math.sqrt on complex, real part

Average Error: 37.6 → 18.6

Time: 18.6s

Precision: 64

Math

\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.1292868428778451 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le -2.053769551615154 \cdot 10^{-273}:\\ \;\;\;\;\left(\frac{\sqrt{\left|im\right|}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot \left(\sqrt{\left|im\right|} \cdot \sqrt{2.0}\right)\right) \cdot 0.5\\ \mathbf{elif}\;re \le 7.239807700907349 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.2290590931535932 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r3239737 = 0.5;
        double r3239738 = 2.0;
        double r3239739 = re;
        double r3239740 = r3239739 * r3239739;
        double r3239741 = im;
        double r3239742 = r3239741 * r3239741;
        double r3239743 = r3239740 + r3239742;
        double r3239744 = sqrt(r3239743);
        double r3239745 = r3239744 + r3239739;
        double r3239746 = r3239738 * r3239745;
        double r3239747 = sqrt(r3239746);
        double r3239748 = r3239737 * r3239747;
        return r3239748;
}

↓

double f(double re, double im) {
        double r3239749 = re;
        double r3239750 = -1.1292868428778451e+139;
        bool r3239751 = r3239749 <= r3239750;
        double r3239752 = 2.0;
        double r3239753 = im;
        double r3239754 = r3239753 * r3239753;
        double r3239755 = r3239752 * r3239754;
        double r3239756 = sqrt(r3239755);
        double r3239757 = -2.0;
        double r3239758 = r3239757 * r3239749;
        double r3239759 = sqrt(r3239758);
        double r3239760 = r3239756 / r3239759;
        double r3239761 = 0.5;
        double r3239762 = r3239760 * r3239761;
        double r3239763 = -2.053769551615154e-273;
        bool r3239764 = r3239749 <= r3239763;
        double r3239765 = fabs(r3239753);
        double r3239766 = sqrt(r3239765);
        double r3239767 = r3239749 * r3239749;
        double r3239768 = r3239754 + r3239767;
        double r3239769 = sqrt(r3239768);
        double r3239770 = r3239769 - r3239749;
        double r3239771 = sqrt(r3239770);
        double r3239772 = r3239766 / r3239771;
        double r3239773 = sqrt(r3239752);
        double r3239774 = r3239766 * r3239773;
        double r3239775 = r3239772 * r3239774;
        double r3239776 = r3239775 * r3239761;
        double r3239777 = 7.239807700907349e-222;
        bool r3239778 = r3239749 <= r3239777;
        double r3239779 = r3239753 + r3239749;
        double r3239780 = r3239779 * r3239752;
        double r3239781 = sqrt(r3239780);
        double r3239782 = r3239781 * r3239761;
        double r3239783 = 1.2290590931535932e+68;
        bool r3239784 = r3239749 <= r3239783;
        double r3239785 = sqrt(r3239769);
        double r3239786 = r3239785 * r3239785;
        double r3239787 = r3239749 + r3239786;
        double r3239788 = r3239787 * r3239752;
        double r3239789 = sqrt(r3239788);
        double r3239790 = r3239761 * r3239789;
        double r3239791 = r3239749 + r3239749;
        double r3239792 = r3239752 * r3239791;
        double r3239793 = sqrt(r3239792);
        double r3239794 = r3239761 * r3239793;
        double r3239795 = r3239784 ? r3239790 : r3239794;
        double r3239796 = r3239778 ? r3239782 : r3239795;
        double r3239797 = r3239764 ? r3239776 : r3239796;
        double r3239798 = r3239751 ? r3239762 : r3239797;
        return r3239798;
}

Error

Bits error versus re

Bits error versus im

Target

Original	37.6
Target	32.6
Herbie	18.6

\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

1. Split input into 5 regimes

2. if re < -1.1292868428778451e+139

   1. Initial program 61.7

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied flip-+ 61.7

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   4. Applied associate-*r/ 61.7

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   5. Applied sqrt-div 61.7

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   6. Simplified 47.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

   7. Taylor expanded around -inf 18.8

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}{\sqrt{\color{blue}{-2 \cdot re}}}\]

   if -1.1292868428778451e+139 < re < -2.053769551615154e-273

   1. Initial program 39.8

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied flip-+ 39.7

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   4. Applied associate-*r/ 39.7

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   5. Applied sqrt-div 39.8

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   6. Simplified 29.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 29.6

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}{\sqrt{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}}\]

   9. Applied sqrt-prod 29.6

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   10. Applied sqrt-prod 29.7

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0} \cdot \sqrt{im \cdot im + 0}}}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

   11. Applied times-frac 29.7

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{2.0}}{\sqrt{1}} \cdot \frac{\sqrt{im \cdot im + 0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\]

   12. Simplified 29.7

       \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{2.0}} \cdot \frac{\sqrt{im \cdot im + 0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\]

   13. Simplified 19.7

       \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \color{blue}{\frac{\left|im\right|}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}}\right)\]
   14. Using strategy rm

   15. Applied *-un-lft-identity 19.7

       \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{im \cdot im + re \cdot re} - \color{blue}{1 \cdot re}}}\right)\]

   16. Applied *-un-lft-identity 19.7

       \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\left|im\right|}{\sqrt{\color{blue}{1 \cdot \sqrt{im \cdot im + re \cdot re}} - 1 \cdot re}}\right)\]

   17. Applied distribute-lft-out-- 19.7

       \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\left|im\right|}{\sqrt{\color{blue}{1 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}}}\right)\]

   18. Applied sqrt-prod 19.7

       \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\left|im\right|}{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re} - re}}}\right)\]

   19. Applied add-sqr-sqrt 19.7

       \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\color{blue}{\sqrt{\left|im\right|} \cdot \sqrt{\left|im\right|}}}{\sqrt{1} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)\]

   20. Applied times-frac 19.7

       \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \color{blue}{\left(\frac{\sqrt{\left|im\right|}}{\sqrt{1}} \cdot \frac{\sqrt{\left|im\right|}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)}\right)\]

   21. Applied associate-*r* 19.7

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2.0} \cdot \frac{\sqrt{\left|im\right|}}{\sqrt{1}}\right) \cdot \frac{\sqrt{\left|im\right|}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)}\]

   22. Simplified 19.7

       \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{2.0} \cdot \sqrt{\left|im\right|}\right)} \cdot \frac{\sqrt{\left|im\right|}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)\]

   if -2.053769551615154e-273 < re < 7.239807700907349e-222

   1. Initial program 30.6

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

   2. Taylor expanded around 0 32.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{im} + re\right)}\]

   if 7.239807700907349e-222 < re < 1.2290590931535932e+68

   1. Initial program 18.4

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 18.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

   if 1.2290590931535932e+68 < re

   1. Initial program 43.1

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

   2. Taylor expanded around inf 10.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1292868428778451 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le -2.053769551615154 \cdot 10^{-273}:\\ \;\;\;\;\left(\frac{\sqrt{\left|im\right|}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot \left(\sqrt{\left|im\right|} \cdot \sqrt{2.0}\right)\right) \cdot 0.5\\ \mathbf{elif}\;re \le 7.239807700907349 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.2290590931535932 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))