\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
Test:
math.sin on complex, imaginary part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 17.7 s
Input Error: 59.0
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\frac{(\frac{1}{60} * \left({im}^{5}\right) + \left((\left({im}^3\right) * \frac{1}{3} + \left(2 \cdot im\right))_*\right))_*}{1} \cdot \left(\cos re \cdot \left(-0.5\right)\right)\)
  1. Started with
    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
    59.0
  2. Applied simplify to get
    \[\color{red}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)} \leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(\cos re \cdot 0.5\right)}\]
    59.0
  3. Applied taylor to get
    \[\left(e^{-im} - e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) \leadsto \left(-\left(\frac{1}{60} \cdot {im}^{5} + \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)\right)\right) \cdot \left(\cos re \cdot 0.5\right)\]
    0.2
  4. Taylor expanded around 0 to get
    \[\color{red}{\left(-\left(\frac{1}{60} \cdot {im}^{5} + \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)\right)\right)} \cdot \left(\cos re \cdot 0.5\right) \leadsto \color{blue}{\left(-\left(\frac{1}{60} \cdot {im}^{5} + \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)\right)\right)} \cdot \left(\cos re \cdot 0.5\right)\]
    0.2
  5. Using strategy rm
    0.2
  6. Applied flip-+ to get
    \[\left(-\color{red}{\left(\frac{1}{60} \cdot {im}^{5} + \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)\right)}\right) \cdot \left(\cos re \cdot 0.5\right) \leadsto \left(-\color{blue}{\frac{{\left(\frac{1}{60} \cdot {im}^{5}\right)}^2 - {\left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}^2}{\frac{1}{60} \cdot {im}^{5} - \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}}\right) \cdot \left(\cos re \cdot 0.5\right)\]
    29.1
  7. Applied distribute-neg-frac to get
    \[\color{red}{\left(-\frac{{\left(\frac{1}{60} \cdot {im}^{5}\right)}^2 - {\left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}^2}{\frac{1}{60} \cdot {im}^{5} - \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}\right)} \cdot \left(\cos re \cdot 0.5\right) \leadsto \color{blue}{\frac{-\left({\left(\frac{1}{60} \cdot {im}^{5}\right)}^2 - {\left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}^2\right)}{\frac{1}{60} \cdot {im}^{5} - \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}} \cdot \left(\cos re \cdot 0.5\right)\]
    29.1
  8. Applied associate-*l/ to get
    \[\color{red}{\frac{-\left({\left(\frac{1}{60} \cdot {im}^{5}\right)}^2 - {\left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}^2\right)}{\frac{1}{60} \cdot {im}^{5} - \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)} \cdot \left(\cos re \cdot 0.5\right)} \leadsto \color{blue}{\frac{\left(-\left({\left(\frac{1}{60} \cdot {im}^{5}\right)}^2 - {\left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}^2\right)\right) \cdot \left(\cos re \cdot 0.5\right)}{\frac{1}{60} \cdot {im}^{5} - \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}}\]
    29.1
  9. Applied taylor to get
    \[\frac{\left(-\left({\left(\frac{1}{60} \cdot {im}^{5}\right)}^2 - {\left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}^2\right)\right) \cdot \left(\cos re \cdot 0.5\right)}{\frac{1}{60} \cdot {im}^{5} - \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)} \leadsto \frac{\left(-\left({\left(\frac{1}{60} \cdot {im}^{5}\right)}^2 - {\left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}^2\right)\right) \cdot \left(\cos re \cdot 0.5\right)}{\frac{1}{60} \cdot {im}^{5} - \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}\]
    29.1
  10. Taylor expanded around 0 to get
    \[\frac{\left(-\left({\left(\frac{1}{60} \cdot {im}^{5}\right)}^2 - {\left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}^2\right)\right) \cdot \left(\cos re \cdot 0.5\right)}{\color{red}{\frac{1}{60} \cdot {im}^{5} - \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}} \leadsto \frac{\left(-\left({\left(\frac{1}{60} \cdot {im}^{5}\right)}^2 - {\left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}^2\right)\right) \cdot \left(\cos re \cdot 0.5\right)}{\color{blue}{\frac{1}{60} \cdot {im}^{5} - \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}}\]
    29.1
  11. Applied simplify to get
    \[\frac{\left(-\left({\left(\frac{1}{60} \cdot {im}^{5}\right)}^2 - {\left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)}^2\right)\right) \cdot \left(\cos re \cdot 0.5\right)}{\frac{1}{60} \cdot {im}^{5} - \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)} \leadsto -\frac{\left(\frac{1}{60} \cdot {im}^{5}\right) \cdot \left(\frac{1}{60} \cdot {im}^{5}\right) - (\left({im}^3\right) * \frac{1}{3} + \left(2 \cdot im\right))_* \cdot (\left({im}^3\right) * \frac{1}{3} + \left(2 \cdot im\right))_*}{\frac{\frac{1}{60} \cdot {im}^{5} - (\left({im}^3\right) * \frac{1}{3} + \left(2 \cdot im\right))_*}{\cos re \cdot 0.5}}\]
    29.1
  12. Applied final simplification
  13. Applied simplify to get
    \[\color{red}{-\frac{\left(\frac{1}{60} \cdot {im}^{5}\right) \cdot \left(\frac{1}{60} \cdot {im}^{5}\right) - (\left({im}^3\right) * \frac{1}{3} + \left(2 \cdot im\right))_* \cdot (\left({im}^3\right) * \frac{1}{3} + \left(2 \cdot im\right))_*}{\frac{\frac{1}{60} \cdot {im}^{5} - (\left({im}^3\right) * \frac{1}{3} + \left(2 \cdot im\right))_*}{\cos re \cdot 0.5}}} \leadsto \color{blue}{\frac{(\frac{1}{60} * \left({im}^{5}\right) + \left((\left({im}^3\right) * \frac{1}{3} + \left(2 \cdot im\right))_*\right))_*}{1} \cdot \left(\cos re \cdot \left(-0.5\right)\right)}\]
    0.2
  14. 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)))))