math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 38.4 → 13.5

Time: 6.0s

Precision: binary64

\[im > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -2.1320242768716358 \cdot 10^{+145}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -1.4186350356885884 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \leq -5.178839982241449 \cdot 10^{-111}:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot im\right)}^{0.5}\\ \mathbf{elif}\;re \leq -1.2571986496294196 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \leq 24805927178.950237:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot im\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -2.1320242768716358e+145)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -1.4186350356885884e-103)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* im im) (* re re))) re))))
     (if (<= re -5.178839982241449e-111)
       (* 0.5 (pow (* 2.0 im) 0.5))
       (if (<= re -1.2571986496294196e-136)
         (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* im im) (* re re))) re))))
         (if (<= re 24805927178.950237)
           (* 0.5 (pow (* 2.0 im) 0.5))
           (* 0.5 (* (* (sqrt 0.5) (* im (sqrt 2.0))) (sqrt (/ 1.0 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 <= -2.1320242768716358e+145) {
		tmp = 0.5 * sqrt(2.0 * (re * -2.0));
	} else if (re <= -1.4186350356885884e-103) {
		tmp = 0.5 * sqrt(2.0 * (sqrt((im * im) + (re * re)) - re));
	} else if (re <= -5.178839982241449e-111) {
		tmp = 0.5 * pow((2.0 * im), 0.5);
	} else if (re <= -1.2571986496294196e-136) {
		tmp = 0.5 * sqrt(2.0 * (sqrt((im * im) + (re * re)) - re));
	} else if (re <= 24805927178.950237) {
		tmp = 0.5 * pow((2.0 * im), 0.5);
	} else {
		tmp = 0.5 * ((sqrt(0.5) * (im * sqrt(2.0))) * sqrt(1.0 / re));
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -2.1320242768716358e145

   1. Initial program 61.9

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

   2. Taylor expanded around -inf 8.1

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

   3. Simplified 8.1

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

   if -2.1320242768716358e145 < re < -1.418635035688588e-103 or -5.178839982241449e-111 < re < -1.25719864962941956e-136

   1. Initial program 15.4

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

   3. Applied +-commutative_binary64 15.4

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

   if -1.418635035688588e-103 < re < -5.178839982241449e-111 or -1.25719864962941956e-136 < re < 24805927178.9502373

   1. Initial program 31.8

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

   2. Taylor expanded around 0 12.9

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

   3. Simplified 12.9

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

   4. Taylor expanded around 0 13.4

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

   5. Simplified 13.4

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{im}\right)}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt_binary64 13.5

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

   8. Applied associate-*l*_binary64 13.4

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

   9. Simplified 13.4

      \[\leadsto 0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \color{blue}{\left(\sqrt{im} \cdot \sqrt{\sqrt{2}}\right)}\right)\]
   10. Using strategy rm

   11. Applied pow1_binary64 13.4

       \[\leadsto 0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{im} \cdot \sqrt{\color{blue}{{\left(\sqrt{2}\right)}^{1}}}\right)\right)\]

   12. Applied sqrt-pow1_binary64 13.4

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

   13. Applied pow1_binary64 13.4

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

   14. Applied sqrt-pow1_binary64 13.4

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

   15. Applied pow-prod-down_binary64 13.3

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

   16. Applied pow1_binary64 13.3

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

   17. Applied sqrt-pow1_binary64 13.3

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

   18. Applied pow-prod-down_binary64 13.5

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

   19. Simplified 13.1

       \[\leadsto 0.5 \cdot {\color{blue}{\left(im \cdot 2\right)}}^{\left(\frac{1}{2}\right)}\]

   if 24805927178.9502373 < re

   1. Initial program 57.3

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

   2. Taylor expanded around 0 15.2

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.1320242768716358 \cdot 10^{+145}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -1.4186350356885884 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \leq -5.178839982241449 \cdot 10^{-111}:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot im\right)}^{0.5}\\ \mathbf{elif}\;re \leq -1.2571986496294196 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \leq 24805927178.950237:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot im\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))