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

Average Error: 38.3 → 26.2

Time: 9.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -1.307585115624309216124260105110207840294 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le -2.628807493432242249890478424299278330008 \cdot 10^{-95}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) - re\right)}\\ \mathbf{elif}\;im \le -9.677770542110420808800061902060254608143 \cdot 10^{-147}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;im \le -8.097842811387572585392563061395395075683 \cdot 10^{-243}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le 1.054568161257482803525852713126222588258 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{2 \cdot re}{im}}}\\ \mathbf{elif}\;im \le 4.068654166237996793746604040555372844878 \cdot 10^{-193}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot im}}{\sqrt{\frac{re + \sqrt{re \cdot re + im \cdot im}}{im}}}\\ \mathbf{elif}\;im \le 2.566682872311100173898771653518908866918 \cdot 10^{-133}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;im \le 1.429821929476942571758399671404759856062 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot im}}{\sqrt{\frac{re + \sqrt{re \cdot re + im \cdot im}}{im}}}\\ \mathbf{elif}\;im \le 3.14127497997689588006514124503829188465 \cdot 10^{-81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) - re\right)}\\ \mathbf{elif}\;im \le 2.341360534110819251436825146816423482641 \cdot 10^{111}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot im}}{\sqrt{\frac{re + \sqrt{re \cdot re + im \cdot im}}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r18719 = 0.5;
        double r18720 = 2.0;
        double r18721 = re;
        double r18722 = r18721 * r18721;
        double r18723 = im;
        double r18724 = r18723 * r18723;
        double r18725 = r18722 + r18724;
        double r18726 = sqrt(r18725);
        double r18727 = r18726 - r18721;
        double r18728 = r18720 * r18727;
        double r18729 = sqrt(r18728);
        double r18730 = r18719 * r18729;
        return r18730;
}

↓

double f(double re, double im) {
        double r18731 = im;
        double r18732 = -1.3075851156243092e-42;
        bool r18733 = r18731 <= r18732;
        double r18734 = 0.5;
        double r18735 = 2.0;
        double r18736 = re;
        double r18737 = r18736 + r18731;
        double r18738 = -r18737;
        double r18739 = r18735 * r18738;
        double r18740 = sqrt(r18739);
        double r18741 = r18734 * r18740;
        double r18742 = -2.6288074934322422e-95;
        bool r18743 = r18731 <= r18742;
        double r18744 = -r18736;
        double r18745 = r18744 - r18736;
        double r18746 = r18735 * r18745;
        double r18747 = sqrt(r18746);
        double r18748 = r18734 * r18747;
        double r18749 = -9.677770542110421e-147;
        bool r18750 = r18731 <= r18749;
        double r18751 = r18731 * r18731;
        double r18752 = r18735 * r18751;
        double r18753 = sqrt(r18752);
        double r18754 = r18736 * r18736;
        double r18755 = r18754 + r18751;
        double r18756 = sqrt(r18755);
        double r18757 = r18736 + r18756;
        double r18758 = sqrt(r18757);
        double r18759 = r18753 / r18758;
        double r18760 = r18734 * r18759;
        double r18761 = -8.097842811387573e-243;
        bool r18762 = r18731 <= r18761;
        double r18763 = 1.0545681612574828e-300;
        bool r18764 = r18731 <= r18763;
        double r18765 = 2.0;
        double r18766 = r18765 * r18736;
        double r18767 = r18766 / r18731;
        double r18768 = r18731 / r18767;
        double r18769 = r18735 * r18768;
        double r18770 = sqrt(r18769);
        double r18771 = r18734 * r18770;
        double r18772 = 4.068654166237997e-193;
        bool r18773 = r18731 <= r18772;
        double r18774 = r18735 * r18731;
        double r18775 = sqrt(r18774);
        double r18776 = r18757 / r18731;
        double r18777 = sqrt(r18776);
        double r18778 = r18775 / r18777;
        double r18779 = r18734 * r18778;
        double r18780 = 2.5666828723111e-133;
        bool r18781 = r18731 <= r18780;
        double r18782 = log(r18756);
        double r18783 = exp(r18782);
        double r18784 = sqrt(r18783);
        double r18785 = sqrt(r18756);
        double r18786 = r18784 * r18785;
        double r18787 = r18786 - r18736;
        double r18788 = r18735 * r18787;
        double r18789 = sqrt(r18788);
        double r18790 = r18734 * r18789;
        double r18791 = 1.4298219294769426e-96;
        bool r18792 = r18731 <= r18791;
        double r18793 = 3.141274979976896e-81;
        bool r18794 = r18731 <= r18793;
        double r18795 = 2.3413605341108193e+111;
        bool r18796 = r18731 <= r18795;
        double r18797 = r18731 - r18736;
        double r18798 = r18735 * r18797;
        double r18799 = sqrt(r18798);
        double r18800 = r18734 * r18799;
        double r18801 = r18796 ? r18779 : r18800;
        double r18802 = r18794 ? r18748 : r18801;
        double r18803 = r18792 ? r18779 : r18802;
        double r18804 = r18781 ? r18790 : r18803;
        double r18805 = r18773 ? r18779 : r18804;
        double r18806 = r18764 ? r18771 : r18805;
        double r18807 = r18762 ? r18741 : r18806;
        double r18808 = r18750 ? r18760 : r18807;
        double r18809 = r18743 ? r18748 : r18808;
        double r18810 = r18733 ? r18741 : r18809;
        return r18810;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 7 regimes

2. if im < -1.3075851156243092e-42 or -9.677770542110421e-147 < im < -8.097842811387573e-243

   1. Initial program 40.6

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

   3. Applied add-sqr-sqrt 40.6

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

   4. Applied sqrt-prod 40.7

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

   6. Applied flip-- 45.6

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

   7. Simplified 42.6

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

   8. Simplified 42.6

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

   9. Taylor expanded around -inf 22.7

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

   if -1.3075851156243092e-42 < im < -2.6288074934322422e-95 or 1.4298219294769426e-96 < im < 3.141274979976896e-81

   1. Initial program 28.7

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

   3. Applied add-sqr-sqrt 28.7

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

   4. Applied sqrt-prod 28.9

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

   5. Taylor expanded around -inf 45.5

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

   6. Simplified 45.5

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

   if -2.6288074934322422e-95 < im < -9.677770542110421e-147

   1. Initial program 30.5

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

   3. Applied add-sqr-sqrt 30.5

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

   4. Applied sqrt-prod 30.8

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

   6. Applied flip-- 43.2

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

   7. Simplified 34.5

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

   8. Simplified 34.5

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

   10. Applied associate-*r/ 34.5

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

   11. Applied sqrt-div 31.0

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

   if -8.097842811387573e-243 < im < 1.0545681612574828e-300

   1. Initial program 41.0

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

   3. Applied add-sqr-sqrt 41.0

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

   4. Applied sqrt-prod 43.6

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

   6. Applied flip-- 57.9

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

   7. Simplified 48.3

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

   8. Simplified 48.6

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

   10. Applied associate-/l* 48.3

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

   11. Taylor expanded around inf 46.5

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

   if 1.0545681612574828e-300 < im < 4.068654166237997e-193 or 2.5666828723111e-133 < im < 1.4298219294769426e-96 or 3.141274979976896e-81 < im < 2.3413605341108193e+111

   1. Initial program 29.6

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

   3. Applied add-sqr-sqrt 29.6

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

   4. Applied sqrt-prod 30.1

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

   6. Applied flip-- 39.5

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

   7. Simplified 32.6

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

   8. Simplified 32.7

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

   10. Applied associate-/l* 32.3

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

   12. Applied associate-*r/ 32.3

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

   13. Applied sqrt-div 29.1

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

   if 4.068654166237997e-193 < im < 2.5666828723111e-133

   1. Initial program 39.0

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

   3. Applied add-sqr-sqrt 39.0

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

   4. Applied sqrt-prod 39.2

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

   6. Applied add-exp-log 40.3

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

   if 2.3413605341108193e+111 < im

   1. Initial program 52.7

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

   2. Taylor expanded around 0 8.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)}\]
3. Recombined 7 regimes into one program.

4. Final simplification 26.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.307585115624309216124260105110207840294 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le -2.628807493432242249890478424299278330008 \cdot 10^{-95}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) - re\right)}\\ \mathbf{elif}\;im \le -9.677770542110420808800061902060254608143 \cdot 10^{-147}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;im \le -8.097842811387572585392563061395395075683 \cdot 10^{-243}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le 1.054568161257482803525852713126222588258 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{2 \cdot re}{im}}}\\ \mathbf{elif}\;im \le 4.068654166237996793746604040555372844878 \cdot 10^{-193}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot im}}{\sqrt{\frac{re + \sqrt{re \cdot re + im \cdot im}}{im}}}\\ \mathbf{elif}\;im \le 2.566682872311100173898771653518908866918 \cdot 10^{-133}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;im \le 1.429821929476942571758399671404759856062 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot im}}{\sqrt{\frac{re + \sqrt{re \cdot re + im \cdot im}}{im}}}\\ \mathbf{elif}\;im \le 3.14127497997689588006514124503829188465 \cdot 10^{-81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) - re\right)}\\ \mathbf{elif}\;im \le 2.341360534110819251436825146816423482641 \cdot 10^{111}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot im}}{\sqrt{\frac{re + \sqrt{re \cdot re + im \cdot im}}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(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)))))