math.sqrt on complex, real part

Average Error: 38.5 → 22.6

Time: 16.9s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r111696 = 0.5;
        double r111697 = 2.0;
        double r111698 = re;
        double r111699 = r111698 * r111698;
        double r111700 = im;
        double r111701 = r111700 * r111700;
        double r111702 = r111699 + r111701;
        double r111703 = sqrt(r111702);
        double r111704 = r111703 + r111698;
        double r111705 = r111697 * r111704;
        double r111706 = sqrt(r111705);
        double r111707 = r111696 * r111706;
        return r111707;
}

↓

double f(double re, double im) {
        double r111708 = re;
        double r111709 = -5.330091552844717e+114;
        bool r111710 = r111708 <= r111709;
        double r111711 = 0.5;
        double r111712 = im;
        double r111713 = fabs(r111712);
        double r111714 = 2.0;
        double r111715 = sqrt(r111714);
        double r111716 = r111713 * r111715;
        double r111717 = -r111708;
        double r111718 = r111717 - r111708;
        double r111719 = sqrt(r111718);
        double r111720 = r111716 / r111719;
        double r111721 = r111711 * r111720;
        double r111722 = -4.2156616274993736e-144;
        bool r111723 = r111708 <= r111722;
        double r111724 = r111708 * r111708;
        double r111725 = r111712 * r111712;
        double r111726 = r111724 + r111725;
        double r111727 = sqrt(r111726);
        double r111728 = r111727 - r111708;
        double r111729 = sqrt(r111728);
        double r111730 = sqrt(r111729);
        double r111731 = r111716 / r111730;
        double r111732 = r111731 / r111730;
        double r111733 = r111711 * r111732;
        double r111734 = 5.124751274050741e-246;
        bool r111735 = r111708 <= r111734;
        double r111736 = r111712 + r111708;
        double r111737 = r111714 * r111736;
        double r111738 = sqrt(r111737);
        double r111739 = r111711 * r111738;
        double r111740 = 1.2802976578175363e-204;
        bool r111741 = r111708 <= r111740;
        double r111742 = 9.727118253535962e-160;
        bool r111743 = r111708 <= r111742;
        double r111744 = !r111743;
        double r111745 = 4.202834506095947e-94;
        bool r111746 = r111708 <= r111745;
        bool r111747 = r111744 && r111746;
        bool r111748 = r111741 || r111747;
        double r111749 = r111728 / r111714;
        double r111750 = r111712 / r111749;
        double r111751 = r111750 * r111712;
        double r111752 = sqrt(r111751);
        double r111753 = r111711 * r111752;
        double r111754 = r111708 + r111708;
        double r111755 = r111714 * r111754;
        double r111756 = sqrt(r111755);
        double r111757 = r111711 * r111756;
        double r111758 = r111748 ? r111753 : r111757;
        double r111759 = r111735 ? r111739 : r111758;
        double r111760 = r111723 ? r111733 : r111759;
        double r111761 = r111710 ? r111721 : r111760;
        return r111761;
}

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)))))