math.sqrt on complex, real part

Average Error: 38.7 → 17.6

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.18864377311358353 \cdot 10^{143}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{-2 \cdot re}}\right|\right)\\ \mathbf{elif}\;re \le 3.84236069908571 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\right)\\ \mathbf{elif}\;re \le 4.03491544026474703 \cdot 10^{36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

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 <= -2.1886437731135835e+143)) {
		VAR = ((double) (0.5 * ((double) (((double) sqrt(2.0)) * ((double) fabs(((double) (im / ((double) sqrt(((double) (-2.0 * re))))))))))));
	} else {
		double VAR_1;
		if ((re <= 3.8423606990857e-310)) {
			VAR_1 = ((double) (0.5 * ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) fabs(((double) (im / ((double) sqrt(((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re))))))))))))));
		} else {
			double VAR_2;
			if ((re <= 4.034915440264747e+36)) {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re))))))));
			} else {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + re))))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.7
Target	33.6
Herbie	17.6

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

Derivation

1. Split input into 4 regimes

2. if re < -2.1886437731135835e+143

   1. Initial program 63.5

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

   3. Applied flip-+ 63.5

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

   4. Simplified 49.3

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

   6. Applied add-sqr-sqrt 49.3

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

   7. Applied times-frac 48.9

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

   9. Applied sqrt-prod 48.9

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

   10. Simplified 47.9

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

   11. Taylor expanded around -inf 9.4

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

   if -2.1886437731135835e+143 < re < 3.8423606990857e-310

   1. Initial program 39.7

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

   3. Applied flip-+ 39.5

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

   4. Simplified 31.0

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

   6. Applied add-sqr-sqrt 31.1

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

   7. Applied times-frac 29.2

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

   9. Applied sqrt-prod 29.2

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

   10. Simplified 20.4

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

   12. Applied add-sqr-sqrt 20.4

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

   13. Applied sqrt-prod 20.5

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

   14. Applied associate-*l* 20.5

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

   if 3.8423606990857e-310 < re < 4.034915440264747e+36

   1. Initial program 21.9

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

   if 4.034915440264747e+36 < re

   1. Initial program 43.0

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

   2. Taylor expanded around inf 12.5

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

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.18864377311358353 \cdot 10^{143}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{-2 \cdot re}}\right|\right)\\ \mathbf{elif}\;re \le 3.84236069908571 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\right)\\ \mathbf{elif}\;re \le 4.03491544026474703 \cdot 10^{36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))