math.sqrt on complex, real part

Average Error: 38.7 → 24.7

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.37964942532106859 \cdot 10^{83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{-2 \cdot re}{im}}}\\ \mathbf{elif}\;re \le -283.23902669347274:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le -1.448710066223221 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{e^{\log \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}{im}}}\\ \mathbf{elif}\;re \le -1.984730296439969 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le 1.573305619337669 \cdot 10^{-228}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{elif}\;re \le 3.9315901045752092 \cdot 10^{-191}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot im}}{\sqrt{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le 5.32364720038125515 \cdot 10^{39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

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 temp;
	if ((re <= -4.3796494253210686e+83)) {
		temp = (0.5 * sqrt((2.0 * (im / ((-2.0 * re) / im)))));
	} else {
		double temp_1;
		if ((re <= -283.23902669347274)) {
			temp_1 = (0.5 * sqrt((2.0 * (im / ((im - re) / im)))));
		} else {
			double temp_2;
			if ((re <= -1.448710066223221e-194)) {
				temp_2 = (0.5 * sqrt((2.0 * (im / (exp(log((sqrt(((re * re) + (im * im))) - re))) / im)))));
			} else {
				double temp_3;
				if ((re <= -1.984730296439969e-262)) {
					temp_3 = (0.5 * sqrt((2.0 * (im / ((im - re) / im)))));
				} else {
					double temp_4;
					if ((re <= 1.573305619337669e-228)) {
						temp_4 = (0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re))));
					} else {
						double temp_5;
						if ((re <= 3.931590104575209e-191)) {
							temp_5 = (0.5 * (sqrt((2.0 * im)) / sqrt(((im - re) / im))));
						} else {
							double temp_6;
							if ((re <= 5.323647200381255e+39)) {
								temp_6 = (0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re))));
							} else {
								temp_6 = (0.5 * sqrt((2.0 * (2.0 * re))));
							}
							temp_5 = temp_6;
						}
						temp_4 = temp_5;
					}
					temp_3 = temp_4;
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.7
Target	33.7
Herbie	24.7

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

Derivation

1. Split input into 6 regimes

2. if re < -4.3796494253210686e+83

   1. Initial program 60.2

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

   3. Applied flip-+ 60.2

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

   4. Simplified 44.4

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

   6. Applied unpow2 44.4

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

   7. Applied associate-/l* 44.1

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

   8. Taylor expanded around -inf 26.4

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

   if -4.3796494253210686e+83 < re < -283.23902669347274 or -1.448710066223221e-194 < re < -1.984730296439969e-262

   1. Initial program 41.0

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

   3. Applied flip-+ 40.7

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

   4. Simplified 31.4

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

   6. Applied unpow2 31.4

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

   7. Applied associate-/l* 30.4

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

   8. Taylor expanded around 0 39.2

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

   if -283.23902669347274 < re < -1.448710066223221e-194

   1. Initial program 37.6

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

   3. Applied flip-+ 37.5

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

   4. Simplified 31.3

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

   6. Applied unpow2 31.3

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

   7. Applied associate-/l* 27.5

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

   9. Applied add-exp-log 29.4

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

   if -1.984730296439969e-262 < re < 1.573305619337669e-228 or 3.931590104575209e-191 < re < 5.323647200381255e+39

   1. Initial program 22.3

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

   if 1.573305619337669e-228 < re < 3.931590104575209e-191

   1. Initial program 30.3

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

   3. Applied flip-+ 32.4

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

   4. Simplified 32.4

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

   6. Applied unpow2 32.4

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

   7. Applied associate-/l* 32.3

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

   8. Taylor expanded around 0 34.8

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

   10. Applied associate-*r/ 34.8

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

   11. Applied sqrt-div 35.0

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

   if 5.323647200381255e+39 < re

   1. Initial program 43.9

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

   3. Applied flip-+ 61.5

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

   4. Simplified 60.8

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

   5. Taylor expanded around 0 13.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 24.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.37964942532106859 \cdot 10^{83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{-2 \cdot re}{im}}}\\ \mathbf{elif}\;re \le -283.23902669347274:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le -1.448710066223221 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{e^{\log \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}{im}}}\\ \mathbf{elif}\;re \le -1.984730296439969 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le 1.573305619337669 \cdot 10^{-228}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{elif}\;re \le 3.9315901045752092 \cdot 10^{-191}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot im}}{\sqrt{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le 5.32364720038125515 \cdot 10^{39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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)))))