math.sqrt on complex, real part

Average Error: 38.9 → 19.3

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.3474626627347847 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-1 \cdot re - re}\right)}\\ \mathbf{elif}\;re \le -1.6126997418755618 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 6.4768545182660149 \cdot 10^{131}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r193814 = 0.5;
        double r193815 = 2.0;
        double r193816 = re;
        double r193817 = r193816 * r193816;
        double r193818 = im;
        double r193819 = r193818 * r193818;
        double r193820 = r193817 + r193819;
        double r193821 = sqrt(r193820);
        double r193822 = r193821 + r193816;
        double r193823 = r193815 * r193822;
        double r193824 = sqrt(r193823);
        double r193825 = r193814 * r193824;
        return r193825;
}

↓

double f(double re, double im) {
        double r193826 = re;
        double r193827 = -1.3474626627347847e+154;
        bool r193828 = r193826 <= r193827;
        double r193829 = 0.5;
        double r193830 = 2.0;
        double r193831 = im;
        double r193832 = -1.0;
        double r193833 = r193832 * r193826;
        double r193834 = r193833 - r193826;
        double r193835 = r193831 / r193834;
        double r193836 = r193831 * r193835;
        double r193837 = r193830 * r193836;
        double r193838 = sqrt(r193837);
        double r193839 = r193829 * r193838;
        double r193840 = -1.612699741875562e-300;
        bool r193841 = r193826 <= r193840;
        double r193842 = sqrt(r193830);
        double r193843 = fabs(r193831);
        double r193844 = r193842 * r193843;
        double r193845 = r193826 * r193826;
        double r193846 = r193831 * r193831;
        double r193847 = r193845 + r193846;
        double r193848 = sqrt(r193847);
        double r193849 = r193848 - r193826;
        double r193850 = sqrt(r193849);
        double r193851 = r193844 / r193850;
        double r193852 = r193829 * r193851;
        double r193853 = 6.476854518266015e+131;
        bool r193854 = r193826 <= r193853;
        double r193855 = r193848 + r193826;
        double r193856 = r193830 * r193855;
        double r193857 = sqrt(r193856);
        double r193858 = r193829 * r193857;
        double r193859 = 2.0;
        double r193860 = r193859 * r193826;
        double r193861 = r193830 * r193860;
        double r193862 = sqrt(r193861);
        double r193863 = r193829 * r193862;
        double r193864 = r193854 ? r193858 : r193863;
        double r193865 = r193841 ? r193852 : r193864;
        double r193866 = r193828 ? r193839 : r193865;
        return r193866;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.9
Target	33.9
Herbie	19.3

\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if re < -1.3474626627347847e+154

   1. Initial program 64.0

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

   3. Applied flip-+ 64.0

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

   4. Simplified 50.6

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

   6. Applied *-un-lft-identity 50.6

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

   7. Applied add-sqr-sqrt 58.1

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

   8. Applied unpow-prod-down 58.1

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

   9. Applied times-frac 57.9

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

   10. Simplified 57.9

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

   11. Simplified 50.2

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

   12. Taylor expanded around -inf 22.7

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

   if -1.3474626627347847e+154 < re < -1.612699741875562e-300

   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 flip-+ 40.4

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

   4. Simplified 31.6

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

   6. Applied *-un-lft-identity 31.6

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

   7. Applied add-sqr-sqrt 47.6

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

   8. Applied unpow-prod-down 47.6

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

   9. Applied times-frac 46.5

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

   10. Simplified 46.4

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

   11. Simplified 29.4

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

   13. Applied associate-*r/ 31.6

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

   14. Applied associate-*r/ 31.6

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

   15. Applied sqrt-div 30.4

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

   17. Applied sqrt-prod 30.4

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

   18. Simplified 20.5

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

   if -1.612699741875562e-300 < re < 6.476854518266015e+131

   1. Initial program 21.1

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

   if 6.476854518266015e+131 < re

   1. Initial program 57.7

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

   3. Applied flip-+ 63.6

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

   4. Simplified 62.6

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

   5. Taylor expanded around 0 8.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 19.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3474626627347847 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-1 \cdot re - re}\right)}\\ \mathbf{elif}\;re \le -1.6126997418755618 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 6.4768545182660149 \cdot 10^{131}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))