math.sqrt on complex, real part

Average Error: 38.7 → 23.3

Time: 12.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -1.979944677356845181284729620626534129512 \cdot 10^{87}:\\ \;\;\;\;\sqrt{\left(re + \left(-im\right)\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;im \le -9.014365595105152305171347650603003238722 \cdot 10^{-259}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2}\\ \mathbf{elif}\;im \le 4.989270859151130549963627132238083124221 \cdot 10^{-124}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \mathbf{elif}\;im \le 1.764070163534531668597418452562937826627 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot 0.5\\ \mathbf{elif}\;im \le 47570234443097721677596864236486656:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2}\\ \mathbf{elif}\;im \le 3.218748827894775859466903593100360257557 \cdot 10^{87}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r127500 = 0.5;
        double r127501 = 2.0;
        double r127502 = re;
        double r127503 = r127502 * r127502;
        double r127504 = im;
        double r127505 = r127504 * r127504;
        double r127506 = r127503 + r127505;
        double r127507 = sqrt(r127506);
        double r127508 = r127507 + r127502;
        double r127509 = r127501 * r127508;
        double r127510 = sqrt(r127509);
        double r127511 = r127500 * r127510;
        return r127511;
}

↓

double f(double re, double im) {
        double r127512 = im;
        double r127513 = -1.9799446773568452e+87;
        bool r127514 = r127512 <= r127513;
        double r127515 = re;
        double r127516 = -r127512;
        double r127517 = r127515 + r127516;
        double r127518 = 2.0;
        double r127519 = r127517 * r127518;
        double r127520 = sqrt(r127519);
        double r127521 = 0.5;
        double r127522 = r127520 * r127521;
        double r127523 = -9.014365595105152e-259;
        bool r127524 = r127512 <= r127523;
        double r127525 = r127512 * r127512;
        double r127526 = r127515 * r127515;
        double r127527 = r127525 + r127526;
        double r127528 = sqrt(r127527);
        double r127529 = r127515 + r127528;
        double r127530 = r127529 * r127518;
        double r127531 = sqrt(r127530);
        double r127532 = r127521 * r127531;
        double r127533 = 4.9892708591511305e-124;
        bool r127534 = r127512 <= r127533;
        double r127535 = 2.0;
        double r127536 = r127535 * r127515;
        double r127537 = r127518 * r127536;
        double r127538 = sqrt(r127537);
        double r127539 = r127521 * r127538;
        double r127540 = 1.7640701635345317e-90;
        bool r127541 = r127512 <= r127540;
        double r127542 = r127525 * r127518;
        double r127543 = sqrt(r127542);
        double r127544 = r127526 + r127525;
        double r127545 = sqrt(r127544);
        double r127546 = r127545 - r127515;
        double r127547 = sqrt(r127546);
        double r127548 = r127543 / r127547;
        double r127549 = r127548 * r127521;
        double r127550 = 4.757023444309772e+34;
        bool r127551 = r127512 <= r127550;
        double r127552 = 3.218748827894776e+87;
        bool r127553 = r127512 <= r127552;
        double r127554 = r127525 / r127546;
        double r127555 = r127554 * r127518;
        double r127556 = sqrt(r127555);
        double r127557 = r127521 * r127556;
        double r127558 = r127515 + r127512;
        double r127559 = r127518 * r127558;
        double r127560 = sqrt(r127559);
        double r127561 = r127521 * r127560;
        double r127562 = r127553 ? r127557 : r127561;
        double r127563 = r127551 ? r127532 : r127562;
        double r127564 = r127541 ? r127549 : r127563;
        double r127565 = r127534 ? r127539 : r127564;
        double r127566 = r127524 ? r127532 : r127565;
        double r127567 = r127514 ? r127522 : r127566;
        return r127567;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.7
Target	33.7
Herbie	23.3

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

Derivation

1. Split input into 6 regimes

2. if im < -1.9799446773568452e+87

   1. Initial program 49.4

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

   3. Applied add-sqr-sqrt 49.4

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

   4. Applied sqrt-prod 49.4

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

   6. Applied *-un-lft-identity 49.4

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

   7. Applied sqrt-prod 49.4

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

   8. Applied sqrt-prod 49.4

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

   9. Applied associate-*l* 49.4

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

   10. Simplified 49.4

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

   11. Taylor expanded around -inf 10.8

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

   12. Simplified 10.8

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

   if -1.9799446773568452e+87 < im < -9.014365595105152e-259 or 1.7640701635345317e-90 < im < 4.757023444309772e+34

   1. Initial program 29.3

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

   3. Applied add-sqr-sqrt 29.3

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

   4. Applied sqrt-prod 29.5

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

   6. Applied *-un-lft-identity 29.5

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

   7. Applied sqrt-prod 29.5

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

   8. Applied sqrt-prod 29.5

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

   9. Applied associate-*l* 29.5

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

   10. Simplified 29.3

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

   if -9.014365595105152e-259 < im < 4.9892708591511305e-124

   1. Initial program 40.4

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

   3. Applied add-sqr-sqrt 40.4

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

   4. Applied sqrt-prod 41.6

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

   5. Taylor expanded around inf 34.9

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

   if 4.9892708591511305e-124 < im < 1.7640701635345317e-90

   1. Initial program 28.2

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

   3. Applied flip-+ 45.1

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

   4. Applied associate-*r/ 45.1

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

   5. Applied sqrt-div 45.2

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

   6. Simplified 27.8

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

   if 4.757023444309772e+34 < im < 3.218748827894776e+87

   1. Initial program 22.4

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

   3. Applied flip-+ 27.0

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

   4. Simplified 21.7

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

   if 3.218748827894776e+87 < im

   1. Initial program 51.0

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

   2. Taylor expanded around 0 10.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + im\right)}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 23.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.979944677356845181284729620626534129512 \cdot 10^{87}:\\ \;\;\;\;\sqrt{\left(re + \left(-im\right)\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;im \le -9.014365595105152305171347650603003238722 \cdot 10^{-259}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2}\\ \mathbf{elif}\;im \le 4.989270859151130549963627132238083124221 \cdot 10^{-124}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \mathbf{elif}\;im \le 1.764070163534531668597418452562937826627 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot 0.5\\ \mathbf{elif}\;im \le 47570234443097721677596864236486656:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2}\\ \mathbf{elif}\;im \le 3.218748827894775859466903593100360257557 \cdot 10^{87}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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)))))