Average Error: 31.9 → 13.0
Time: 56.4s
Precision: 64
Internal Precision: 1344
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.im \le -1.6666144695324475 \cdot 10^{+157}:\\ \;\;\;\;\frac{{\left(-x.im\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{if}\;x.im \le -2.317071040327287 \cdot 10^{-154}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;x.im \le -2.4766512115837237 \cdot 10^{-222}:\\ \;\;\;\;\frac{{\left(-x.im\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{if}\;x.im \le -5.110982781221942 \cdot 10^{-290}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;x.im \le 1.60724677888386 \cdot 10^{-310}:\\ \;\;\;\;\frac{{\left(-x.im\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{if}\;x.im \le 1.713707274235917 \cdot 10^{-142} \lor \neg \left(x.im \le 1.2876188257639362 \cdot 10^{+90}\right):\\ \;\;\;\;\log \left(e^{\frac{{x.im}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.im \cdot y.im\right)\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if x.im < -1.6666144695324475e+157 or -2.317071040327287e-154 < x.im < -2.4766512115837237e-222 or -5.110982781221942e-290 < x.im < 1.60724677888386e-310

    1. Initial program 51.5

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around 0 27.8

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]

    3. Applied simplify31.6

      \[\leadsto \color{blue}{{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}}\]

    4. Taylor expanded around -inf 13.6

      \[\leadsto {\color{blue}{\left(-1 \cdot x.im\right)}}^{y.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\]

    5. Applied simplify13.8

      \[\leadsto \color{blue}{\frac{{\left(-x.im\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\]

    if -1.6666144695324475e+157 < x.im < -2.317071040327287e-154 or -2.4766512115837237e-222 < x.im < -5.110982781221942e-290

    1. Initial program 20.7

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around 0 13.8

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]

    3. Applied simplify18.6

      \[\leadsto \color{blue}{{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}}\]
    4. Using strategy rm

    5. Applied add-exp-log18.6

      \[\leadsto {\color{blue}{\left(e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}\right)}}^{y.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\]

    6. Applied pow-exp18.6

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re}} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\]

    7. Applied prod-exp13.8

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re + \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}}\]

    8. Applied simplify13.8

      \[\leadsto e^{\color{blue}{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\]

    if 1.60724677888386e-310 < x.im < 1.713707274235917e-142 or 1.2876188257639362e+90 < x.im

    1. Initial program 40.5

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around 0 24.4

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]

    3. Applied simplify28.2

      \[\leadsto \color{blue}{{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}}\]

    4. Taylor expanded around inf 12.8

      \[\leadsto {\color{blue}{x.im}}^{y.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\]

    5. Applied simplify13.3

      \[\leadsto \color{blue}{\frac{{x.im}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\]
    6. Using strategy rm

    7. Applied add-log-exp13.3

      \[\leadsto \color{blue}{\log \left(e^{\frac{{x.im}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\right)}\]

    if 1.713707274235917e-142 < x.im < 1.2876188257639362e+90

    1. Initial program 16.9

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around 0 10.6

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \color{blue}{x.im} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify13.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x.im \le -1.6666144695324475 \cdot 10^{+157}:\\ \;\;\;\;\frac{{\left(-x.im\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{if}\;x.im \le -2.317071040327287 \cdot 10^{-154}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;x.im \le -2.4766512115837237 \cdot 10^{-222}:\\ \;\;\;\;\frac{{\left(-x.im\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{if}\;x.im \le -5.110982781221942 \cdot 10^{-290}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;x.im \le 1.60724677888386 \cdot 10^{-310}:\\ \;\;\;\;\frac{{\left(-x.im\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{if}\;x.im \le 1.713707274235917 \cdot 10^{-142} \lor \neg \left(x.im \le 1.2876188257639362 \cdot 10^{+90}\right):\\ \;\;\;\;\log \left(e^{\frac{{x.im}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.im \cdot y.im\right)\\ \end{array}}\]

Runtime

Time bar (total: 56.4s)Debug logProfile

herbie shell --seed 2018207 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))