\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
Test:
math.sqrt on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 15.0 s
Input Error: 37.9
Output Error: 18.2
Log: ⚲
Profile: 🕒
\(\begin{cases} 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\left(-re\right) - re}} & \text{when } re \le -2.7522388519686937 \cdot 10^{+80} \\ 0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{{re}^2 + im \cdot im} - re}} & \text{when } re \le -8.68041930647669 \cdot 10^{-252} \\ 0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)} & \text{when } re \le 1.9727183986347288 \cdot 10^{-223} \\ 0.5 \cdot \sqrt{2.0 \cdot \left({\left(\sqrt{\sqrt{{re}^2 + im \cdot im}}\right)}^2 + re\right)} & \text{when } re \le 3.873554890065949 \cdot 10^{+113} \\ 0.5 \cdot \sqrt{\left(2 \cdot re + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\right) \cdot 2.0} & \text{otherwise} \end{cases}\)

    if re < -2.7522388519686937e+80

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      59.9
    2. Using strategy rm
      59.9
    3. Applied add-sqr-sqrt to get
      \[0.5 \cdot \color{red}{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\right)}^2}\]
      59.9
    4. Using strategy rm
      59.9
    5. Applied flip-+ to get
      \[0.5 \cdot {\left(\sqrt{\sqrt{2.0 \cdot \color{red}{\left(\sqrt{re \cdot re + im \cdot im} + re\right)}}}\right)}^2 \leadsto 0.5 \cdot {\left(\sqrt{\sqrt{2.0 \cdot \color{blue}{\frac{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}{\sqrt{re \cdot re + im \cdot im} - re}}}}\right)}^2\]
      59.9
    6. Applied associate-*r/ to get
      \[0.5 \cdot {\left(\sqrt{\sqrt{\color{red}{2.0 \cdot \frac{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}{\sqrt{re \cdot re + im \cdot im} - re}}}}\right)}^2 \leadsto 0.5 \cdot {\left(\sqrt{\sqrt{\color{blue}{\frac{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}}\right)}^2\]
      59.9
    7. Applied sqrt-div to get
      \[0.5 \cdot {\left(\sqrt{\color{red}{\sqrt{\frac{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}}\right)}^2 \leadsto 0.5 \cdot {\left(\sqrt{\color{blue}{\frac{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}\right)}^2\]
      59.9
    8. Applied sqrt-div to get
      \[0.5 \cdot {\color{red}{\left(\sqrt{\frac{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)}}^2 \leadsto 0.5 \cdot {\color{blue}{\left(\frac{\sqrt{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)}}^2\]
      59.9
    9. Applied simplify to get
      \[0.5 \cdot {\left(\frac{\color{red}{\sqrt{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)}^2 \leadsto 0.5 \cdot {\left(\frac{\color{blue}{\sqrt{\sqrt{\left(im \cdot im\right) \cdot 2.0}}}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)}^2\]
      43.1
    10. Applied simplify to get
      \[0.5 \cdot {\left(\frac{\sqrt{\sqrt{\left(im \cdot im\right) \cdot 2.0}}}{\color{red}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}\right)}^2 \leadsto 0.5 \cdot {\left(\frac{\sqrt{\sqrt{\left(im \cdot im\right) \cdot 2.0}}}{\color{blue}{\sqrt{\sqrt{\sqrt{{re}^2 + im \cdot im} - re}}}}\right)}^2\]
      43.1
    11. Applied taylor to get
      \[0.5 \cdot {\left(\frac{\sqrt{\sqrt{\left(im \cdot im\right) \cdot 2.0}}}{\sqrt{\sqrt{\sqrt{{re}^2 + im \cdot im} - re}}}\right)}^2 \leadsto 0.5 \cdot {\left(\frac{\sqrt{\sqrt{\left(im \cdot im\right) \cdot 2.0}}}{\sqrt{\sqrt{-1 \cdot re - re}}}\right)}^2\]
      21.7
    12. Taylor expanded around -inf to get
      \[0.5 \cdot {\left(\frac{\sqrt{\sqrt{\left(im \cdot im\right) \cdot 2.0}}}{\sqrt{\sqrt{\color{red}{-1 \cdot re} - re}}}\right)}^2 \leadsto 0.5 \cdot {\left(\frac{\sqrt{\sqrt{\left(im \cdot im\right) \cdot 2.0}}}{\sqrt{\sqrt{\color{blue}{-1 \cdot re} - re}}}\right)}^2\]
      21.7
    13. Applied simplify to get
      \[0.5 \cdot {\left(\frac{\sqrt{\sqrt{\left(im \cdot im\right) \cdot 2.0}}}{\sqrt{\sqrt{-1 \cdot re - re}}}\right)}^2 \leadsto 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\left(-re\right) - re}}\]
      21.5
    14. Applied final simplification

    if -2.7522388519686937e+80 < re < -8.68041930647669e-252

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      37.5
    2. Using strategy rm
      37.5
    3. Applied flip-+ to get
      \[0.5 \cdot \sqrt{2.0 \cdot \color{red}{\left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}{\sqrt{re \cdot re + im \cdot im} - re}}}\]
      37.4
    4. Applied simplify to get
      \[0.5 \cdot \sqrt{2.0 \cdot \frac{\color{red}{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}}{\sqrt{re \cdot re + im \cdot im} - re}} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
      28.8
    5. Applied simplify to get
      \[0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\color{red}{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\color{blue}{\sqrt{{re}^2 + im \cdot im} - re}}}\]
      28.8

    if -8.68041930647669e-252 < re < 1.9727183986347288e-223

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      27.4
    2. Applied taylor to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\]
      0.0
    3. Taylor expanded around 0 to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\color{red}{im} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{im} + re\right)}\]
      0.0

    if 1.9727183986347288e-223 < re < 3.873554890065949e+113

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      17.6
    2. Using strategy rm
      17.6
    3. Applied add-sqr-sqrt to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\color{red}{\sqrt{re \cdot re + im \cdot im}} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}^2} + re\right)}\]
      17.7
    4. Applied simplify to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left({\color{red}{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}}^2 + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left({\color{blue}{\left(\sqrt{\sqrt{{re}^2 + im \cdot im}}\right)}}^2 + re\right)}\]
      17.7

    if 3.873554890065949e+113 < re

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      51.4
    2. Using strategy rm
      51.4
    3. Applied add-sqr-sqrt to get
      \[0.5 \cdot \color{red}{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\right)}^2}\]
      51.5
    4. Using strategy rm
      51.5
    5. Applied add-cube-cbrt to get
      \[0.5 \cdot {\left(\sqrt{\sqrt{2.0 \cdot \left(\color{red}{\sqrt{re \cdot re + im \cdot im}} + re\right)}}\right)}^2 \leadsto 0.5 \cdot {\left(\sqrt{\sqrt{2.0 \cdot \left(\color{blue}{{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^3} + re\right)}}\right)}^2\]
      51.5
    6. Applied simplify to get
      \[0.5 \cdot {\left(\sqrt{\sqrt{2.0 \cdot \left({\color{red}{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}^3 + re\right)}}\right)}^2 \leadsto 0.5 \cdot {\left(\sqrt{\sqrt{2.0 \cdot \left({\color{blue}{\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}^3 + re\right)}}\right)}^2\]
      51.5
    7. Applied taylor to get
      \[0.5 \cdot {\left(\sqrt{\sqrt{2.0 \cdot \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3 + re\right)}}\right)}^2 \leadsto 0.5 \cdot {\left(\sqrt{\sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}}\right)}^2\]
      15.8
    8. Taylor expanded around 0 to get
      \[0.5 \cdot {\left(\sqrt{\sqrt{2.0 \cdot \color{red}{\left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}}}\right)}^2 \leadsto 0.5 \cdot {\left(\sqrt{\sqrt{2.0 \cdot \color{blue}{\left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}}}\right)}^2\]
      15.8
    9. Applied simplify to get
      \[0.5 \cdot {\left(\sqrt{\sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}}\right)}^2 \leadsto 0.5 \cdot \sqrt{\left(2 \cdot re + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\right) \cdot 2.0}\]
      5.1
    10. Applied final simplification
  1. Removed slow pow expressions

Original test:


(lambda ((re default) (im default))
  #:name "math.sqrt on complex, real part"
  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))))
  #:target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (sqr im) (- (sqrt (+ (sqr re) (sqr im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))))))