math.sqrt on complex, real part

Average Error: 38.6 → 22.6

Time: 6.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.96976657093792319 \cdot 10^{-263}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\\ \mathbf{elif}\;re \le 5.0746073328416246 \cdot 10^{77}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

C

double code(double re, double im) {
	return ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re))))))));
}

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -3.969766570937923e-263)) {
		VAR = ((double) (0.5 * ((double) (((double) sqrt(2.0)) * ((double) fabs(((double) (im / ((double) sqrt(((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re))))))))))));
	} else {
		double VAR_1;
		if ((re <= 5.074607332841625e+77)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) (((double) sqrt(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) * ((double) sqrt(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))) + re))))))));
		} else {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + re))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.6
Target	33.3
Herbie	22.6

\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if re < -3.969766570937923e-263

   1. Initial program 47.5

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

   3. Applied flip-+ 47.4

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

   4. Simplified 35.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 36.0

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

   7. Applied add-sqr-sqrt 50.6

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

   8. Applied unpow-prod-down 50.6

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

   9. Applied times-frac 49.8

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

   10. Simplified 49.8

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

   11. Simplified 34.3

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

   13. Applied sqrt-prod 34.3

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

   14. Simplified 27.1

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

   if -3.969766570937923e-263 < re < 5.074607332841625e+77

   1. Initial program 22.0

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

   3. Applied add-sqr-sqrt 22.0

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

   4. Applied sqrt-prod 22.1

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

   if 5.074607332841625e+77 < re

   1. Initial program 49.2

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

   2. Taylor expanded around inf 12.2

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

4. Final simplification 22.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.96976657093792319 \cdot 10^{-263}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\\ \mathbf{elif}\;re \le 5.0746073328416246 \cdot 10^{77}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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)))))