math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 38.1 → 20.2

Time: 4.6s

Precision: 64

Math

\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.133313689967870834445494013025035211846 \cdot 10^{114}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -2.994155833278531802388158297765977679156 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le -1.435841514367864448151505165102991137359 \cdot 10^{-274}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 1.04092089734393064684781899004829158297 \cdot 10^{-280}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{im + re}\right)}\\ \mathbf{elif}\;re \le 7.670159887084534463468521621657291548674 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 1.546110109845155683121455826441872957199 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{re + re}\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r20776 = 0.5;
        double r20777 = 2.0;
        double r20778 = re;
        double r20779 = r20778 * r20778;
        double r20780 = im;
        double r20781 = r20780 * r20780;
        double r20782 = r20779 + r20781;
        double r20783 = sqrt(r20782);
        double r20784 = r20783 - r20778;
        double r20785 = r20777 * r20784;
        double r20786 = sqrt(r20785);
        double r20787 = r20776 * r20786;
        return r20787;
}

↓

double f(double re, double im) {
        double r20788 = re;
        double r20789 = -3.133313689967871e+114;
        bool r20790 = r20788 <= r20789;
        double r20791 = 0.5;
        double r20792 = 2.0;
        double r20793 = -1.0;
        double r20794 = r20793 * r20788;
        double r20795 = r20794 - r20788;
        double r20796 = r20792 * r20795;
        double r20797 = sqrt(r20796);
        double r20798 = r20791 * r20797;
        double r20799 = -2.9941558332785318e-164;
        bool r20800 = r20788 <= r20799;
        double r20801 = r20788 * r20788;
        double r20802 = im;
        double r20803 = r20802 * r20802;
        double r20804 = r20801 + r20803;
        double r20805 = sqrt(r20804);
        double r20806 = cbrt(r20805);
        double r20807 = r20806 * r20806;
        double r20808 = r20807 * r20806;
        double r20809 = r20808 - r20788;
        double r20810 = r20792 * r20809;
        double r20811 = sqrt(r20810);
        double r20812 = r20791 * r20811;
        double r20813 = -1.4358415143678644e-274;
        bool r20814 = r20788 <= r20813;
        double r20815 = r20788 + r20802;
        double r20816 = -r20815;
        double r20817 = r20792 * r20816;
        double r20818 = sqrt(r20817);
        double r20819 = r20791 * r20818;
        double r20820 = 1.0409208973439306e-280;
        bool r20821 = r20788 <= r20820;
        double r20822 = r20802 + r20788;
        double r20823 = r20802 / r20822;
        double r20824 = r20802 * r20823;
        double r20825 = r20792 * r20824;
        double r20826 = sqrt(r20825);
        double r20827 = r20791 * r20826;
        double r20828 = 7.670159887084534e-217;
        bool r20829 = r20788 <= r20828;
        double r20830 = 1.5461101098451557e+118;
        bool r20831 = r20788 <= r20830;
        double r20832 = sqrt(r20792);
        double r20833 = r20805 + r20788;
        double r20834 = sqrt(r20833);
        double r20835 = r20802 / r20834;
        double r20836 = fabs(r20835);
        double r20837 = r20832 * r20836;
        double r20838 = r20791 * r20837;
        double r20839 = r20788 + r20788;
        double r20840 = r20802 / r20839;
        double r20841 = r20802 * r20840;
        double r20842 = r20792 * r20841;
        double r20843 = sqrt(r20842);
        double r20844 = r20791 * r20843;
        double r20845 = r20831 ? r20838 : r20844;
        double r20846 = r20829 ? r20819 : r20845;
        double r20847 = r20821 ? r20827 : r20846;
        double r20848 = r20814 ? r20819 : r20847;
        double r20849 = r20800 ? r20812 : r20848;
        double r20850 = r20790 ? r20798 : r20849;
        return r20850;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 6 regimes

2. if re < -3.133313689967871e+114

   1. Initial program 53.5

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

   2. Taylor expanded around -inf 9.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} - re\right)}\]

   if -3.133313689967871e+114 < re < -2.9941558332785318e-164

   1. Initial program 16.0

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 16.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\]

   if -2.9941558332785318e-164 < re < -1.4358415143678644e-274 or 1.0409208973439306e-280 < re < 7.670159887084534e-217

   1. Initial program 31.6

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
   2. Using strategy rm

   3. Applied flip-- 32.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   4. Simplified 32.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

   5. Taylor expanded around -inf 34.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-\left(re + im\right)\right)}}\]

   if -1.4358415143678644e-274 < re < 1.0409208973439306e-280

   1. Initial program 28.3

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
   2. Using strategy rm

   3. Applied flip-- 28.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   4. Simplified 28.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 28.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}}\]

   7. Applied add-sqr-sqrt 47.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)}}^{2}}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\]

   8. Applied unpow-prod-down 47.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{\left(\sqrt{im}\right)}^{2} \cdot {\left(\sqrt{im}\right)}^{2}}}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\]

   9. Applied times-frac 47.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{\left(\sqrt{im}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt{im}\right)}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\]

   10. Simplified 47.0

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} \cdot \frac{{\left(\sqrt{im}\right)}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\]

   11. Simplified 27.9

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(im \cdot \color{blue}{\frac{im}{\sqrt{re \cdot re + im \cdot im} + re}}\right)}\]

   12. Taylor expanded around 0 32.6

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\color{blue}{im} + re}\right)}\]

   if 7.670159887084534e-217 < re < 1.5461101098451557e+118

   1. Initial program 40.0

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
   2. Using strategy rm

   3. Applied flip-- 39.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   4. Simplified 30.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 30.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}}\]

   7. Applied add-sqr-sqrt 47.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)}}^{2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   8. Applied unpow-prod-down 47.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{\left(\sqrt{im}\right)}^{2} \cdot {\left(\sqrt{im}\right)}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   9. Applied times-frac 46.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{\left(\sqrt{im}\right)}^{2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}} \cdot \frac{{\left(\sqrt{im}\right)}^{2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right)}}\]

   10. Simplified 46.2

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}} \cdot \frac{{\left(\sqrt{im}\right)}^{2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right)}\]

   11. Simplified 28.1

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}} \cdot \color{blue}{\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\right)}\]
   12. Using strategy rm

   13. Applied sqrt-prod 28.1

       \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}} \cdot \frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\right)}\]

   14. Simplified 17.8

       \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right|}\right)\]

   if 1.5461101098451557e+118 < re

   1. Initial program 61.1

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
   2. Using strategy rm

   3. Applied flip-- 61.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   4. Simplified 45.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 45.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}}\]

   7. Applied add-sqr-sqrt 55.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)}}^{2}}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\]

   8. Applied unpow-prod-down 55.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{\left(\sqrt{im}\right)}^{2} \cdot {\left(\sqrt{im}\right)}^{2}}}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\]

   9. Applied times-frac 55.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{\left(\sqrt{im}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt{im}\right)}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\]

   10. Simplified 55.3

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} \cdot \frac{{\left(\sqrt{im}\right)}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\]

   11. Simplified 44.6

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(im \cdot \color{blue}{\frac{im}{\sqrt{re \cdot re + im \cdot im} + re}}\right)}\]

   12. Taylor expanded around inf 23.4

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\color{blue}{re} + re}\right)}\]
3. Recombined 6 regimes into one program.

4. Final simplification 20.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.133313689967870834445494013025035211846 \cdot 10^{114}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -2.994155833278531802388158297765977679156 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le -1.435841514367864448151505165102991137359 \cdot 10^{-274}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 1.04092089734393064684781899004829158297 \cdot 10^{-280}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{im + re}\right)}\\ \mathbf{elif}\;re \le 7.670159887084534463468521621657291548674 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 1.546110109845155683121455826441872957199 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{re + re}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))