\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
Test:
math.sqrt on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 11.6 s
Input Error: 18.3
Output Error: 10.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \sqrt{{im}^2 \cdot 2.0} \cdot \frac{0.5}{\sqrt{re \cdot -2}} & \text{when } re \le -1.9909943f-05 \\ 0.5 \cdot \sqrt{2.0 \cdot \left({\left(\sqrt{\sqrt{{re}^2 + im \cdot im}}\right)}^{2} + re\right)} & \text{when } re \le 1.6052693f+16 \\ 0.5 \cdot \sqrt{\left(2 \cdot re + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\right) \cdot 2.0} & \text{otherwise} \end{cases}\)

    if re < -1.9909943f-05

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      28.3
    2. Using strategy rm
      28.3
    3. Applied flip-+ to get
      \[0.5 \cdot \sqrt{2.0 \cdot \color{red}{\left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}{\sqrt{re \cdot re + im \cdot im} - re}}}\]
      29.1
    4. Applied associate-*r/ to get
      \[0.5 \cdot \sqrt{\color{red}{2.0 \cdot \frac{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]
      29.1
    5. Applied sqrt-div to get
      \[0.5 \cdot \color{red}{\sqrt{\frac{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]
      29.1
    6. Applied simplify to get
      \[0.5 \cdot \frac{\color{red}{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(2.0 \cdot im\right) \cdot im}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
      20.3
    7. Applied simplify to get
      \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\color{red}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\color{blue}{\sqrt{\sqrt{{re}^2 + im \cdot im} - re}}}\]
      20.3
    8. Applied taylor to get
      \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\sqrt{{re}^2 + im \cdot im} - re}} \leadsto 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{-2 \cdot re}}\]
      11.1
    9. Taylor expanded around -inf to get
      \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\color{red}{-2 \cdot re}}} \leadsto 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\color{blue}{-2 \cdot re}}}\]
      11.1
    10. Applied simplify to get
      \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{-2 \cdot re}} \leadsto \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\frac{\sqrt{-2 \cdot re}}{0.5}}\]
      11.2
    11. Applied final simplification
    12. Applied simplify to get
      \[\color{red}{\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\frac{\sqrt{-2 \cdot re}}{0.5}}} \leadsto \color{blue}{\sqrt{{im}^2 \cdot 2.0} \cdot \frac{0.5}{\sqrt{re \cdot -2}}}\]
      11.2

    if -1.9909943f-05 < re < 1.6052693f+16

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      11.3
    2. Using strategy rm
      11.3
    3. Applied add-sqr-sqrt to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\color{red}{\sqrt{re \cdot re + im \cdot im}} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}^2} + re\right)}\]
      11.4
    4. Applied simplify to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left({\color{red}{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}}^2 + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left({\color{blue}{\left(\sqrt{\sqrt{{re}^2 + im \cdot im}}\right)}}^2 + re\right)}\]
      11.4
    5. Using strategy rm
      11.4
    6. Applied pow2 to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\color{red}{{\left(\sqrt{\sqrt{{re}^2 + im \cdot im}}\right)}^2} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{{\left(\sqrt{\sqrt{{re}^2 + im \cdot im}}\right)}^{2}} + re\right)}\]
      11.4

    if 1.6052693f+16 < re

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      26.7
    2. Using strategy rm
      26.7
    3. Applied add-cube-cbrt to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{red}{re \cdot re + im \cdot im}} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{{\left(\sqrt[3]{re \cdot re + im \cdot im}\right)}^3}} + re\right)}\]
      26.7
    4. Applied simplify to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{{\color{red}{\left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}^3} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{{\color{blue}{\left(\sqrt[3]{{re}^2 + im \cdot im}\right)}}^3} + re\right)}\]
      26.7
    5. Applied taylor to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{{\left(\sqrt[3]{{re}^2 + im \cdot im}\right)}^3} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}\]
      8.0
    6. Taylor expanded around 0 to get
      \[0.5 \cdot \sqrt{2.0 \cdot \color{red}{\left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}}\]
      8.0
    7. Applied taylor to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}\]
      8.0
    8. Taylor expanded around 0 to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \color{red}{\frac{{im}^2}{re}}\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \color{blue}{\frac{{im}^2}{re}}\right)}\]
      8.0
    9. Applied simplify to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)} \leadsto 0.5 \cdot \sqrt{\left(2 \cdot re + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\right) \cdot 2.0}\]
      2.8
    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))))))