math.sqrt on complex, real part

Average Error: 37.3 → 22.5

Time: 42.5s

Precision: 64

Math

\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.1481475150762484 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le 3.9393847958873876 \cdot 10^{-306}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}{\sqrt{\left(im \cdot im\right) \cdot 2.0}}}\\ \mathbf{elif}\;re \le 3.154040392752644 \cdot 10^{-193}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(im + re\right) \cdot 2.0}\\ \mathbf{elif}\;re \le 1.5286717285691206 \cdot 10^{-168}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re} \cdot 2.0}\\ \mathbf{elif}\;re \le 3.714487936304632 \cdot 10^{+58}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{e^{\log \left(\sqrt[3]{im \cdot im + re \cdot re}\right)}} \cdot \left|\sqrt[3]{im \cdot im + re \cdot re}\right|} + re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]

C

double f(double re, double im) {
        double r30487638 = 0.5;
        double r30487639 = 2.0;
        double r30487640 = re;
        double r30487641 = r30487640 * r30487640;
        double r30487642 = im;
        double r30487643 = r30487642 * r30487642;
        double r30487644 = r30487641 + r30487643;
        double r30487645 = sqrt(r30487644);
        double r30487646 = r30487645 + r30487640;
        double r30487647 = r30487639 * r30487646;
        double r30487648 = sqrt(r30487647);
        double r30487649 = r30487638 * r30487648;
        return r30487649;
}

↓

double f(double re, double im) {
        double r30487650 = re;
        double r30487651 = -1.1481475150762484e+139;
        bool r30487652 = r30487650 <= r30487651;
        double r30487653 = im;
        double r30487654 = r30487653 * r30487653;
        double r30487655 = 2.0;
        double r30487656 = r30487654 * r30487655;
        double r30487657 = sqrt(r30487656);
        double r30487658 = -2.0;
        double r30487659 = r30487658 * r30487650;
        double r30487660 = sqrt(r30487659);
        double r30487661 = r30487657 / r30487660;
        double r30487662 = 0.5;
        double r30487663 = r30487661 * r30487662;
        double r30487664 = 3.9393847958873876e-306;
        bool r30487665 = r30487650 <= r30487664;
        double r30487666 = 1.0;
        double r30487667 = r30487650 * r30487650;
        double r30487668 = r30487654 + r30487667;
        double r30487669 = sqrt(r30487668);
        double r30487670 = r30487669 - r30487650;
        double r30487671 = sqrt(r30487670);
        double r30487672 = r30487671 / r30487657;
        double r30487673 = r30487666 / r30487672;
        double r30487674 = r30487662 * r30487673;
        double r30487675 = 3.154040392752644e-193;
        bool r30487676 = r30487650 <= r30487675;
        double r30487677 = r30487653 + r30487650;
        double r30487678 = r30487677 * r30487655;
        double r30487679 = sqrt(r30487678);
        double r30487680 = r30487662 * r30487679;
        double r30487681 = 1.5286717285691206e-168;
        bool r30487682 = r30487650 <= r30487681;
        double r30487683 = r30487654 / r30487670;
        double r30487684 = r30487683 * r30487655;
        double r30487685 = sqrt(r30487684);
        double r30487686 = r30487662 * r30487685;
        double r30487687 = 3.714487936304632e+58;
        bool r30487688 = r30487650 <= r30487687;
        double r30487689 = sqrt(r30487669);
        double r30487690 = cbrt(r30487668);
        double r30487691 = log(r30487690);
        double r30487692 = exp(r30487691);
        double r30487693 = sqrt(r30487692);
        double r30487694 = fabs(r30487690);
        double r30487695 = r30487693 * r30487694;
        double r30487696 = sqrt(r30487695);
        double r30487697 = r30487689 * r30487696;
        double r30487698 = r30487697 + r30487650;
        double r30487699 = r30487655 * r30487698;
        double r30487700 = sqrt(r30487699);
        double r30487701 = r30487700 * r30487662;
        double r30487702 = r30487650 + r30487650;
        double r30487703 = r30487655 * r30487702;
        double r30487704 = sqrt(r30487703);
        double r30487705 = r30487704 * r30487662;
        double r30487706 = r30487688 ? r30487701 : r30487705;
        double r30487707 = r30487682 ? r30487686 : r30487706;
        double r30487708 = r30487676 ? r30487680 : r30487707;
        double r30487709 = r30487665 ? r30487674 : r30487708;
        double r30487710 = r30487652 ? r30487663 : r30487709;
        return r30487710;
}

Error

Bits error versus re

Bits error versus im

Target

Original	37.3
Target	32.0
Herbie	22.5

\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

1. Split input into 6 regimes

2. if re < -1.1481475150762484e+139

   1. Initial program 61.7

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied flip-+ 61.7

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   4. Applied associate-*r/ 61.8

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   5. Applied sqrt-div 61.8

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   6. Simplified 46.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

   7. Taylor expanded around -inf 19.1

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\color{blue}{-2 \cdot re}}}\]

   if -1.1481475150762484e+139 < re < 3.9393847958873876e-306

   1. Initial program 38.7

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied flip-+ 38.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   4. Applied associate-*r/ 38.6

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   5. Applied sqrt-div 38.7

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   6. Simplified 28.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
   7. Using strategy rm

   8. Applied clear-num 28.6

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}}\]

   if 3.9393847958873876e-306 < re < 3.154040392752644e-193

   1. Initial program 28.6

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 28.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

   4. Applied sqrt-prod 28.8

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

   5. Taylor expanded around 0 34.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{im} + re\right)}\]

   if 3.154040392752644e-193 < re < 1.5286717285691206e-168

   1. Initial program 32.8

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 32.8

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

   4. Applied sqrt-prod 32.9

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]
   5. Using strategy rm

   6. Applied flip-+ 36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right) - re \cdot re}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re}}}\]

   7. Simplified 36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re}}\]

   8. Simplified 36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\color{blue}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   if 1.5286717285691206e-168 < re < 3.714487936304632e+58

   1. Initial program 14.3

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 14.3

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

   4. Applied sqrt-prod 14.3

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 14.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

   7. Applied sqrt-prod 14.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\color{blue}{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

   8. Simplified 14.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\color{blue}{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\]
   9. Using strategy rm

   10. Applied add-exp-log 15.3

       \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\color{blue}{e^{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}}} + re\right)}\]

   if 3.714487936304632e+58 < re

   1. Initial program 43.6

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 43.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

   4. Applied sqrt-prod 43.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

   5. Taylor expanded around inf 13.2

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
3. Recombined 6 regimes into one program.

4. Final simplification 22.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1481475150762484 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le 3.9393847958873876 \cdot 10^{-306}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}{\sqrt{\left(im \cdot im\right) \cdot 2.0}}}\\ \mathbf{elif}\;re \le 3.154040392752644 \cdot 10^{-193}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(im + re\right) \cdot 2.0}\\ \mathbf{elif}\;re \le 1.5286717285691206 \cdot 10^{-168}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re} \cdot 2.0}\\ \mathbf{elif}\;re \le 3.714487936304632 \cdot 10^{+58}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{e^{\log \left(\sqrt[3]{im \cdot im + re \cdot re}\right)}} \cdot \left|\sqrt[3]{im \cdot im + re \cdot re}\right|} + re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (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)))))