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

Average Error: 38.7 → 14.0

Time: 5.2s

Precision: binary64

Cost: 21195

\[im > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -3.695071676918518 \cdot 10^{+120}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -1.704786349615486 \cdot 10^{-131}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \leq 8.443019886679552 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - re\right)}\\ \mathbf{elif}\;re \leq 5.594187968714849 \cdot 10^{+35} \lor \neg \left(re \leq 2.6425163668929388 \cdot 10^{+91}\right):\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array}\]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -3.695071676918518e+120)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -1.704786349615486e-131)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* im im) (* re re))) re))))
     (if (<= re 8.443019886679552e-21)
       (* 0.5 (sqrt (* 2.0 (- (+ im (* 0.5 (/ (* re re) im))) re))))
       (if (or (<= re 5.594187968714849e+35)
               (not (<= re 2.6425163668929388e+91)))
         (* 0.5 (* (* (sqrt 0.5) (* im (sqrt 2.0))) (sqrt (/ 1.0 re))))
         (* 0.5 (sqrt (* 2.0 im))))))))

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 <= -3.695071676918518e+120) {
		tmp = 0.5 * sqrt(2.0 * (re * -2.0));
	} else if (re <= -1.704786349615486e-131) {
		tmp = 0.5 * sqrt(2.0 * (sqrt((im * im) + (re * re)) - re));
	} else if (re <= 8.443019886679552e-21) {
		tmp = 0.5 * sqrt(2.0 * ((im + (0.5 * ((re * re) / im))) - re));
	} else if ((re <= 5.594187968714849e+35) || !(re <= 2.6425163668929388e+91)) {
		tmp = 0.5 * ((sqrt(0.5) * (im * sqrt(2.0))) * sqrt(1.0 / re));
	} else {
		tmp = 0.5 * sqrt(2.0 * im);
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Alternatives

Alternative 1
Error	17.6
Cost	14274

\[\begin{array}{l} \mathbf{if}\;re \leq -6.749403418164483 \cdot 10^{+115}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -7.949631741057429 \cdot 10^{-133}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \leq 1.5189952251471227 \cdot 10^{+91}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{im \cdot im}{re}\right)}\\ \end{array}\]

Alternative 2
Error	19.6
Cost	7746

\[\begin{array}{l} \mathbf{if}\;re \leq -2.194337097190631 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.5189952251471227 \cdot 10^{+91}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{im \cdot im}{re}\right)}\\ \end{array}\]

Alternative 3
Error	22.5
Cost	7490

\[\begin{array}{l} \mathbf{if}\;re \leq -2.194337097190631 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.0870141299270512 \cdot 10^{+184}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 4
Error	22.4
Cost	7362

\[\begin{array}{l} \mathbf{if}\;re \leq -7.5266266125105474 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.6890995397918132 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 5
Error	30.0
Cost	7041

\[\begin{array}{l} \mathbf{if}\;re \leq 2.2140007932492494 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 6
Error	58.3
Cost	385

\[\begin{array}{l} \mathbf{if}\;im \leq 4.8736104130745854 \cdot 10^{-170}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 7
Error	59.8
Cost	64

\[1\]

Derivation

1. Split input into 5 regimes

2. if re < -3.69507167691851811e120

   1. Initial program 56.1

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

   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

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

   4. Simplified 8.1

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

   if -3.69507167691851811e120 < re < -1.7047863496154861e-131

   1. Initial program 16.0

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

   2. Simplified 16.0

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

   if -1.7047863496154861e-131 < re < 8.4430198866795517e-21

   1. Initial program 31.6

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

   2. Taylor expanded around 0 12.0

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

   3. Simplified 12.0

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

   4. Simplified 12.0

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

   if 8.4430198866795517e-21 < re < 5.594187968714849e35 or 2.6425163668929388e91 < re

   1. Initial program 57.5

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

   2. Taylor expanded around 0 15.4

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

   3. Simplified 15.4

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

   if 5.594187968714849e35 < re < 2.6425163668929388e91

   1. Initial program 49.0

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

   2. Taylor expanded around 0 35.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{im}}\]

   3. Simplified 35.5

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot im}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.695071676918518 \cdot 10^{+120}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -1.704786349615486 \cdot 10^{-131}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \leq 8.443019886679552 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - re\right)}\\ \mathbf{elif}\;re \leq 5.594187968714849 \cdot 10^{+35} \lor \neg \left(re \leq 2.6425163668929388 \cdot 10^{+91}\right):\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))))