\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
Test:
math.sin on complex, imaginary part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 20.4 s
Input Error: 58.8
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\left(\left(-0.5\right) \cdot \cos re\right) \cdot \frac{\left(\left(im \cdot \frac{1}{3}\right) \cdot \left(im \cdot im\right) + \frac{1}{60} \cdot {im}^{5}\right) + 2 \cdot im}{1}\)
  1. Started with
    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
    58.8
  2. Applied simplify to get
    \[\color{red}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)} \leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(\cos re \cdot 0.5\right)}\]
    58.8
  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 add-cube-cbrt to get
    \[\color{red}{\left(-\left(\frac{1}{60} \cdot {im}^{5} + \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)\right)\right) \cdot \left(\cos re \cdot 0.5\right)} \leadsto \color{blue}{{\left(\sqrt[3]{\left(-\left(\frac{1}{60} \cdot {im}^{5} + \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)\right)\right) \cdot \left(\cos re \cdot 0.5\right)}\right)}^3}\]
    1.5
  7. Applied simplify to get
    \[{\color{red}{\left(\sqrt[3]{\left(-\left(\frac{1}{60} \cdot {im}^{5} + \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)\right)\right) \cdot \left(\cos re \cdot 0.5\right)}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{\left(\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) + \frac{1}{3} \cdot {im}^3\right) \cdot \left(\left(-0.5\right) \cdot \cos re\right)}\right)}}^3\]
    1.5
  8. Using strategy rm
    1.5
  9. Applied flip-+ to get
    \[{\left(\sqrt[3]{\color{red}{\left(\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) + \frac{1}{3} \cdot {im}^3\right)} \cdot \left(\left(-0.5\right) \cdot \cos re\right)}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)}^2 - {\left(\frac{1}{3} \cdot {im}^3\right)}^2}{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) - \frac{1}{3} \cdot {im}^3}} \cdot \left(\left(-0.5\right) \cdot \cos re\right)}\right)}^3\]
    29.8
  10. Applied associate-*l/ to get
    \[{\left(\sqrt[3]{\color{red}{\frac{{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)}^2 - {\left(\frac{1}{3} \cdot {im}^3\right)}^2}{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) - \frac{1}{3} \cdot {im}^3} \cdot \left(\left(-0.5\right) \cdot \cos re\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{\left({\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)}^2 - {\left(\frac{1}{3} \cdot {im}^3\right)}^2\right) \cdot \left(\left(-0.5\right) \cdot \cos re\right)}{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) - \frac{1}{3} \cdot {im}^3}}}\right)}^3\]
    29.8
  11. Applied taylor to get
    \[{\left(\sqrt[3]{\frac{\left({\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)}^2 - {\left(\frac{1}{3} \cdot {im}^3\right)}^2\right) \cdot \left(\left(-0.5\right) \cdot \cos re\right)}{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) - \frac{1}{3} \cdot {im}^3}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{\left({\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)}^2 - {\left(\frac{1}{3} \cdot {im}^3\right)}^2\right) \cdot \left(\left(-0.5\right) \cdot \cos re\right)}{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) - \frac{1}{3} \cdot {im}^3}}\right)}^3\]
    29.8
  12. Taylor expanded around 0 to get
    \[{\left(\sqrt[3]{\frac{\left({\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)}^2 - {\left(\frac{1}{3} \cdot {im}^3\right)}^2\right) \cdot \left(\left(-0.5\right) \cdot \cos re\right)}{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) - \color{red}{\frac{1}{3} \cdot {im}^3}}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{\left({\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)}^2 - {\left(\frac{1}{3} \cdot {im}^3\right)}^2\right) \cdot \left(\left(-0.5\right) \cdot \cos re\right)}{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) - \color{blue}{\frac{1}{3} \cdot {im}^3}}}\right)}^3\]
    29.8
  13. Applied simplify to get
    \[{\left(\sqrt[3]{\frac{\left({\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)}^2 - {\left(\frac{1}{3} \cdot {im}^3\right)}^2\right) \cdot \left(\left(-0.5\right) \cdot \cos re\right)}{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) - \frac{1}{3} \cdot {im}^3}}\right)}^3 \leadsto \frac{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) \cdot \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) - {\left(\frac{1}{3} \cdot {im}^3\right)}^2}{\frac{2 \cdot im + \left(\frac{1}{60} \cdot {im}^{5} - \frac{1}{3} \cdot {im}^3\right)}{\cos re \cdot \left(-0.5\right)}}\]
    29.2
  14. Applied final simplification
  15. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) \cdot \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right) - {\left(\frac{1}{3} \cdot {im}^3\right)}^2}{\frac{2 \cdot im + \left(\frac{1}{60} \cdot {im}^{5} - \frac{1}{3} \cdot {im}^3\right)}{\cos re \cdot \left(-0.5\right)}}} \leadsto \color{blue}{\left(\left(-0.5\right) \cdot \cos re\right) \cdot \frac{\left(\left(im \cdot \frac{1}{3}\right) \cdot \left(im \cdot im\right) + \frac{1}{60} \cdot {im}^{5}\right) + 2 \cdot im}{1}}\]
    0.2
  16. 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)))))