Average Error: 32.4 → 6.0
Time: 50.4s
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}\;\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}{\frac{(\left(\frac{1}{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\left(y.im \cdot y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \left((y.im \cdot \left(\tan^{-1}_* \frac{x.im}{x.re}\right) + 1)_*\right))_*}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}} \le -2.595986232179534 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sin \left((\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{\sqrt[3]{{\left({\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}}}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\\ \mathbf{if}\;\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}{\frac{(\left(\frac{1}{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\left(y.im \cdot y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \left((y.im \cdot \left(\tan^{-1}_* \frac{x.im}{x.re}\right) + 1)_*\right))_*}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}} \le -0.0:\\ \;\;\;\;\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}{\frac{(\left(\frac{1}{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\left(y.im \cdot y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \left((y.im \cdot \left(\tan^{-1}_* \frac{x.im}{x.re}\right) + 1)_*\right))_*}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left((\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{\sqrt[3]{{\left({\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}}}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\\ \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 2 regimes

  2. if (/ (sin (fma y.im (log (hypot x.im x.re)) (* y.re (atan2 x.im x.re)))) (/ (fma (* 1/2 (atan2 x.im x.re)) (* (* y.im y.im) (atan2 x.im x.re)) (fma y.im (atan2 x.im x.re) 1)) (pow (hypot x.im x.re) y.re))) < -2.595986232179534e-309 or -0.0 < (/ (sin (fma y.im (log (hypot x.im x.re)) (* y.re (atan2 x.im x.re)))) (/ (fma (* 1/2 (atan2 x.im x.re)) (* (* y.im y.im) (atan2 x.im x.re)) (fma y.im (atan2 x.im x.re) 1)) (pow (hypot x.im x.re) y.re)))

    1. Initial program 33.6

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

    2. Applied simplify9.7

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

    4. Applied add-cbrt-cube9.8

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

    5. Applied simplify9.8

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

    if -2.595986232179534e-309 < (/ (sin (fma y.im (log (hypot x.im x.re)) (* y.re (atan2 x.im x.re)))) (/ (fma (* 1/2 (atan2 x.im x.re)) (* (* y.im y.im) (atan2 x.im x.re)) (fma y.im (atan2 x.im x.re) 1)) (pow (hypot x.im x.re) y.re))) < -0.0

    1. Initial program 30.8

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

    2. Applied simplify8.3

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

    4. Applied add-cbrt-cube8.4

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

    5. Applied simplify8.4

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

    6. Taylor expanded around 0 2.9

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

    7. Applied simplify1.4

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

Runtime

Time bar (total: 50.4s)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' +o rules:numerics
(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)))))