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

Average Error: 38.6 → 0.3

Time: 3.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le 2.11590261065708125 \cdot 10^{-251}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(im, re\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{2}{re + \mathsf{hypot}\left(re, im\right)}} \cdot \left|im\right|\right)\\ \end{array}\]

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 VAR;
	if ((re <= 2.1159026106570812e-251)) {
		VAR = (0.5 * sqrt((2.0 * (hypot(im, re) - re))));
	} else {
		VAR = (0.5 * (sqrt((2.0 / (re + hypot(re, im)))) * fabs(im)));
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 2 regimes

2. if re < 2.1159026106570812e-251

   1. Initial program 31.8

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

   3. Applied +-commutative 31.8

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

   4. Applied hypot-def 0.3

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

   if 2.1159026106570812e-251 < re

   1. Initial program 46.9

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

   3. Applied flip-- 46.9

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

   4. Simplified 36.2

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

   5. Simplified 31.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{re + \mathsf{hypot}\left(re, im\right)}}}\]
   6. Using strategy rm

   7. Applied sqr-pow 31.4

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

   8. Applied associate-/l* 12.2

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

   9. Applied associate-*r/ 12.2

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

   10. Simplified 12.2

       \[\leadsto 0.5 \cdot \sqrt{\frac{\color{blue}{im \cdot 2}}{\frac{re + \mathsf{hypot}\left(re, im\right)}{{im}^{\left(\frac{2}{2}\right)}}}}\]
   11. Using strategy rm

   12. Applied div-inv 12.3

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

   13. Applied *-commutative 12.3

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

   14. Applied times-frac 31.4

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

   15. Applied sqrt-prod 27.2

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

   16. Simplified 0.3

       \[\leadsto 0.5 \cdot \left(\sqrt{\frac{2}{re + \mathsf{hypot}\left(re, im\right)}} \cdot \color{blue}{\left|im\right|}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 2.11590261065708125 \cdot 10^{-251}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(im, re\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{2}{re + \mathsf{hypot}\left(re, im\right)}} \cdot \left|im\right|\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))