\[\sqrt{re \cdot re + im \cdot im}\]
Test:
math.abs on complex
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 3.7 s
Input Error: 40.2
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\left|re\right| + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\)
  1. Started with
    \[\sqrt{re \cdot re + im \cdot im}\]
    40.2
  2. Applied simplify to get
    \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
    40.2
  3. Using strategy rm
    40.2
  4. Applied add-exp-log to get
    \[\color{red}{\sqrt{{re}^2 + im \cdot im}} \leadsto \color{blue}{e^{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}}\]
    41.8
  5. Using strategy rm
    41.8
  6. Applied pow1/2 to get
    \[e^{\log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)}} \leadsto e^{\log \color{blue}{\left({\left({re}^2 + im \cdot im\right)}^{\frac{1}{2}}\right)}}\]
    41.8
  7. Applied log-pow to get
    \[e^{\color{red}{\log \left({\left({re}^2 + im \cdot im\right)}^{\frac{1}{2}}\right)}} \leadsto e^{\color{blue}{\frac{1}{2} \cdot \log \left({re}^2 + im \cdot im\right)}}\]
    41.8
  8. Applied exp-prod to get
    \[\color{red}{e^{\frac{1}{2} \cdot \log \left({re}^2 + im \cdot im\right)}} \leadsto \color{blue}{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left({re}^2 + im \cdot im\right)\right)}}\]
    42.0
  9. Using strategy rm
    42.0
  10. Applied add-cube-cbrt to get
    \[{\left(e^{\frac{1}{2}}\right)}^{\color{red}{\left(\log \left({re}^2 + im \cdot im\right)\right)}} \leadsto {\left(e^{\frac{1}{2}}\right)}^{\color{blue}{\left({\left(\sqrt[3]{\log \left({re}^2 + im \cdot im\right)}\right)}^3\right)}}\]
    42.7
  11. Applied taylor to get
    \[{\left(e^{\frac{1}{2}}\right)}^{\left({\left(\sqrt[3]{\log \left({re}^2 + im \cdot im\right)}\right)}^3\right)} \leadsto {\left({re}^2\right)}^{\frac{1}{2}} + \frac{1}{2} \cdot \frac{{im}^2}{re}\]
    24.3
  12. Taylor expanded around 0 to get
    \[\color{red}{{\left({re}^2\right)}^{\frac{1}{2}} + \frac{1}{2} \cdot \frac{{im}^2}{re}} \leadsto \color{blue}{{\left({re}^2\right)}^{\frac{1}{2}} + \frac{1}{2} \cdot \frac{{im}^2}{re}}\]
    24.3
  13. Applied simplify to get
    \[{\left({re}^2\right)}^{\frac{1}{2}} + \frac{1}{2} \cdot \frac{{im}^2}{re} \leadsto \left|re\right| + \frac{im \cdot \frac{1}{2}}{\frac{re}{im}}\]
    0.1
  14. Applied final simplification

Original test:


(lambda ((re default) (im default))
  #:name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))