-\left(\frac{1}{1920} \cdot {im}^{5} + \left(im + \frac{1}{24} \cdot {im}^{3}\right)\right)\right)}\right) \cdot \left(\cos re \cdot 0.5\right)\]
    0.3
  9. Applied simplify to get
    \[\left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \left(-\left(\frac{1}{1920} \cdot {im}^{5} + \left(im + \frac{1}{24} \cdot {im}^{3}\right)\right)\right)\right) \cdot \left(\cos re \cdot 0.5\right) \leadsto \left(\cos re \cdot 0.5\right) \cdot \left(\left(-(\left({im}^{5}\right) * \frac{1}{1920} + \left((\frac{1}{24} * \left({im}^3\right) + im)_*\right))_*\right) \cdot \left(\sqrt{e^{im}} + \sqrt{e^{-im}}\right)\right)\]
    0.3
  10. Applied final simplification
  11. Removed slow pow expressions

Original test:


(lambda ((re default) (im default))
  #:name "math.sin on complex, imaginary part"
  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im)))
  #:target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im)))))