Average Error: 35.6 → 19.5
Time: 1.8m
Precision: 64
Internal precision: 128
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
⬇
\[\begin{array}{l} \mathbf{if}\;x.im \le -4.452822703070531 \cdot 10^{+62}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.im\right)\right) \cdot \frac{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{if}\;x.im \le -7.459641188452852 \cdot 10^{-162}:\\ \;\;\;\;{\left(\sqrt[3]{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(\sqrt{{x.im}^2 + x.re \cdot x.re}\right)\right)}\right)}^3 \cdot \frac{{\left(\sqrt{{x.im}^2 + x.re \cdot x.re}\right)}^{y.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{if}\;x.im \le 4.110283407852792 \cdot 10^{-299}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.im\right)\right) \cdot \frac{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log x.im\right) \cdot \frac{{\left(\sqrt{{x.im}^2 + x.re \cdot x.re}\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \end{array}\]

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 < -4.452822703070531e+62 or -7.459641188452852e-162 < x.im < 4.110283407852792e-299

    1. Initial program 44.4

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

    2. Applied simplify 43.8

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

    3. Applied taylor 22.6

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

    4. Taylor expanded around -inf 22.6

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

    5. Applied simplify 22.6

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

    if -4.452822703070531e+62 < x.im < -7.459641188452852e-162

    1. Initial program 19.1

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

    2. Applied simplify 19.7

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

    4. Applied add-cube-cbrt 20.1

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

    6. Applied pow-exp 19.6

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

    if 4.110283407852792e-299 < x.im

    1. Initial program 36.1

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

    2. Applied simplify 35.6

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

    3. Applied taylor 17.5

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

    4. Taylor expanded around inf 17.5

      \[\leadsto \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \color{blue}{x.im}\right) \cdot \frac{{\left(\sqrt{{x.im}^2 + x.re \cdot x.re}\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 1.8m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(3256694377 330064089 4260074272 2213997589 1457464827 1960738255)'
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))