math.sqrt on complex, real part

Average Error: 39.0 → 26.3

Time: 12.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -7.240673218843522327919250789136495189333 \cdot 10^{150}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{-2 \cdot re}\right)}\\ \mathbf{elif}\;re \le 3.787000431377413829039638453986283883044 \cdot 10^{-283}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {\left(\left|im\right|\right)}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 8.736784500225597746587986871686332519603 \cdot 10^{-188}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{im - re}\right)}\\ \mathbf{elif}\;re \le 1.153451770878403765578063868666891624253 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re} \cdot \left|im\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r131519 = 0.5;
        double r131520 = 2.0;
        double r131521 = re;
        double r131522 = r131521 * r131521;
        double r131523 = im;
        double r131524 = r131523 * r131523;
        double r131525 = r131522 + r131524;
        double r131526 = sqrt(r131525);
        double r131527 = r131526 + r131521;
        double r131528 = r131520 * r131527;
        double r131529 = sqrt(r131528);
        double r131530 = r131519 * r131529;
        return r131530;
}

↓

double f(double re, double im) {
        double r131531 = re;
        double r131532 = -7.240673218843522e+150;
        bool r131533 = r131531 <= r131532;
        double r131534 = 0.5;
        double r131535 = 2.0;
        double r131536 = im;
        double r131537 = fabs(r131536);
        double r131538 = -2.0;
        double r131539 = r131538 * r131531;
        double r131540 = r131537 / r131539;
        double r131541 = r131537 * r131540;
        double r131542 = r131535 * r131541;
        double r131543 = sqrt(r131542);
        double r131544 = r131534 * r131543;
        double r131545 = 3.787000431377414e-283;
        bool r131546 = r131531 <= r131545;
        double r131547 = 2.0;
        double r131548 = pow(r131537, r131547);
        double r131549 = r131535 * r131548;
        double r131550 = sqrt(r131549);
        double r131551 = r131531 * r131531;
        double r131552 = r131536 * r131536;
        double r131553 = r131551 + r131552;
        double r131554 = sqrt(r131553);
        double r131555 = r131554 - r131531;
        double r131556 = sqrt(r131555);
        double r131557 = r131550 / r131556;
        double r131558 = r131534 * r131557;
        double r131559 = 8.736784500225598e-188;
        bool r131560 = r131531 <= r131559;
        double r131561 = r131536 - r131531;
        double r131562 = r131537 / r131561;
        double r131563 = r131537 * r131562;
        double r131564 = r131535 * r131563;
        double r131565 = sqrt(r131564);
        double r131566 = r131534 * r131565;
        double r131567 = 1.1534517708784038e-53;
        bool r131568 = r131531 <= r131567;
        double r131569 = r131537 / r131555;
        double r131570 = r131569 * r131537;
        double r131571 = r131535 * r131570;
        double r131572 = sqrt(r131571);
        double r131573 = r131534 * r131572;
        double r131574 = r131547 * r131531;
        double r131575 = r131535 * r131574;
        double r131576 = sqrt(r131575);
        double r131577 = r131534 * r131576;
        double r131578 = r131568 ? r131573 : r131577;
        double r131579 = r131560 ? r131566 : r131578;
        double r131580 = r131546 ? r131558 : r131579;
        double r131581 = r131533 ? r131544 : r131580;
        return r131581;
}

Error

Bits error versus re

Bits error versus im

Target

Original	39.0
Target	33.9
Herbie	26.3

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

Derivation

1. Split input into 5 regimes

2. if re < -7.240673218843522e+150

   1. Initial program 63.9

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

   3. Applied flip-+ 63.9

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

   4. Simplified 50.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im + 0}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 50.5

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

   7. Applied add-sqr-sqrt 50.5

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

   8. Applied times-frac 50.5

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

   9. Simplified 50.5

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

   10. Simplified 50.1

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

   11. Taylor expanded around -inf 24.7

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

   if -7.240673218843522e+150 < re < 3.787000431377414e-283

   1. Initial program 39.7

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

   3. Applied flip-+ 39.6

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

   4. Simplified 30.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im + 0}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 30.9

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

   7. Applied add-sqr-sqrt 30.9

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

   8. Applied times-frac 30.9

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

   9. Simplified 30.8

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

   10. Simplified 28.8

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

   12. Applied associate-*r/ 30.9

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

   13. Applied associate-*r/ 30.9

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

   14. Applied sqrt-div 29.9

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

   15. Simplified 29.9

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

   if 3.787000431377414e-283 < re < 8.736784500225598e-188

   1. Initial program 29.5

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

   3. Applied flip-+ 31.1

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

   4. Simplified 31.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im + 0}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 31.1

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

   7. Applied add-sqr-sqrt 31.1

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

   8. Applied times-frac 31.1

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

   9. Simplified 31.0

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

   10. Simplified 31.2

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

   11. Taylor expanded around 0 35.6

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

   if 8.736784500225598e-188 < re < 1.1534517708784038e-53

   1. Initial program 18.4

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

   3. Applied flip-+ 34.6

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

   4. Simplified 34.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im + 0}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 34.6

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

   7. Applied add-sqr-sqrt 34.6

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

   8. Applied times-frac 34.6

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

   9. Simplified 34.5

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

   10. Simplified 34.3

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

   12. Applied *-commutative 34.3

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

   if 1.1534517708784038e-53 < re

   1. Initial program 37.2

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

   2. Taylor expanded around inf 17.1

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

4. Final simplification 26.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.240673218843522327919250789136495189333 \cdot 10^{150}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{-2 \cdot re}\right)}\\ \mathbf{elif}\;re \le 3.787000431377413829039638453986283883044 \cdot 10^{-283}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {\left(\left|im\right|\right)}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 8.736784500225597746587986871686332519603 \cdot 10^{-188}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{im - re}\right)}\\ \mathbf{elif}\;re \le 1.153451770878403765578063868666891624253 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re} \cdot \left|im\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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)))))