math.sqrt on complex, real part

Average Error: 38.4 → 22.7

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -8.1561596166685901 \cdot 10^{125}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{-2 \cdot re}}\\ \mathbf{elif}\;re \le -1.10443298905324865 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{2} \cdot \left|im\right|\right) \cdot \frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{elif}\;re \le 4.9104618261116892 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -8.15615961666859e+125)) {
		VAR = (0.5 * ((sqrt(2.0) * fabs(im)) / sqrt((-2.0 * re))));
	} else {
		double VAR_1;
		if ((re <= -1.1044329890532487e-116)) {
			VAR_1 = (0.5 * ((sqrt(2.0) * fabs(im)) * (1.0 / sqrt((sqrt(((re * re) + (im * im))) - re)))));
		} else {
			double VAR_2;
			if ((re <= 4.910461826111689e-93)) {
				VAR_2 = (0.5 * sqrt((2.0 * (im + re))));
			} else {
				VAR_2 = (0.5 * sqrt((2.0 * (re + re))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.4
Target	33.2
Herbie	22.7

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

Derivation

1. Split input into 4 regimes

2. if re < -8.15615961666859e+125

   1. Initial program 61.9

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

   3. Applied flip-+ 61.9

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

   4. Simplified 46.9

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

   6. Applied associate-*r/ 46.9

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

   7. Applied sqrt-div 45.7

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

   9. Applied sqrt-prod 45.7

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

   10. Simplified 43.9

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

   11. Taylor expanded around -inf 9.3

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

   if -8.15615961666859e+125 < re < -1.1044329890532487e-116

   1. Initial program 45.6

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

   3. Applied flip-+ 45.6

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

   4. Simplified 30.8

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

   6. Applied associate-*r/ 30.8

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

   7. Applied sqrt-div 29.2

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

   9. Applied sqrt-prod 29.2

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

   10. Simplified 16.2

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

   12. Applied div-inv 16.2

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

   if -1.1044329890532487e-116 < re < 4.910461826111689e-93

   1. Initial program 28.8

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

   2. Taylor expanded around 0 36.2

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

   if 4.910461826111689e-93 < re

   1. Initial program 33.3

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

   2. Taylor expanded around inf 18.8

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

4. Final simplification 22.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.1561596166685901 \cdot 10^{125}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{-2 \cdot re}}\\ \mathbf{elif}\;re \le -1.10443298905324865 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{2} \cdot \left|im\right|\right) \cdot \frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{elif}\;re \le 4.9104618261116892 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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)))))