math.sqrt on complex, real part

Average Error: 38.5 → 13.1

Time: 2.4min

Precision: binary64

Cost: 46530

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.199222094282219 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{re \cdot -2}}\right)\\ \mathbf{elif}\;re \leq -1.4106756593205677 \cdot 10^{-77}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \left(\sqrt{\sqrt{2}} \cdot \frac{\sqrt{\sqrt{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\right)\\ \mathbf{elif}\;re \leq 3.214172448806604 \cdot 10^{-141}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|\frac{re \cdot re}{im} \cdot -0.5 - im\right|\right)}\\ \mathbf{elif}\;re \leq 1.809200894956933 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{elif}\;re \leq 9.542964692004831 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|im\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -1.199222094282219e+126)
   (* 0.5 (* (fabs im) (/ (sqrt 2.0) (sqrt (* re -2.0)))))
   (if (<= re -1.4106756593205677e-77)
     (*
      0.5
      (*
       (fabs im)
       (*
        (sqrt (sqrt 2.0))
        (/ (sqrt (sqrt 2.0)) (sqrt (- (sqrt (+ (* re re) (* im im))) re))))))
     (if (<= re 3.214172448806604e-141)
       (* 0.5 (sqrt (* 2.0 (+ re (fabs (- (* (/ (* re re) im) -0.5) im))))))
       (if (<= re 1.809200894956933e+46)
         (* 0.5 (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))))
         (if (<= re 9.542964692004831e+126)
           (* 0.5 (sqrt (* 2.0 (+ re (fabs im)))))
           (* 0.5 (* 2.0 (sqrt re)))))))))

C

double code(double re, double im) {
	return 0.5 * sqrt(2.0 * (sqrt((re * re) + (im * im)) + re));
}

↓

double code(double re, double im) {
	double tmp;
	if (re <= -1.199222094282219e+126) {
		tmp = 0.5 * (fabs(im) * (sqrt(2.0) / sqrt(re * -2.0)));
	} else if (re <= -1.4106756593205677e-77) {
		tmp = 0.5 * (fabs(im) * (sqrt(sqrt(2.0)) * (sqrt(sqrt(2.0)) / sqrt(sqrt((re * re) + (im * im)) - re))));
	} else if (re <= 3.214172448806604e-141) {
		tmp = 0.5 * sqrt(2.0 * (re + fabs((((re * re) / im) * -0.5) - im)));
	} else if (re <= 1.809200894956933e+46) {
		tmp = 0.5 * sqrt(2.0 * (re + sqrt((re * re) + (im * im))));
	} else if (re <= 9.542964692004831e+126) {
		tmp = 0.5 * sqrt(2.0 * (re + fabs(im)));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.5
Target	33.1
Herbie	13.1

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

Alternatives

Alternative 1
Error	13.1
Cost	20802

\[\begin{array}{l} \mathbf{if}\;re \leq -1.3496433026116003 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{re \cdot -2}}\right)\\ \mathbf{elif}\;re \leq -1.1261129844060676 \cdot 10^{-77}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{elif}\;re \leq 3.162926000560335 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|\frac{re \cdot re}{im} \cdot -0.5 - im\right|\right)}\\ \mathbf{elif}\;re \leq 5.717442834118705 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{elif}\;re \leq 1.0075467371755552 \cdot 10^{+127}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|im\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

Alternative 2
Error	14.9
Cost	20097

\[\begin{array}{l} \mathbf{if}\;re \leq -2.0806375032771606 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{re \cdot -2}}\right)\\ \mathbf{elif}\;re \leq 1.7845848739333778 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|\frac{re \cdot re}{im} \cdot -0.5 - im\right|\right)}\\ \mathbf{elif}\;re \leq 8.629321367885434 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{elif}\;re \leq 9.542964692004831 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|im\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

Alternative 3
Error	17.1
Cost	14595

\[\begin{array}{l} \mathbf{if}\;re \leq -1.132560508771453 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq 4.416560081930134 \cdot 10^{-141}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|\frac{re \cdot re}{im} \cdot -0.5 - im\right|\right)}\\ \mathbf{elif}\;re \leq 1.1017477864781767 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{elif}\;re \leq 8.269551164031008 \cdot 10^{+127}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|im\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

Alternative 4
Error	17.4
Cost	14595

\[\begin{array}{l} \mathbf{if}\;re \leq -1.2549082425860653 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq 7.705216962722596 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|im\right|\right)}\\ \mathbf{elif}\;re \leq 1.8539930993574273 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{elif}\;re \leq 8.269551164031008 \cdot 10^{+127}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|im\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

Alternative 5
Error	19.0
Cost	14029

\[\begin{array}{l} \mathbf{if}\;re \leq -5.574209862081529 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq 4.7126399157857446 \cdot 10^{-82} \lor \neg \left(re \leq 5.4168310351630246 \cdot 10^{+45}\right) \land re \leq 1.0075467371755552 \cdot 10^{+127}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|im\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

Alternative 6
Error	20.7
Cost	14029

\[\begin{array}{l} \mathbf{if}\;re \leq -1.4408258177403338 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{im \cdot im}{re}\right)}\\ \mathbf{elif}\;re \leq 2.750311134892316 \cdot 10^{-82} \lor \neg \left(re \leq 5.5288115461642606 \cdot 10^{+45}\right) \land re \leq 1.6775231267361767 \cdot 10^{+127}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|im\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

Alternative 7
Error	25.2
Cost	8002

\[\begin{array}{l} \mathbf{if}\;im \leq -3.7840109269703677 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 3.0364793013828164 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{im \cdot im}{re} + re \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]

Alternative 8
Error	25.2
Cost	7490

\[\begin{array}{l} \mathbf{if}\;im \leq -9.925624596920426 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 1.7012003094525194 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]

Alternative 9
Error	25.5
Cost	7490

\[\begin{array}{l} \mathbf{if}\;im \leq -1.7840492392198191 \cdot 10^{-127}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 2.743907599676933 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]

Alternative 10
Error	25.8
Cost	7362

\[\begin{array}{l} \mathbf{if}\;im \leq -3.1071524292899965 \cdot 10^{-130}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 3.510349781693256 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array}\]

Alternative 11
Error	35.9
Cost	7362

\[\begin{array}{l} \mathbf{if}\;re \leq -1.391174801252767 \cdot 10^{+98}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 5.7862847256276915 \cdot 10^{-182}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

Alternative 12
Error	44.1
Cost	7362

\[\begin{array}{l} \mathbf{if}\;im \leq -2.3356729153207394 \cdot 10^{-227}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.5437260940449797 \cdot 10^{-194}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array}\]

Alternative 13
Error	58.2
Cost	385

\[\begin{array}{l} \mathbf{if}\;re \leq -4.467215323160996 \cdot 10^{+120}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 14
Error	60.0
Cost	64

\[0\]

Derivation

1. Split input into 6 regimes

2. if re < -1.19922209428221903e126

   1. Initial program 62.3

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

   3. Applied flip-+_binary64_2780 62.3

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

   4. Applied associate-*r/_binary64_2748 62.3

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

   5. Applied sqrt-div_binary64_2823 62.3

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

   6. Simplified 44.6

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

   8. Applied *-un-lft-identity_binary64_2806 44.6

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

   9. Applied sqrt-prod_binary64_2822 44.6

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

   10. Applied sqrt-prod_binary64_2822 44.6

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

   11. Applied times-frac_binary64_2812 44.6

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

   12. Simplified 42.9

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

   13. Taylor expanded around -inf 9.3

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

   14. Simplified 9.3

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

   15. Simplified 9.3

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

   if -1.19922209428221903e126 < re < -1.41067565932056774e-77

   1. Initial program 47.2

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

   3. Applied flip-+_binary64_2780 47.2

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

   4. Applied associate-*r/_binary64_2748 47.2

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

   5. Applied sqrt-div_binary64_2823 47.3

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

   6. Simplified 29.3

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

   8. Applied *-un-lft-identity_binary64_2806 29.3

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

   9. Applied sqrt-prod_binary64_2822 29.3

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

   10. Applied sqrt-prod_binary64_2822 29.3

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

   11. Applied times-frac_binary64_2812 29.3

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

   12. Simplified 16.9

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

   14. Applied *-un-lft-identity_binary64_2806 16.9

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

   15. Applied sqrt-prod_binary64_2822 16.9

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

   16. Applied add-sqr-sqrt_binary64_2828 17.0

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

   17. Applied times-frac_binary64_2812 16.9

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

   18. Simplified 16.9

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

   19. Simplified 16.9

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

   if -1.41067565932056774e-77 < re < 3.2141724488066041e-141

   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-+_binary64_2780 46.9

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

   4. Simplified 46.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\frac{\color{blue}{{re}^{4} - {im}^{4}}}{re \cdot re - im \cdot im}} + re\right)}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt_binary64_2828 46.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\sqrt{\frac{{re}^{4} - {im}^{4}}{re \cdot re - im \cdot im}} \cdot \sqrt{\frac{{re}^{4} - {im}^{4}}{re \cdot re - im \cdot im}}}} + re\right)}\]

   7. Applied rem-sqrt-square_binary64_2819 46.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left|\sqrt{\frac{{re}^{4} - {im}^{4}}{re \cdot re - im \cdot im}}\right|} + re\right)}\]

   8. Taylor expanded around -inf 9.6

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

   9. Simplified 9.6

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

   10. Simplified 9.6

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

   if 3.2141724488066041e-141 < re < 1.8092008949569329e46

   1. Initial program 16.0

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

   3. Applied +-commutative_binary64_2736 16.0

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

   4. Simplified 16.0

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

   if 1.8092008949569329e46 < re < 9.54296469200483094e126

   1. Initial program 17.0

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

   3. Applied flip-+_binary64_2780 49.5

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

   4. Simplified 49.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\frac{\color{blue}{{re}^{4} - {im}^{4}}}{re \cdot re - im \cdot im}} + re\right)}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt_binary64_2828 49.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\sqrt{\frac{{re}^{4} - {im}^{4}}{re \cdot re - im \cdot im}} \cdot \sqrt{\frac{{re}^{4} - {im}^{4}}{re \cdot re - im \cdot im}}}} + re\right)}\]

   7. Applied rem-sqrt-square_binary64_2819 49.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left|\sqrt{\frac{{re}^{4} - {im}^{4}}{re \cdot re - im \cdot im}}\right|} + re\right)}\]

   8. Taylor expanded around 0 32.2

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

   9. Simplified 32.2

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

   if 9.54296469200483094e126 < re

   1. Initial program 57.2

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

   2. Taylor expanded around 0 10.4

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

   3. Simplified 9.4

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

   4. Simplified 9.4

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

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.199222094282219 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{re \cdot -2}}\right)\\ \mathbf{elif}\;re \leq -1.4106756593205677 \cdot 10^{-77}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \left(\sqrt{\sqrt{2}} \cdot \frac{\sqrt{\sqrt{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\right)\\ \mathbf{elif}\;re \leq 3.214172448806604 \cdot 10^{-141}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|\frac{re \cdot re}{im} \cdot -0.5 - im\right|\right)}\\ \mathbf{elif}\;re \leq 1.809200894956933 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{elif}\;re \leq 9.542964692004831 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left|im\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (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)))))