\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
Test:
math.sqrt on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 10.2 s
Input Error: 42.2
Output Error: 24.5
Log: ⚲
Profile: 🕒
\(\begin{cases} 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\sqrt{{re}^2 + im \cdot im} - re}} & \text{when } re \le 2.2955426280380868 \cdot 10^{-209} \\ 0.5 \cdot \sqrt{\left(2 \cdot re + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\right) \cdot 2.0} & \text{when } re \le 1.3480729395074825 \cdot 10^{+78} \\ 0.5 \cdot \sqrt{\left(2 \cdot re + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\right) \cdot 2.0} & \text{otherwise} \end{cases}\)

    if re < 2.2955426280380868e-209

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      43.2
    2. Using strategy rm
      43.2
    3. Applied flip-+ to get
      \[0.5 \cdot \sqrt{2.0 \cdot \color{red}{\left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}{\sqrt{re \cdot re + im \cdot im} - re}}}\]
      43.4
    4. Applied associate-*r/ to get
      \[0.5 \cdot \sqrt{\color{red}{2.0 \cdot \frac{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]
      43.4
    5. Applied sqrt-div to get
      \[0.5 \cdot \color{red}{\sqrt{\frac{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]
      43.5
    6. Applied simplify to get
      \[0.5 \cdot \frac{\color{red}{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(2.0 \cdot im\right) \cdot im}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
      34.8
    7. Applied simplify to get
      \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\color{red}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\color{blue}{\sqrt{\sqrt{{re}^2 + im \cdot im} - re}}}\]
      34.8

    if 2.2955426280380868e-209 < re < 1.3480729395074825e+78

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      35.9
    2. Using strategy rm
      35.9
    3. Applied add-cube-cbrt to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\color{red}{\sqrt{re \cdot re + im \cdot im}} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^3} + re\right)}\]
      36.1
    4. Applied simplify to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left({\color{red}{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}^3 + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left({\color{blue}{\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}^3 + re\right)}\]
      36.1
    5. Using strategy rm
      36.1
    6. Applied add-cbrt-cube to get
      \[0.5 \cdot \color{red}{\sqrt{2.0 \cdot \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3 + re\right)}} \leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(\sqrt{2.0 \cdot \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3 + re\right)}\right)}^3}}\]
      36.2
    7. Applied taylor to get
      \[0.5 \cdot \sqrt[3]{{\left(\sqrt{2.0 \cdot \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3 + re\right)}\right)}^3} \leadsto 0.5 \cdot \sqrt[3]{{\left(\sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}\right)}^3}\]
      21.2
    8. Taylor expanded around 0 to get
      \[0.5 \cdot \sqrt[3]{{\left(\sqrt{2.0 \cdot \color{red}{\left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}}\right)}^3} \leadsto 0.5 \cdot \sqrt[3]{{\left(\sqrt{2.0 \cdot \color{blue}{\left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}}\right)}^3}\]
      21.2
    9. Applied simplify to get
      \[\color{red}{0.5 \cdot \sqrt[3]{{\left(\sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}\right)}^3}} \leadsto \color{blue}{0.5 \cdot \sqrt{\left(2 \cdot re + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\right) \cdot 2.0}}\]
      15.1

    if 1.3480729395074825e+78 < re

    1. Started with
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
      46.4
    2. Using strategy rm
      46.4
    3. Applied add-cube-cbrt to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left(\color{red}{\sqrt{re \cdot re + im \cdot im}} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^3} + re\right)}\]
      46.5
    4. Applied simplify to get
      \[0.5 \cdot \sqrt{2.0 \cdot \left({\color{red}{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}^3 + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left({\color{blue}{\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}^3 + re\right)}\]
      46.5
    5. Using strategy rm
      46.5
    6. Applied add-cbrt-cube to get
      \[0.5 \cdot \color{red}{\sqrt{2.0 \cdot \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3 + re\right)}} \leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(\sqrt{2.0 \cdot \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3 + re\right)}\right)}^3}}\]
      46.6
    7. Applied taylor to get
      \[0.5 \cdot \sqrt[3]{{\left(\sqrt{2.0 \cdot \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3 + re\right)}\right)}^3} \leadsto 0.5 \cdot \sqrt[3]{{\left(\sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}\right)}^3}\]
      34.8
    8. Taylor expanded around 0 to get
      \[0.5 \cdot \sqrt[3]{{\left(\sqrt{2.0 \cdot \color{red}{\left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}}\right)}^3} \leadsto 0.5 \cdot \sqrt[3]{{\left(\sqrt{2.0 \cdot \color{blue}{\left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}}\right)}^3}\]
      34.8
    9. Applied simplify to get
      \[\color{red}{0.5 \cdot \sqrt[3]{{\left(\sqrt{2.0 \cdot \left(2 \cdot re + \frac{1}{2} \cdot \frac{{im}^2}{re}\right)}\right)}^3}} \leadsto \color{blue}{0.5 \cdot \sqrt{\left(2 \cdot re + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\right) \cdot 2.0}}\]
      5.1
  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))))))