math.sqrt on complex, real part

Average Error: 38.7 → 26.9

Time: 7.1s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.61334022276189894 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 4.1753668252846765 \cdot 10^{-213}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 7.06982122822797643 \cdot 10^{116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\left(\sqrt[3]{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}} \cdot \sqrt[3]{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}}\right) \cdot \sqrt[3]{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

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 <= -1.613340222761899e-296)) {
		VAR = ((double) ((((double) sqrt(((double) (2.0 * ((double) (im * im)))))) / ((double) sqrt(((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re))))) * 0.5));
	} else {
		double VAR_1;
		if ((re <= 4.1753668252846765e-213)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im + re))))))));
		} else {
			double VAR_2;
			if ((re <= 7.069821228227976e+116)) {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) (((double) sqrt(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) * ((double) sqrt(((double) (((double) (((double) cbrt(((double) pow(((double) exp(0.5)), ((double) log(((double) (((double) (re * re)) + ((double) (im * im)))))))))) * ((double) cbrt(((double) pow(((double) exp(0.5)), ((double) log(((double) (((double) (re * re)) + ((double) (im * im)))))))))))) * ((double) cbrt(((double) pow(((double) exp(0.5)), ((double) log(((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	26.9

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

Derivation

1. Split input into 4 regimes

2. if re < -1.61334022276189894e-296

   1. Initial program 45.9

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

   3. Applied flip-+ 45.8

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

   4. Applied associate-*r/ 45.8

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

   5. Applied sqrt-div 45.8

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

   6. Simplified 34.7

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(0 + im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

   if -1.61334022276189894e-296 < re < 4.1753668252846765e-213

   1. Initial program 28.7

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

   2. Taylor expanded around 0 31.4

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

   if 4.1753668252846765e-213 < re < 7.06982122822797643e116

   1. Initial program 18.5

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

   3. Applied add-sqr-sqrt 18.5

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

   4. Applied sqrt-prod 18.6

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

   6. Applied add-exp-log 20.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} + re\right)}\]
   7. Using strategy rm

   8. Applied pow1/2 20.2

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

   9. Applied log-pow 20.2

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

   10. Applied exp-prod 20.5

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\color{blue}{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}}} + re\right)}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 20.5

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\color{blue}{\left(\sqrt[3]{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}} \cdot \sqrt[3]{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}}\right) \cdot \sqrt[3]{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}}}} + re\right)}\]

   if 7.06982122822797643e116 < re

   1. Initial program 55.0

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

   2. Taylor expanded around inf 9.9

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

4. Final simplification 26.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.61334022276189894 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 4.1753668252846765 \cdot 10^{-213}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 7.06982122822797643 \cdot 10^{116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\left(\sqrt[3]{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}} \cdot \sqrt[3]{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}}\right) \cdot \sqrt[3]{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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)))))