math.sqrt on complex, real part

Average Error: 38.7 → 5.6

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le 1.3940086170973382 \cdot 10^{-278}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{\mathsf{hypot}\left(re, im\right) - re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(im, re\right) + re}\right)\\ \end{array}\]

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 ((re <= 1.3940086170973382e-278)) {
		VAR = (0.5 * sqrt((2.0 * (pow(im, 1.0) / ((hypot(re, im) - re) / im)))));
	} else {
		VAR = (0.5 * (sqrt(2.0) * sqrt((hypot(im, re) + re))));
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.7
Target	33.7
Herbie	5.6

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

Derivation

1. Split input into 2 regimes

2. if re < 1.3940086170973382e-278

   1. Initial program 44.8

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

   3. Applied flip-+ 44.7

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

   4. Simplified 35.0

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

   5. Simplified 30.3

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

   7. Applied sqr-pow 30.3

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

   8. Applied associate-/l* 10.3

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

   9. Simplified 10.3

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

   if 1.3940086170973382e-278 < re

   1. Initial program 32.2

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

   3. Applied +-commutative 32.2

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

   4. Applied hypot-def 0.1

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

   6. Applied sqrt-prod 0.5

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

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 1.3940086170973382 \cdot 10^{-278}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{\mathsf{hypot}\left(re, im\right) - re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(im, re\right) + re}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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)))))