Average Error: 32.2 → 14.4
Time: 45.2s
Precision: 64
Internal Precision: 1408
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.im \le -1.8195698843937282 \cdot 10^{-247}:\\ \;\;\;\;\frac{{\left(-x.im\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{if}\;x.im \le 3.8709927373610617 \cdot 10^{-268}:\\ \;\;\;\;{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re} \cdot {\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{\left(-y.im\right)}\\ \mathbf{if}\;x.im \le 5.865960667014197 \cdot 10^{-239}:\\ \;\;\;\;\frac{{x.im}^{y.re}}{\left(\sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\right) \cdot \sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\\ \mathbf{if}\;x.im \le 8.85610940636531 \cdot 10^{-70}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log x.im \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x.im}^{y.re}}{\left(\sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\right) \cdot \sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Split input into 5 regimes

  2. if x.im < -1.8195698843937282e-247

    1. Initial program 31.8

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

    2. Taylor expanded around 0 19.3

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

    3. Applied simplify23.1

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

    4. Taylor expanded around -inf 13.3

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

    5. Applied simplify13.7

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

    if -1.8195698843937282e-247 < x.im < 3.8709927373610617e-268

    1. Initial program 36.1

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

    2. Taylor expanded around 0 24.9

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

    3. Applied simplify27.8

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

    5. Applied add-log-exp27.8

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

    6. Applied exp-to-pow27.8

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

    if 3.8709927373610617e-268 < x.im < 5.865960667014197e-239

    1. Initial program 32.8

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

    2. Taylor expanded around 0 18.2

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

    3. Applied simplify20.7

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

    4. Taylor expanded around inf 22.9

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

    5. Applied simplify23.6

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

    7. Applied add-cube-cbrt23.6

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

    if 5.865960667014197e-239 < x.im < 8.85610940636531e-70

    1. Initial program 24.4

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

    2. Taylor expanded around 0 15.9

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

    if 8.85610940636531e-70 < x.im

    1. Initial program 35.2

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

    2. Taylor expanded around 0 20.8

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

    3. Applied simplify25.0

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

    4. Taylor expanded around inf 9.8

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

    5. Applied simplify10.1

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

    7. Applied add-cube-cbrt10.1

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

Runtime

Time bar (total: 45.2s)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' 
(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)))))