Average Error: 33.0 → 22.5
Time: 35.9s
Precision: 64
Internal Precision: 2112
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -5.471148339882993 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(-x.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\\ \mathbf{elif}\;x.re \le -9.556075680129787 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\\ \mathbf{elif}\;x.re \le 2.6342583104113 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(-x.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\\ \end{array}\]

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 3 regimes

  2. if x.re < -5.471148339882993e-38 or -9.556075680129787e-186 < x.re < 2.6342583104113e-310

    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. Initial simplification36.1

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

    3. Taylor expanded around -inf 22.4

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

    4. Simplified22.4

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

    if -5.471148339882993e-38 < x.re < -9.556075680129787e-186

    1. Initial program 17.8

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

    2. Initial simplification17.8

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

    if 2.6342583104113e-310 < x.re

    1. Initial program 34.0

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

    2. Initial simplification34.0

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

    3. Taylor expanded around inf 23.5

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

    4. Simplified23.4

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -5.471148339882993 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(-x.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\\ \mathbf{elif}\;x.re \le -9.556075680129787 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\\ \mathbf{elif}\;x.re \le 2.6342583104113 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(-x.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\\ \end{array}\]

Runtime

Time bar (total: 35.9s)Debug logProfile

herbie shell --seed 2018234 
(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)))))