Average Error: 38.7 → 17.3
Time: 9.2s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -8.962122735460895 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot \left(\frac{1}{\sqrt{re \cdot -2}} \cdot \left(\left|im\right| \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;re \leq 4.071140785912515 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\left|im\right|}}\\ \mathbf{elif}\;re \leq 3.963621779117234 \cdot 10^{-220}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)\right)}\\ \mathbf{elif}\;re \leq 3.4795012242168694 \cdot 10^{-174}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(\frac{re \cdot re}{im} \cdot -0.5 - im\right)\right)}\\ \mathbf{elif}\;re \leq 5.743696651812785 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -8.962122735460895 \cdot 10^{+136}:\\
\;\;\;\;0.5 \cdot \left(\frac{1}{\sqrt{re \cdot -2}} \cdot \left(\left|im\right| \cdot \sqrt{2}\right)\right)\\

\mathbf{elif}\;re \leq 4.071140785912515 \cdot 10^{-303}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\left|im\right|}}\\

\mathbf{elif}\;re \leq 3.963621779117234 \cdot 10^{-220}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)\right)}\\

\mathbf{elif}\;re \leq 3.4795012242168694 \cdot 10^{-174}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(\frac{re \cdot re}{im} \cdot -0.5 - im\right)\right)}\\

\mathbf{elif}\;re \leq 5.743696651812785 \cdot 10^{+111}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{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 -8.962122735460895e+136)
   (* 0.5 (* (/ 1.0 (sqrt (* re -2.0))) (* (fabs im) (sqrt 2.0))))
   (if (<= re 4.071140785912515e-303)
     (*
      0.5
      (/
       (sqrt 2.0)
       (/ (sqrt (- (sqrt (+ (* re re) (* im im))) re)) (fabs im))))
     (if (<= re 3.963621779117234e-220)
       (* 0.5 (sqrt (* 2.0 (+ re (+ im (* 0.5 (/ (* re re) im)))))))
       (if (<= re 3.4795012242168694e-174)
         (* 0.5 (sqrt (* 2.0 (+ re (- (* (/ (* re re) im) -0.5) im)))))
         (if (<= re 5.743696651812785e+111)
           (* 0.5 (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))))
           (* 0.5 (* 2.0 (sqrt re)))))))))
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 tmp;
	if (re <= -8.962122735460895e+136) {
		tmp = 0.5 * ((1.0 / sqrt(re * -2.0)) * (fabs(im) * sqrt(2.0)));
	} else if (re <= 4.071140785912515e-303) {
		tmp = 0.5 * (sqrt(2.0) / (sqrt(sqrt((re * re) + (im * im)) - re) / fabs(im)));
	} else if (re <= 3.963621779117234e-220) {
		tmp = 0.5 * sqrt(2.0 * (re + (im + (0.5 * ((re * re) / im)))));
	} else if (re <= 3.4795012242168694e-174) {
		tmp = 0.5 * sqrt(2.0 * (re + ((((re * re) / im) * -0.5) - im)));
	} else if (re <= 5.743696651812785e+111) {
		tmp = 0.5 * sqrt(2.0 * (re + sqrt((re * re) + (im * im))));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	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

Original38.7
Target33.3
Herbie17.3
\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 6 regimes

  2. if re < -8.96212273546089458e136

    1. Initial program 63.0

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

    3. Applied flip-+_binary64_209863.0

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

    4. Applied associate-*r/_binary64_206663.0

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

    5. Applied sqrt-div_binary64_214163.0

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

    6. Simplified47.7

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

    8. Applied sqrt-prod_binary64_214047.7

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

    9. Applied associate-/l*_binary64_206947.7

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

    10. Simplified46.3

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

    12. Applied div-inv_binary64_212146.3

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

    13. Applied *-un-lft-identity_binary64_212446.3

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

    14. Applied sqrt-prod_binary64_214046.3

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

    15. Applied times-frac_binary64_213046.1

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

    16. Simplified46.1

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

    17. Simplified46.1

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

    18. Taylor expanded around -inf 9.0

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

    19. Simplified9.0

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

    if -8.96212273546089458e136 < re < 4.07114078591251498e-303

    1. Initial program 39.9

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

    3. Applied flip-+_binary64_209839.7

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

    4. Applied associate-*r/_binary64_206639.7

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

    5. Applied sqrt-div_binary64_214139.8

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

    6. Simplified29.9

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

    8. Applied sqrt-prod_binary64_214030.0

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

    9. Applied associate-/l*_binary64_206930.0

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

    10. Simplified20.5

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

    if 4.07114078591251498e-303 < re < 3.96362177911723373e-220

    1. Initial program 29.1

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

    2. Taylor expanded around 0 33.6

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

    3. Simplified33.6

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

    if 3.96362177911723373e-220 < re < 3.47950122421686942e-174

    1. Initial program 25.6

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

    2. Taylor expanded around -inf 34.0

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

    3. Simplified34.0

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

    if 3.47950122421686942e-174 < re < 5.74369665181278522e111

    1. Initial program 16.2

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

    if 5.74369665181278522e111 < re

    1. Initial program 54.3

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

    2. Taylor expanded around 0 9.8

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

    3. Simplified8.7

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.962122735460895 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot \left(\frac{1}{\sqrt{re \cdot -2}} \cdot \left(\left|im\right| \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;re \leq 4.071140785912515 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\left|im\right|}}\\ \mathbf{elif}\;re \leq 3.963621779117234 \cdot 10^{-220}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)\right)}\\ \mathbf{elif}\;re \leq 3.4795012242168694 \cdot 10^{-174}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(\frac{re \cdot re}{im} \cdot -0.5 - im\right)\right)}\\ \mathbf{elif}\;re \leq 5.743696651812785 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021045 
(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)))))