math.sqrt on complex, real part

Average Error: 38.9 → 13.4

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \cdot im \le 4.0275550664508406 \cdot 10^{48}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right)\\ \mathbf{elif}\;im \cdot im \le 4.9142277901571849 \cdot 10^{293}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

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 (((im * im) <= 4.0275550664508406e+48)) {
		VAR = (0.5 * (sqrt(2.0) * sqrt((hypot(re, im) + re))));
	} else {
		double VAR_1;
		if (((im * im) <= 4.914227790157185e+293)) {
			VAR_1 = (0.5 * sqrt((2.0 * ((0.0 + pow(im, 2.0)) / (hypot(re, im) - re)))));
		} else {
			VAR_1 = (0.5 * sqrt((2.0 * (hypot(re, im) + re))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.9
Target	33.8
Herbie	13.4

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

Derivation

1. Split input into 3 regimes

2. if (* im im) < 4.0275550664508406e+48

   1. Initial program 34.7

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

   3. Applied hypot-def 19.0

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

   5. Applied sqrt-prod 19.2

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

   if 4.0275550664508406e+48 < (* im im) < 4.914227790157185e+293

   1. Initial program 19.3

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

   3. Applied flip-+ 22.1

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

   4. Simplified 18.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{0 + {im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]

   5. Simplified 10.9

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

   if 4.914227790157185e+293 < (* im im)

   1. Initial program 62.0

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

   3. Applied hypot-def 3.7

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

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \le 4.0275550664508406 \cdot 10^{48}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right)\\ \mathbf{elif}\;im \cdot im \le 4.9142277901571849 \cdot 10^{293}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(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)))))