Average Error: 38.9 → 24.5
Time: 4.1s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -2.649308197943826 \cdot 10^{+162}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq -3.2410829879680627 \cdot 10^{-270}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)\\ \mathbf{elif}\;re \leq 3.3363403467192465 \cdot 10^{-186}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 4.000710555951358 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)\\ \mathbf{elif}\;re \leq 1.4310037079643518 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{elif}\;re \leq 9.810431200447886 \cdot 10^{+56}:\\ \;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \leq 8.99727301172237 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -2.649308197943826 \cdot 10^{+162}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{re \cdot -2}}\\

\mathbf{elif}\;re \leq -3.2410829879680627 \cdot 10^{-270}:\\
\;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)\\

\mathbf{elif}\;re \leq 3.3363403467192465 \cdot 10^{-186}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \leq 4.000710555951358 \cdot 10^{-118}:\\
\;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)\\

\mathbf{elif}\;re \leq 1.4310037079643518 \cdot 10^{-78}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\

\mathbf{elif}\;re \leq 9.810431200447886 \cdot 10^{+56}:\\
\;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\

\mathbf{elif}\;re \leq 8.99727301172237 \cdot 10^{+57}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\

\end{array}
(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -2.649308197943826e+162)
   (* 0.5 (/ (sqrt (* (* im im) 2.0)) (sqrt (* re -2.0))))
   (if (<= re -3.2410829879680627e-270)
     (*
      0.5
      (*
       (fabs im)
       (/ (sqrt 2.0) (sqrt (- (sqrt (+ (* im im) (* re re))) re)))))
     (if (<= re 3.3363403467192465e-186)
       (* 0.5 (sqrt (* 2.0 (+ re im))))
       (if (<= re 4.000710555951358e-118)
         (*
          0.5
          (*
           (fabs im)
           (/ (sqrt 2.0) (sqrt (- (sqrt (+ (* im im) (* re re))) re)))))
         (if (<= re 1.4310037079643518e-78)
           (* 0.5 (sqrt (* 2.0 (+ re re))))
           (if (<= re 9.810431200447886e+56)
             (*
              0.5
              (/
               (* (fabs im) (sqrt 2.0))
               (sqrt (- (sqrt (+ (* im im) (* re re))) re))))
             (if (<= re 8.99727301172237e+57)
               (* 0.5 (sqrt (* 2.0 (+ re im))))
               (* 0.5 (sqrt (* 2.0 (+ re re))))))))))))
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 tmp;
	if ((re <= -2.649308197943826e+162)) {
		tmp = ((double) (0.5 * (((double) sqrt(((double) (((double) (im * im)) * 2.0)))) / ((double) sqrt(((double) (re * -2.0)))))));
	} else {
		double tmp_1;
		if ((re <= -3.2410829879680627e-270)) {
			tmp_1 = ((double) (0.5 * ((double) (((double) fabs(im)) * (((double) sqrt(2.0)) / ((double) sqrt(((double) (((double) sqrt(((double) (((double) (im * im)) + ((double) (re * re)))))) - re)))))))));
		} else {
			double tmp_2;
			if ((re <= 3.3363403467192465e-186)) {
				tmp_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + im))))))));
			} else {
				double tmp_3;
				if ((re <= 4.000710555951358e-118)) {
					tmp_3 = ((double) (0.5 * ((double) (((double) fabs(im)) * (((double) sqrt(2.0)) / ((double) sqrt(((double) (((double) sqrt(((double) (((double) (im * im)) + ((double) (re * re)))))) - re)))))))));
				} else {
					double tmp_4;
					if ((re <= 1.4310037079643518e-78)) {
						tmp_4 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + re))))))));
					} else {
						double tmp_5;
						if ((re <= 9.810431200447886e+56)) {
							tmp_5 = ((double) (0.5 * (((double) (((double) fabs(im)) * ((double) sqrt(2.0)))) / ((double) sqrt(((double) (((double) sqrt(((double) (((double) (im * im)) + ((double) (re * re)))))) - re)))))));
						} else {
							double tmp_6;
							if ((re <= 8.99727301172237e+57)) {
								tmp_6 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + im))))))));
							} else {
								tmp_6 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + re))))))));
							}
							tmp_5 = tmp_6;
						}
						tmp_4 = tmp_5;
					}
					tmp_3 = tmp_4;
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target34.0
Herbie24.5
\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -2.64930819794382591e162

    1. Initial program Error: 64.0 bits

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

    3. Applied flip-+Error: 64.0 bits

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

    4. Applied associate-*r/Error: 64.0 bits

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

    5. Applied sqrt-divError: 64.0 bits

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

    6. SimplifiedError: 50.7 bits

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

    7. Taylor expanded around -inf Error: 21.1 bits

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

    8. SimplifiedError: 21.1 bits

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

    if -2.64930819794382591e162 < re < -3.24108298796806271e-270 or 3.3363403467192465e-186 < re < 4.0007105559513578e-118

    1. Initial program Error: 38.0 bits

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

    3. Applied flip-+Error: 39.5 bits

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

    4. Applied associate-*r/Error: 39.5 bits

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

    5. Applied sqrt-divError: 39.7 bits

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

    6. SimplifiedError: 30.5 bits

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

    8. Applied *-un-lft-identityError: 30.5 bits

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

    9. Applied sqrt-prodError: 30.5 bits

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

    10. Applied sqrt-prodError: 30.6 bits

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{im \cdot im} \cdot \sqrt{2}}}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    11. Applied times-fracError: 30.5 bits

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

    12. SimplifiedError: 22.0 bits

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

    if -3.24108298796806271e-270 < re < 3.3363403467192465e-186 or 9.8104312004478862e56 < re < 8.9972730117223705e57

    1. Initial program Error: 29.9 bits

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

    2. Taylor expanded around 0 Error: 32.2 bits

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

    if 4.0007105559513578e-118 < re < 1.43100370796435177e-78 or 8.9972730117223705e57 < re

    1. Initial program Error: 42.1 bits

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

    2. Taylor expanded around inf Error: 16.7 bits

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

    if 1.43100370796435177e-78 < re < 9.8104312004478862e56

    1. Initial program Error: 18.6 bits

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

    3. Applied flip-+Error: 46.3 bits

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

    4. Applied associate-*r/Error: 46.3 bits

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

    5. Applied sqrt-divError: 46.4 bits

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

    6. SimplifiedError: 46.4 bits

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

    8. Applied sqrt-prodError: 46.5 bits

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

    9. SimplifiedError: 46.0 bits

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\left|im\right|} \cdot \sqrt{2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplificationError: 24.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.649308197943826 \cdot 10^{+162}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq -3.2410829879680627 \cdot 10^{-270}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)\\ \mathbf{elif}\;re \leq 3.3363403467192465 \cdot 10^{-186}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 4.000710555951358 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)\\ \mathbf{elif}\;re \leq 1.4310037079643518 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{elif}\;re \leq 9.810431200447886 \cdot 10^{+56}:\\ \;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \leq 8.99727301172237 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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)))))