math.sqrt on complex, real part

Average Error: 38.8 → 23.4

Time: 5.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -2.3838929425074713 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq -6.527935055640734 \cdot 10^{-301}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2}{\sqrt{im \cdot im + re \cdot re} - re}}\right)\\ \mathbf{elif}\;re \leq 1.1339651548786607 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 2.3897943661054227 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2}{\left(-im\right) - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -2.3838929425074713e+136)
   (* 0.5 (/ (sqrt (* (* im im) 2.0)) (sqrt (* re -2.0))))
   (if (<= re -6.527935055640734e-301)
     (* 0.5 (* (fabs im) (sqrt (/ 2.0 (- (sqrt (+ (* im im) (* re re))) re)))))
     (if (<= re 1.1339651548786607e-217)
       (* 0.5 (sqrt (* 2.0 (+ re im))))
       (if (<= re 2.3897943661054227e-18)
         (* 0.5 (* (fabs im) (sqrt (/ 2.0 (- (- 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 <= -2.3838929425074713e+136) {
		tmp = 0.5 * (sqrt((im * im) * 2.0) / sqrt(re * -2.0));
	} else if (re <= -6.527935055640734e-301) {
		tmp = 0.5 * (fabs(im) * sqrt(2.0 / (sqrt((im * im) + (re * re)) - re)));
	} else if (re <= 1.1339651548786607e-217) {
		tmp = 0.5 * sqrt(2.0 * (re + im));
	} else if (re <= 2.3897943661054227e-18) {
		tmp = 0.5 * (fabs(im) * sqrt(2.0 / (-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	23.4

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

Derivation

1. Split input into 5 regimes

2. if re < -2.3838929425074713e136

   1. Initial program 63.0

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

   3. Applied flip-+_binary64_1429 63.0

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

   4. Applied associate-*r/_binary64_1397 63.0

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

   5. Applied sqrt-div_binary64_1472 63.0

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

   6. Simplified 46.7

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

   7. Taylor expanded around -inf 18.6

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

   8. Simplified 18.6

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

   if -2.3838929425074713e136 < re < -6.527935055640734e-301

   1. Initial program 39.5

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

   3. Applied flip-+_binary64_1429 39.3

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

   4. Applied associate-*r/_binary64_1397 39.3

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

   5. Applied sqrt-div_binary64_1472 39.4

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

   6. Simplified 30.1

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

   8. Applied *-un-lft-identity_binary64_1455 30.1

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

   9. Applied sqrt-prod_binary64_1471 30.1

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

   10. Applied sqrt-prod_binary64_1471 30.2

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

   11. Applied times-frac_binary64_1461 30.2

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

   12. Simplified 20.5

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

   13. Simplified 20.5

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

   15. Applied sqrt-undiv_binary64_1476 20.3

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

   if -6.527935055640734e-301 < re < 1.1339651548786607e-217

   1. Initial program 28.0

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

   2. Taylor expanded around 0 31.9

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

   if 1.1339651548786607e-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_1429 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_1397 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_1472 37.7

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

   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_1455 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_1471 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_1471 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_1461 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 37.3

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

   13. Simplified 37.3

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

   15. Applied sqrt-undiv_binary64_1476 37.3

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

   16. Taylor expanded around -inf 43.2

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

   17. Simplified 43.2

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

   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 23.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.3838929425074713 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq -6.527935055640734 \cdot 10^{-301}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2}{\sqrt{im \cdot im + re \cdot re} - re}}\right)\\ \mathbf{elif}\;re \leq 1.1339651548786607 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 2.3897943661054227 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2}{\left(-im\right) - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

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)))))