Average Error: 32.5 → 13.3
Time: 53.6s
Precision: 64
Internal Precision: 1600
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -9.587737600049712 \cdot 10^{+131}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\right) \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(\frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2} \cdot {y.im}^{2}\right) + 1\right)}}\\ \mathbf{if}\;x.re \le -9.528397543937879 \cdot 10^{-79}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{-1}{x.re}\right) \cdot y.im\right)\\ \mathbf{if}\;x.re \le 2.20165564976506 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\right) \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(\frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2} \cdot {y.im}^{2}\right) + 1\right)}}\\ \mathbf{if}\;x.re \le 1.4828000225841304 \cdot 10^{-136} \lor \neg \left(x.re \le 1.7242911125239578 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{{x.re}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)\\ \end{array}\]

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 5 regimes

  2. if x.re < -9.587737600049712e+131

    1. Initial program 56.7

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

    2. Taylor expanded around 0 31.2

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

    3. Applied simplify35.7

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

    4. Taylor expanded around -inf 7.9

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

    5. Applied simplify7.9

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

    7. Applied add-cube-cbrt8.0

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\right) \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}}\]

    8. Taylor expanded around 0 8.0

      \[\leadsto \left(\sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\right) \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + \left(\frac{1}{2} \cdot \left({y.im}^{2} \cdot {\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}\right) + 1\right)}}}\]

    if -9.587737600049712e+131 < x.re < -9.528397543937879e-79

    1. Initial program 16.5

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

    2. Taylor expanded around -inf 9.1

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

    if -9.528397543937879e-79 < x.re < 2.20165564976506e-310

    1. Initial program 25.3

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

    2. Taylor expanded around 0 14.3

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

    3. Applied simplify19.9

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

    4. Taylor expanded around -inf 16.7

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

    5. Applied simplify16.7

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

    7. Applied add-cube-cbrt16.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\right) \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}}\]

    8. Taylor expanded around 0 16.8

      \[\leadsto \left(\sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\right) \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + \left(\frac{1}{2} \cdot \left({y.im}^{2} \cdot {\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}\right) + 1\right)}}}\]

    if 2.20165564976506e-310 < x.re < 1.4828000225841304e-136 or 1.7242911125239578e+17 < x.re

    1. Initial program 38.5

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

    2. Taylor expanded around 0 23.8

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

    3. Applied simplify26.6

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

    4. Taylor expanded around inf 14.1

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

    5. Applied simplify15.7

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

    if 1.4828000225841304e-136 < x.re < 1.7242911125239578e+17

    1. Initial program 19.4

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

    2. Taylor expanded around inf 13.0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \color{blue}{x.re} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
  3. Recombined 5 regimes into one program.

  4. Applied simplify13.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x.re \le -9.587737600049712 \cdot 10^{+131}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\right) \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(\frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2} \cdot {y.im}^{2}\right) + 1\right)}}\\ \mathbf{if}\;x.re \le -9.528397543937879 \cdot 10^{-79}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{-1}{x.re}\right) \cdot y.im\right)\\ \mathbf{if}\;x.re \le 2.20165564976506 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\right) \cdot \sqrt[3]{\frac{{\left(-x.re\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(\frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2} \cdot {y.im}^{2}\right) + 1\right)}}\\ \mathbf{if}\;x.re \le 1.4828000225841304 \cdot 10^{-136} \lor \neg \left(x.re \le 1.7242911125239578 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{{x.re}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)\\ \end{array}}\]

Runtime

Time bar (total: 53.6s)Debug logProfile

herbie shell --seed 2019053 
(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)))))