\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
Test:
math.sqrt on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 8.8 s
Input Error: 18.4
Output Error: 10.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{0.5 \cdot \sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\left(-re\right) - re}} & \text{when } re \le -1.4391974f-06 \\ 0.5 \cdot \sqrt{2.0 \cdot \left({\left(\sqrt{\sqrt{{re}^2 + im \cdot im}}\right)}^2 + re\right)} & \text{when } re \le 2.5568826f+12 \\ 0.5 \cdot \sqrt{\left(2 \cdot re + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\right) \cdot 2.0} & \text{otherwise} \end{cases}\)

    if re < -1.4391974f-06

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      28.1
    2. Using strategy rm
      28.1
    3. Applied flip-+ to get
      \[0.5 \cdot \sqrt{2.0 \cdot \color{red}{\left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}{\sqrt{re \cdot re + im \cdot im} - re}}}\]
      28.9
    4. Applied associate-*r/ to get
      \[0.5 \cdot \sqrt{\color{red}{2.0 \cdot \frac{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]
      28.9
    5. Applied sqrt-div to get
      \[0.5 \cdot \color{red}{\sqrt{\frac{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]
      28.9
    6. Applied simplify to get
      \[0.5 \cdot \frac{\color{red}{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(2.0 \cdot im\right) \cdot im}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
      19.8
    7. Applied simplify to get
      \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\color{red}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\color{blue}{\sqrt{\sqrt{{re}^2 + im \cdot im} - re}}}\]
      19.8
    8. Applied taylor to get
      \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\sqrt{{re}^2 + im \cdot im} - re}} \leadsto 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{-1 \cdot re - re}}\]
      10.8
    9. Taylor expanded around -inf to get
      \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\color{red}{-1 \cdot re} - re}} \leadsto 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\color{blue}{-1 \cdot re} - re}}\]
      10.8
    10. Applied simplify to get
      \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{-1 \cdot re - re}} \leadsto \frac{0.5 \cdot \sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\left(-re\right) - re}}\]
      10.9
    11. Applied final simplification

    if -1.4391974f-06 < re < 2.5568826f+12

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      11.4
    2. Using strategy rm
      11.4
    3. Applied add-sqr-sqrt to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\color{red}{\sqrt{re \cdot re + im \cdot im}} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}^2} + re\right)}\]
      11.6
    4. Applied simplify to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left({\color{red}{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}}^2 + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left({\color{blue}{\left(\sqrt{\sqrt{{re}^2 + im \cdot im}}\right)}}^2 + re\right)}\]
      11.6

    if 2.5568826f+12 < re

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      24.3
    2. Using strategy rm
      24.3
    3. Applied add-cube-cbrt to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\color{red}{\sqrt{re \cdot re + im \cdot im}} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^3} + re\right)}\]
      24.3
    4. Applied simplify to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left({\color{red}{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}^3 + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left({\color{blue}{\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}^3 + re\right)}\]
      24.3
    5. Applied taylor to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3 + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}\]
      7.7
    6. Taylor expanded around 0 to get
      \[0.5 \cdot \sqrt{2.0 \cdot \color{red}{\left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}}\]
      7.7
    7. Applied simplify to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)} \leadsto 0.5 \cdot \sqrt{\left(2 \cdot re + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\right) \cdot 2.0}\]
      3.2
    8. 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))))))