\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
Test:
math.cos on complex, imaginary part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 16.1 s
Input Error: 18.2
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(0.5 \cdot \sin re\right) \cdot \left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \left(\sqrt{e^{-im}} - \sqrt{e^{im}}\right)\right) & \text{when } im \le -0.21930791f0 \\ \left(-(\left({im}^{5}\right) * \frac{1}{1920} + \left((\left({im}^3\right) * \frac{1}{24} + im)_*\right))_*\right) \cdot (\left(\sin re \cdot 0.5\right) * \left(\sqrt{e^{-im}}\right) + \left(\left(\sin re \cdot 0.5\right) \cdot \sqrt{e^{im}}\right))_* & \text{otherwise} \end{cases}\)

    if im < -0.21930791f0

    1. Started with
      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
      1.7
    2. Using strategy rm
      1.7
    3. Applied add-sqr-sqrt to get
      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - \color{red}{e^{im}}\right) \leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - \color{blue}{{\left(\sqrt{e^{im}}\right)}^2}\right)\]
      1.8
    4. Applied add-sqr-sqrt to get
      \[\left(0.5 \cdot \sin re\right) \cdot \left(\color{red}{e^{-im}} - {\left(\sqrt{e^{im}}\right)}^2\right) \leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{{\left(\sqrt{e^{-im}}\right)}^2} - {\left(\sqrt{e^{im}}\right)}^2\right)\]
      2.1
    5. Applied difference-of-squares to get
      \[\left(0.5 \cdot \sin re\right) \cdot \color{red}{\left({\left(\sqrt{e^{-im}}\right)}^2 - {\left(\sqrt{e^{im}}\right)}^2\right)} \leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \left(\sqrt{e^{-im}} - \sqrt{e^{im}}\right)\right)}\]
      2.1

    if -0.21930791f0 < im

    1. Started with
      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
      18.7
    2. Using strategy rm
      18.7
    3. Applied add-sqr-sqrt to get
      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - \color{red}{e^{im}}\right) \leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - \color{blue}{{\left(\sqrt{e^{im}}\right)}^2}\right)\]
      18.9
    4. Applied add-sqr-sqrt to get
      \[\left(0.5 \cdot \sin re\right) \cdot \left(\color{red}{e^{-im}} - {\left(\sqrt{e^{im}}\right)}^2\right) \leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{{\left(\sqrt{e^{-im}}\right)}^2} - {\left(\sqrt{e^{im}}\right)}^2\right)\]
      18.9
    5. Applied difference-of-squares to get
      \[\left(0.5 \cdot \sin re\right) \cdot \color{red}{\left({\left(\sqrt{e^{-im}}\right)}^2 - {\left(\sqrt{e^{im}}\right)}^2\right)} \leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \left(\sqrt{e^{-im}} - \sqrt{e^{im}}\right)\right)}\]
      18.9
    6. Applied taylor to get
      \[\left(0.5 \cdot \sin re\right) \cdot \left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \left(\sqrt{e^{-im}} - \sqrt{e^{im}}\right)\right) \leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \left(-\left(\frac{1}{1920} \cdot {im}^{5} + \left(im + \frac{1}{24} \cdot {im}^{3}\right)\right)\right)\right)\]
      0.2
    7. Taylor expanded around 0 to get
      \[\left(0.5 \cdot \sin re\right) \cdot \left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \color{red}{\left(-\left(\frac{1}{1920} \cdot {im}^{5} + \left(im + \frac{1}{24} \cdot {im}^{3}\right)\right)\right)}\right) \leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \color{blue}{\left(-\left(\frac{1}{1920} \cdot {im}^{5} + \left(im + \frac{1}{24} \cdot {im}^{3}\right)\right)\right)}\right)\]
      0.2
    8. Applied simplify to get
      \[\left(0.5 \cdot \sin re\right) \cdot \left(\left(\sqrt{e^{-im}} + \sqrt{e^{im}}\right) \cdot \left(-\left(\frac{1}{1920} \cdot {im}^{5} + \left(im + \frac{1}{24} \cdot {im}^{3}\right)\right)\right)\right) \leadsto \left(-(\left({im}^{5}\right) * \frac{1}{1920} + \left((\left({im}^3\right) * \frac{1}{24} + im)_*\right))_*\right) \cdot (\left(\sin re \cdot 0.5\right) * \left(\sqrt{e^{-im}}\right) + \left(\left(\sin re \cdot 0.5\right) \cdot \sqrt{e^{im}}\right))_*\]
      0.3
    9. Applied final simplification
  1. Removed slow pow expressions

Original test:


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