math.sqrt on complex, real part

Average Error: 38.7 → 27.0

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 7.78698804780065138 \cdot 10^{76}:\\ \;\;\;\;0.5 \cdot \sqrt[3]{\sqrt{{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}^{3}} \cdot \sqrt{{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \end{array}\]

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 ((((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re)))))) <= 0.0)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) (im * im)) / ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re))))))))));
	} else {
		double VAR_1;
		if ((((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re)))))) <= 7.786988047800651e+76)) {
			VAR_1 = ((double) (0.5 * ((double) cbrt(((double) (((double) sqrt(((double) pow(((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re)))))), 3.0)))) * ((double) sqrt(((double) pow(((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re)))))), 3.0))))))))));
		} else {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im + re))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.7
Target	33.8
Herbie	27.0

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

Derivation

1. Split input into 3 regimes

2. if (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))) < 0.0

   1. Initial program 57.7

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

   3. Applied flip-+ 57.7

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

   4. Simplified 29.6

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

   if 0.0 < (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))) < 7.78698804780065138e76

   1. Initial program 4.6

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

   3. Applied add-cbrt-cube 5.4

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

   4. Simplified 5.4

      \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}^{3}}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 5.4

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

   if 7.78698804780065138e76 < (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))

   1. Initial program 63.8

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

   2. Taylor expanded around 0 45.2

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

4. Final simplification 27.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 7.78698804780065138 \cdot 10^{76}:\\ \;\;\;\;0.5 \cdot \sqrt[3]{\sqrt{{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}^{3}} \cdot \sqrt{{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(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)))))