math.sqrt on complex, real part

Average Error: 38.1 → 20.4

Time: 16.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -6.05252701517827769261902701918403618702 \cdot 10^{150}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{-2 \cdot re}}\right)\\ \mathbf{elif}\;re \le 1.859645186058447714528239489016906729797 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r9559607 = 0.5;
        double r9559608 = 2.0;
        double r9559609 = re;
        double r9559610 = r9559609 * r9559609;
        double r9559611 = im;
        double r9559612 = r9559611 * r9559611;
        double r9559613 = r9559610 + r9559612;
        double r9559614 = sqrt(r9559613);
        double r9559615 = r9559614 + r9559609;
        double r9559616 = r9559608 * r9559615;
        double r9559617 = sqrt(r9559616);
        double r9559618 = r9559607 * r9559617;
        return r9559618;
}

↓

double f(double re, double im) {
        double r9559619 = re;
        double r9559620 = -6.052527015178278e+150;
        bool r9559621 = r9559619 <= r9559620;
        double r9559622 = 0.5;
        double r9559623 = im;
        double r9559624 = fabs(r9559623);
        double r9559625 = 2.0;
        double r9559626 = sqrt(r9559625);
        double r9559627 = -2.0;
        double r9559628 = r9559627 * r9559619;
        double r9559629 = sqrt(r9559628);
        double r9559630 = r9559626 / r9559629;
        double r9559631 = r9559624 * r9559630;
        double r9559632 = r9559622 * r9559631;
        double r9559633 = 1.8596451860584477e-128;
        bool r9559634 = r9559619 <= r9559633;
        double r9559635 = r9559619 * r9559619;
        double r9559636 = r9559623 * r9559623;
        double r9559637 = r9559635 + r9559636;
        double r9559638 = sqrt(r9559637);
        double r9559639 = r9559638 - r9559619;
        double r9559640 = r9559625 / r9559639;
        double r9559641 = sqrt(r9559640);
        double r9559642 = r9559624 * r9559641;
        double r9559643 = r9559622 * r9559642;
        double r9559644 = r9559619 + r9559619;
        double r9559645 = r9559625 * r9559644;
        double r9559646 = sqrt(r9559645);
        double r9559647 = r9559622 * r9559646;
        double r9559648 = r9559634 ? r9559643 : r9559647;
        double r9559649 = r9559621 ? r9559632 : r9559648;
        return r9559649;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.1
Target	33.2
Herbie	20.4

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

Derivation

1. Split input into 3 regimes

2. if re < -6.052527015178278e+150

   1. Initial program 63.7

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

   3. Applied flip-+ 63.7

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

   4. Applied associate-*r/ 63.7

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

   5. Applied sqrt-div 63.7

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

   6. Simplified 50.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(im \cdot im + 0\right) \cdot 2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 50.6

      \[\leadsto 0.5 \cdot \frac{\sqrt{\left(im \cdot im + 0\right) \cdot 2}}{\sqrt{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}}\]

   9. Applied sqrt-prod 50.6

      \[\leadsto 0.5 \cdot \frac{\sqrt{\left(im \cdot im + 0\right) \cdot 2}}{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   10. Applied sqrt-prod 50.6

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{im \cdot im + 0} \cdot \sqrt{2}}}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

   11. Applied times-frac 50.6

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{im \cdot im + 0}}{\sqrt{1}} \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\]

   12. Simplified 50.1

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

   13. Taylor expanded around -inf 8.7

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

   if -6.052527015178278e+150 < re < 1.8596451860584477e-128

   1. Initial program 35.4

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

   3. Applied flip-+ 36.4

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

   4. Applied associate-*r/ 36.4

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

   5. Applied sqrt-div 36.7

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

   6. Simplified 29.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(im \cdot im + 0\right) \cdot 2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 29.6

      \[\leadsto 0.5 \cdot \frac{\sqrt{\left(im \cdot im + 0\right) \cdot 2}}{\sqrt{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}}\]

   9. Applied sqrt-prod 29.6

      \[\leadsto 0.5 \cdot \frac{\sqrt{\left(im \cdot im + 0\right) \cdot 2}}{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   10. Applied sqrt-prod 29.7

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{im \cdot im + 0} \cdot \sqrt{2}}}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

   11. Applied times-frac 29.7

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{im \cdot im + 0}}{\sqrt{1}} \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\]

   12. Simplified 23.1

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

   14. Applied sqrt-undiv 22.9

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

   if 1.8596451860584477e-128 < re

   1. Initial program 33.0

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

   2. Taylor expanded around inf 20.9

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

4. Final simplification 20.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.05252701517827769261902701918403618702 \cdot 10^{150}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{-2 \cdot re}}\right)\\ \mathbf{elif}\;re \le 1.859645186058447714528239489016906729797 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (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)))))