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

Average Error: 39.3 → 23.7

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.5777813673616158 \cdot 10^{127}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -2.5763601950204309 \cdot 10^{-265}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 1.27662858127337166 \cdot 10^{-281}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{re + im}\right)}\\ \mathbf{elif}\;re \le 2.694808318755371 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\left|im\right| \cdot \frac{\sqrt[3]{\left|im\right|} \cdot \sqrt[3]{\left|im\right|}}{\sqrt[3]{\sqrt{re \cdot re + im \cdot im} + re} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im} + re}}\right) \cdot \frac{\sqrt[3]{\left|im\right|}}{\sqrt[3]{\sqrt{re \cdot re + im \cdot im} + re}}\right)}\\ \mathbf{elif}\;re \le 8.6973947796722868 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 4.94013962552010715 \cdot 10^{141}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(\left|im\right| \cdot \left|im\right|\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{re + re}\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r20974 = 0.5;
        double r20975 = 2.0;
        double r20976 = re;
        double r20977 = r20976 * r20976;
        double r20978 = im;
        double r20979 = r20978 * r20978;
        double r20980 = r20977 + r20979;
        double r20981 = sqrt(r20980);
        double r20982 = r20981 - r20976;
        double r20983 = r20975 * r20982;
        double r20984 = sqrt(r20983);
        double r20985 = r20974 * r20984;
        return r20985;
}

↓

double f(double re, double im) {
        double r20986 = re;
        double r20987 = -5.577781367361616e+127;
        bool r20988 = r20986 <= r20987;
        double r20989 = 0.5;
        double r20990 = 2.0;
        double r20991 = -1.0;
        double r20992 = r20991 * r20986;
        double r20993 = r20992 - r20986;
        double r20994 = r20990 * r20993;
        double r20995 = sqrt(r20994);
        double r20996 = r20989 * r20995;
        double r20997 = -2.576360195020431e-265;
        bool r20998 = r20986 <= r20997;
        double r20999 = r20986 * r20986;
        double r21000 = im;
        double r21001 = r21000 * r21000;
        double r21002 = r20999 + r21001;
        double r21003 = cbrt(r21002);
        double r21004 = fabs(r21003);
        double r21005 = sqrt(r21003);
        double r21006 = r21004 * r21005;
        double r21007 = r21006 - r20986;
        double r21008 = r20990 * r21007;
        double r21009 = sqrt(r21008);
        double r21010 = r20989 * r21009;
        double r21011 = 1.2766285812733717e-281;
        bool r21012 = r20986 <= r21011;
        double r21013 = fabs(r21000);
        double r21014 = r20986 + r21000;
        double r21015 = r21013 / r21014;
        double r21016 = r21013 * r21015;
        double r21017 = r20990 * r21016;
        double r21018 = sqrt(r21017);
        double r21019 = r20989 * r21018;
        double r21020 = 2.694808318755371e-253;
        bool r21021 = r20986 <= r21020;
        double r21022 = cbrt(r21013);
        double r21023 = r21022 * r21022;
        double r21024 = sqrt(r21002);
        double r21025 = r21024 + r20986;
        double r21026 = cbrt(r21025);
        double r21027 = r21026 * r21026;
        double r21028 = r21023 / r21027;
        double r21029 = r21013 * r21028;
        double r21030 = r21022 / r21026;
        double r21031 = r21029 * r21030;
        double r21032 = r20990 * r21031;
        double r21033 = sqrt(r21032);
        double r21034 = r20989 * r21033;
        double r21035 = 8.697394779672287e-128;
        bool r21036 = r20986 <= r21035;
        double r21037 = -r21014;
        double r21038 = r20990 * r21037;
        double r21039 = sqrt(r21038);
        double r21040 = r20989 * r21039;
        double r21041 = 4.940139625520107e+141;
        bool r21042 = r20986 <= r21041;
        double r21043 = r21013 * r21013;
        double r21044 = r20990 * r21043;
        double r21045 = sqrt(r21044);
        double r21046 = sqrt(r21025);
        double r21047 = r21045 / r21046;
        double r21048 = r20989 * r21047;
        double r21049 = r20986 + r20986;
        double r21050 = r21013 / r21049;
        double r21051 = r21013 * r21050;
        double r21052 = r20990 * r21051;
        double r21053 = sqrt(r21052);
        double r21054 = r20989 * r21053;
        double r21055 = r21042 ? r21048 : r21054;
        double r21056 = r21036 ? r21040 : r21055;
        double r21057 = r21021 ? r21034 : r21056;
        double r21058 = r21012 ? r21019 : r21057;
        double r21059 = r20998 ? r21010 : r21058;
        double r21060 = r20988 ? r20996 : r21059;
        return r21060;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 7 regimes

2. if re < -5.577781367361616e+127

   1. Initial program 56.4

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

   2. Taylor expanded around -inf 9.0

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

   if -5.577781367361616e+127 < re < -2.576360195020431e-265

   1. Initial program 20.6

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

   3. Applied add-cube-cbrt 20.8

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

   4. Applied sqrt-prod 20.8

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

   5. Simplified 20.8

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

   if -2.576360195020431e-265 < re < 1.2766285812733717e-281

   1. Initial program 30.9

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

   3. Applied flip-- 31.2

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

   4. Simplified 31.2

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

   6. Applied *-un-lft-identity 31.2

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

   7. Applied add-sqr-sqrt 31.2

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

   8. Applied times-frac 31.2

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

   9. Simplified 30.8

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

   10. Simplified 30.9

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

   11. Taylor expanded around 0 33.5

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

   if 1.2766285812733717e-281 < re < 2.694808318755371e-253

   1. Initial program 34.2

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

   3. Applied flip-- 33.5

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

   4. Simplified 33.5

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

   6. Applied *-un-lft-identity 33.5

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

   7. Applied add-sqr-sqrt 33.5

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

   8. Applied times-frac 33.5

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

   9. Simplified 33.3

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

   10. Simplified 32.0

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

   12. Applied add-cube-cbrt 32.4

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

   13. Applied add-cube-cbrt 32.0

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

   14. Applied times-frac 32.0

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

   15. Applied associate-*r* 32.0

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

   if 2.694808318755371e-253 < re < 8.697394779672287e-128

   1. Initial program 32.4

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

   3. Applied flip-- 31.9

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

   4. Simplified 31.6

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

   5. Taylor expanded around -inf 36.8

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

   if 8.697394779672287e-128 < re < 4.940139625520107e+141

   1. Initial program 45.0

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

   3. Applied flip-- 45.0

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

   4. Simplified 30.2

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

   6. Applied *-un-lft-identity 30.2

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

   7. Applied add-sqr-sqrt 30.2

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

   8. Applied times-frac 30.2

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

   9. Simplified 30.2

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

   10. Simplified 28.5

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

   12. Applied associate-*r/ 30.2

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

   13. Applied associate-*r/ 30.2

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

   14. Applied sqrt-div 28.4

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

   if 4.940139625520107e+141 < re

   1. Initial program 63.4

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

   3. Applied flip-- 63.4

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

   4. Simplified 48.9

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

   6. Applied *-un-lft-identity 48.9

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

   7. Applied add-sqr-sqrt 48.9

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

   8. Applied times-frac 48.9

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

   9. Simplified 48.9

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

   10. Simplified 48.5

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

   11. Taylor expanded around inf 23.4

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

4. Final simplification 23.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.5777813673616158 \cdot 10^{127}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -2.5763601950204309 \cdot 10^{-265}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 1.27662858127337166 \cdot 10^{-281}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{re + im}\right)}\\ \mathbf{elif}\;re \le 2.694808318755371 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\left|im\right| \cdot \frac{\sqrt[3]{\left|im\right|} \cdot \sqrt[3]{\left|im\right|}}{\sqrt[3]{\sqrt{re \cdot re + im \cdot im} + re} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im} + re}}\right) \cdot \frac{\sqrt[3]{\left|im\right|}}{\sqrt[3]{\sqrt{re \cdot re + im \cdot im} + re}}\right)}\\ \mathbf{elif}\;re \le 8.6973947796722868 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 4.94013962552010715 \cdot 10^{141}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(\left|im\right| \cdot \left|im\right|\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{re + re}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))))