math.sqrt on complex, real part

Average Error: 38.5 → 22.6

Time: 17.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\left(-re\right) - re}}\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\\ \mathbf{elif}\;re \le 5.124751274050741168628571362640123162884 \cdot 10^{-246}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.280297657817536289043603160829670533045 \cdot 10^{-204} \lor \neg \left(re \le 9.727118253535961652403013059453411638468 \cdot 10^{-160}\right) \land re \le 4.202834506095946744840619038062984088453 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{2}} \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r109586 = 0.5;
        double r109587 = 2.0;
        double r109588 = re;
        double r109589 = r109588 * r109588;
        double r109590 = im;
        double r109591 = r109590 * r109590;
        double r109592 = r109589 + r109591;
        double r109593 = sqrt(r109592);
        double r109594 = r109593 + r109588;
        double r109595 = r109587 * r109594;
        double r109596 = sqrt(r109595);
        double r109597 = r109586 * r109596;
        return r109597;
}

↓

double f(double re, double im) {
        double r109598 = re;
        double r109599 = -5.330091552844717e+114;
        bool r109600 = r109598 <= r109599;
        double r109601 = 0.5;
        double r109602 = im;
        double r109603 = fabs(r109602);
        double r109604 = 2.0;
        double r109605 = sqrt(r109604);
        double r109606 = r109603 * r109605;
        double r109607 = -r109598;
        double r109608 = r109607 - r109598;
        double r109609 = sqrt(r109608);
        double r109610 = r109606 / r109609;
        double r109611 = r109601 * r109610;
        double r109612 = -4.2156616274993736e-144;
        bool r109613 = r109598 <= r109612;
        double r109614 = r109598 * r109598;
        double r109615 = r109602 * r109602;
        double r109616 = r109614 + r109615;
        double r109617 = sqrt(r109616);
        double r109618 = r109617 - r109598;
        double r109619 = sqrt(r109618);
        double r109620 = sqrt(r109619);
        double r109621 = r109606 / r109620;
        double r109622 = r109621 / r109620;
        double r109623 = r109601 * r109622;
        double r109624 = 5.124751274050741e-246;
        bool r109625 = r109598 <= r109624;
        double r109626 = r109602 + r109598;
        double r109627 = r109604 * r109626;
        double r109628 = sqrt(r109627);
        double r109629 = r109601 * r109628;
        double r109630 = 1.2802976578175363e-204;
        bool r109631 = r109598 <= r109630;
        double r109632 = 9.727118253535962e-160;
        bool r109633 = r109598 <= r109632;
        double r109634 = !r109633;
        double r109635 = 4.202834506095947e-94;
        bool r109636 = r109598 <= r109635;
        bool r109637 = r109634 && r109636;
        bool r109638 = r109631 || r109637;
        double r109639 = r109618 / r109604;
        double r109640 = r109602 / r109639;
        double r109641 = r109640 * r109602;
        double r109642 = sqrt(r109641);
        double r109643 = r109601 * r109642;
        double r109644 = r109598 + r109598;
        double r109645 = r109604 * r109644;
        double r109646 = sqrt(r109645);
        double r109647 = r109601 * r109646;
        double r109648 = r109638 ? r109643 : r109647;
        double r109649 = r109625 ? r109629 : r109648;
        double r109650 = r109613 ? r109623 : r109649;
        double r109651 = r109600 ? r109611 : r109650;
        return r109651;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.5
Target	33.3
Herbie	22.6

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

Derivation

1. Split input into 5 regimes

2. if re < -5.330091552844717e+114

   1. Initial program 61.8

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

   3. Applied flip-+ 61.8

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

   4. Applied associate-*r/ 61.9

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

   5. Applied sqrt-div 61.9

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

   6. Simplified 45.4

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

   8. Applied sqrt-prod 45.3

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

   9. Simplified 43.1

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

   10. Taylor expanded around -inf 8.9

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

   11. Simplified 8.9

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

   if -5.330091552844717e+114 < re < -4.2156616274993736e-144

   1. Initial program 43.2

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

   3. Applied flip-+ 43.2

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

   4. Applied associate-*r/ 43.4

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

   5. Applied sqrt-div 43.5

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

   6. Simplified 28.4

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

   8. Applied sqrt-prod 28.3

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

   9. Simplified 15.5

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

   11. Applied add-sqr-sqrt 15.5

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

   12. Applied sqrt-prod 15.7

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

   13. Applied associate-/r* 15.7

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

   if -4.2156616274993736e-144 < re < 5.124751274050741e-246

   1. Initial program 31.6

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

   2. Taylor expanded around 0 36.2

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

   if 5.124751274050741e-246 < re < 1.2802976578175363e-204 or 9.727118253535962e-160 < re < 4.202834506095947e-94

   1. Initial program 20.9

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

   3. Applied flip-+ 33.1

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

   4. Applied associate-*r/ 33.2

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

   5. Applied sqrt-div 33.6

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

   6. Simplified 33.6

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

   8. Applied sqrt-undiv 33.2

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

   9. Simplified 32.8

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

   if 1.2802976578175363e-204 < re < 9.727118253535962e-160 or 4.202834506095947e-94 < re

   1. Initial program 33.3

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

   2. Taylor expanded around inf 23.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} + re\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 22.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\left(-re\right) - re}}\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\\ \mathbf{elif}\;re \le 5.124751274050741168628571362640123162884 \cdot 10^{-246}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.280297657817536289043603160829670533045 \cdot 10^{-204} \lor \neg \left(re \le 9.727118253535961652403013059453411638468 \cdot 10^{-160}\right) \land re \le 4.202834506095946744840619038062984088453 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{2}} \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))))