math.sqrt on complex, real part

Average Error: 38.5 → 18.4

Time: 11.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.11696641977354674 \cdot 10^{59}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{-2 \cdot re}}\\ \mathbf{elif}\;re \le -4.65770807539910302 \cdot 10^{-252}:\\ \;\;\;\;0.5 \cdot \frac{\left(\sqrt{2} \cdot \sqrt{\left|im\right|}\right) \cdot \sqrt{\left|im\right|}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.28573546673855928 \cdot 10^{-281}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.29297700493325525 \cdot 10^{48}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

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 <= -4.116966419773547e+59)) {
		VAR = ((double) (0.5 * ((double) (((double) (((double) sqrt(2.0)) * ((double) fabs(im)))) / ((double) sqrt(((double) (-2.0 * re))))))));
	} else {
		double VAR_1;
		if ((re <= -4.657708075399103e-252)) {
			VAR_1 = ((double) (0.5 * ((double) (((double) (((double) (((double) sqrt(2.0)) * ((double) sqrt(((double) fabs(im)))))) * ((double) sqrt(((double) fabs(im)))))) / ((double) sqrt(((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re))))))));
		} else {
			double VAR_2;
			if ((re <= 1.2857354667385593e-281)) {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im + re))))))));
			} else {
				double VAR_3;
				if ((re <= 1.2929770049332553e+48)) {
					VAR_3 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re))))))));
				} else {
					VAR_3 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + re))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.5
Target	33.4
Herbie	18.4

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

Derivation

1. Split input into 5 regimes

2. if re < -4.11696641977354674e59

   1. Initial program 59.7

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

   3. Applied flip-+ 59.7

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

   4. Simplified 42.8

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

   6. Applied associate-*r/ 42.9

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

   7. Applied sqrt-div 40.7

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

   9. Applied sqrt-prod 40.7

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

   10. Simplified 36.8

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

   11. Taylor expanded around -inf 13.1

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

   if -4.11696641977354674e59 < re < -4.65770807539910302e-252

   1. Initial program 36.8

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

   3. Applied flip-+ 36.6

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

   4. Simplified 29.6

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

   6. Applied associate-*r/ 29.6

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

   7. Applied sqrt-div 29.6

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

   9. Applied sqrt-prod 29.6

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

   10. Simplified 19.1

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

   12. Applied add-sqr-sqrt 19.2

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

   13. Applied associate-*r* 19.1

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

   if -4.65770807539910302e-252 < re < 1.28573546673855928e-281

   1. Initial program 32.7

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

   2. Taylor expanded around 0 34.3

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

   if 1.28573546673855928e-281 < re < 1.29297700493325525e48

   1. Initial program 21.8

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

   if 1.29297700493325525e48 < re

   1. Initial program 44.0

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

   2. Taylor expanded around inf 13.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} + re\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.11696641977354674 \cdot 10^{59}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{-2 \cdot re}}\\ \mathbf{elif}\;re \le -4.65770807539910302 \cdot 10^{-252}:\\ \;\;\;\;0.5 \cdot \frac{\left(\sqrt{2} \cdot \sqrt{\left|im\right|}\right) \cdot \sqrt{\left|im\right|}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.28573546673855928 \cdot 10^{-281}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.29297700493325525 \cdot 10^{48}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(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)))))