Average Error: 32.0 → 12.7
Time: 45.3s
Precision: 64
Internal Precision: 1408
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.im \le -1.5780120459582932 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\frac{{\left(-x.im\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \sqrt{\frac{{\left(-x.im\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\\ \mathbf{if}\;x.im \le 2.6331407899701736 \cdot 10^{-42}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x.im}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \end{array}\]

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 3 regimes

  2. if x.im < -1.5780120459582932e+41

    1. Initial program 43.0

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

    2. Taylor expanded around 0 23.7

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

    3. Applied simplify28.4

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

    4. Taylor expanded around -inf 8.6

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

    5. Applied simplify8.6

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

    7. Applied add-sqr-sqrt8.6

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

    if -1.5780120459582932e+41 < x.im < 2.6331407899701736e-42

    1. Initial program 25.1

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

    2. Taylor expanded around 0 16.3

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

    3. Applied simplify21.2

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

    5. Applied pow-exp20.2

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

    6. Applied add-exp-log20.2

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

    7. Applied pow-exp20.2

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

    8. Applied div-exp16.3

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

    if 2.6331407899701736e-42 < x.im

    1. Initial program 36.6

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

    2. Taylor expanded around 0 20.2

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

    3. Applied simplify25.1

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

    4. Taylor expanded around inf 9.1

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

Runtime

Time bar (total: 45.3s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(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)))))