math.sqrt on complex, real part

Average Error: 38.8 → 23.5

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -6.549533459362007 \cdot 10^{+160}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq -5.543930521314922 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{\sqrt{2}} \cdot \left|im\right|\right) \cdot \sqrt{\frac{\sqrt{2}}{\sqrt{im \cdot im + re \cdot re} - re}}\right)\\ \mathbf{elif}\;re \leq 1.4495720106119306 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 2.3897943661054227 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{\sqrt{2}} \cdot \left|im\right|\right) \cdot \frac{\sqrt{\sqrt{2}}}{\sqrt{\left(-im\right) - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -6.549533459362007e+160)
   (* 0.5 (/ (sqrt (* (* im im) 2.0)) (sqrt (* re -2.0))))
   (if (<= re -5.543930521314922e-300)
     (*
      0.5
      (*
       (* (sqrt (sqrt 2.0)) (fabs im))
       (sqrt (/ (sqrt 2.0) (- (sqrt (+ (* im im) (* re re))) re)))))
     (if (<= re 1.4495720106119306e-217)
       (* 0.5 (sqrt (* 2.0 (+ re im))))
       (if (<= re 2.3897943661054227e-18)
         (*
          0.5
          (*
           (* (sqrt (sqrt 2.0)) (fabs im))
           (/ (sqrt (sqrt 2.0)) (sqrt (- (- im) re)))))
         (* 0.5 (sqrt (* 2.0 (+ re 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 <= -6.549533459362007e+160) {
		tmp = 0.5 * (sqrt((im * im) * 2.0) / sqrt(re * -2.0));
	} else if (re <= -5.543930521314922e-300) {
		tmp = 0.5 * ((sqrt(sqrt(2.0)) * fabs(im)) * sqrt(sqrt(2.0) / (sqrt((im * im) + (re * re)) - re)));
	} else if (re <= 1.4495720106119306e-217) {
		tmp = 0.5 * sqrt(2.0 * (re + im));
	} else if (re <= 2.3897943661054227e-18) {
		tmp = 0.5 * ((sqrt(sqrt(2.0)) * fabs(im)) * (sqrt(sqrt(2.0)) / sqrt(-im - re)));
	} else {
		tmp = 0.5 * sqrt(2.0 * (re + re));
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.8
Target	33.6
Herbie	23.5

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

Derivation

1. Split input into 5 regimes

2. if re < -6.5495334593620073e160

   1. Initial program 64.0

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

   3. Applied flip-+_binary64_2410 64.0

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

   4. Applied associate-*r/_binary64_2380 64.0

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

   5. Applied sqrt-div_binary64_2452 64.0

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

   6. Simplified 48.6

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

   7. Taylor expanded around -inf 17.6

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

   8. Simplified 17.6

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

   if -6.5495334593620073e160 < re < -5.543930521314922e-300

   1. Initial program 40.5

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

   3. Applied flip-+_binary64_2410 40.4

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

   4. Applied associate-*r/_binary64_2380 40.4

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

   5. Applied sqrt-div_binary64_2452 40.5

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

   6. Simplified 30.4

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

   8. Applied *-un-lft-identity_binary64_2436 30.4

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

   9. Applied sqrt-prod_binary64_2451 30.4

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

   10. Applied sqrt-prod_binary64_2451 30.5

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

   11. Applied times-frac_binary64_2442 30.5

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

   12. Simplified 20.9

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

   13. Simplified 20.9

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

   15. Applied *-un-lft-identity_binary64_2436 20.9

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

   16. Applied sqrt-prod_binary64_2451 20.9

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

   17. Applied add-sqr-sqrt_binary64_2457 20.9

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

   18. Applied times-frac_binary64_2442 20.9

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

   19. Applied associate-*r*_binary64_2378 20.9

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

   20. Simplified 20.9

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

   22. Applied sqrt-undiv_binary64_2456 20.8

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

   if -5.543930521314922e-300 < re < 1.4495720106119306e-217

   1. Initial program 28.1

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

   2. Taylor expanded around 0 31.9

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

   if 1.4495720106119306e-217 < re < 2.3897943661054227e-18

   1. Initial program 21.4

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

   3. Applied flip-+_binary64_2410 37.4

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

   4. Applied associate-*r/_binary64_2380 37.5

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

   5. Applied sqrt-div_binary64_2452 37.8

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

   6. Simplified 37.8

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

   8. Applied *-un-lft-identity_binary64_2436 37.8

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

   9. Applied sqrt-prod_binary64_2451 37.8

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

   10. Applied sqrt-prod_binary64_2451 37.8

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

   11. Applied times-frac_binary64_2442 37.8

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

   12. Simplified 37.4

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

   13. Simplified 37.4

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

   15. Applied *-un-lft-identity_binary64_2436 37.4

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

   16. Applied sqrt-prod_binary64_2451 37.4

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

   17. Applied add-sqr-sqrt_binary64_2457 37.4

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

   18. Applied times-frac_binary64_2442 37.4

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

   19. Applied associate-*r*_binary64_2378 37.4

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

   20. Simplified 37.4

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

   21. Taylor expanded around -inf 43.2

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

   22. Simplified 43.2

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

   if 2.3897943661054227e-18 < re

   1. Initial program 38.5

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

   2. Taylor expanded around inf 15.4

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

4. Final simplification 23.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.549533459362007 \cdot 10^{+160}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq -5.543930521314922 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{\sqrt{2}} \cdot \left|im\right|\right) \cdot \sqrt{\frac{\sqrt{2}}{\sqrt{im \cdot im + re \cdot re} - re}}\right)\\ \mathbf{elif}\;re \leq 1.4495720106119306 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 2.3897943661054227 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{\sqrt{2}} \cdot \left|im\right|\right) \cdot \frac{\sqrt{\sqrt{2}}}{\sqrt{\left(-im\right) - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(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)))))