Average Error: 38.3 → 18.1
Time: 17.8s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.3360753823747522 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{\left|im\right|} \cdot \sqrt{\sqrt{2}}\right) \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\sqrt{\left(-re\right) - re}}{\sqrt{\left|im\right|}}}\right)\\ \mathbf{elif}\;re \leq 6.993600331950945 \cdot 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\left|im\right| \cdot \sqrt{2}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\sqrt{\left|im\right|}}}\right)\\ \mathbf{elif}\;re \leq 9.000466420424405 \cdot 10^{-202}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)\right)}\\ \mathbf{elif}\;re \leq 1.0073950938736855 \cdot 10^{-164} \lor \neg \left(re \leq 2.632424878302857 \cdot 10^{+113}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.3360753823747522 \cdot 10^{+154}:\\
\;\;\;\;0.5 \cdot \left(\left(\sqrt{\left|im\right|} \cdot \sqrt{\sqrt{2}}\right) \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\sqrt{\left(-re\right) - re}}{\sqrt{\left|im\right|}}}\right)\\

\mathbf{elif}\;re \leq 6.993600331950945 \cdot 10^{-261}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{\left|im\right| \cdot \sqrt{2}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\sqrt{\left|im\right|}}}\right)\\

\mathbf{elif}\;re \leq 9.000466420424405 \cdot 10^{-202}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)\right)}\\

\mathbf{elif}\;re \leq 1.0073950938736855 \cdot 10^{-164} \lor \neg \left(re \leq 2.632424878302857 \cdot 10^{+113}\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\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 -1.3360753823747522e+154)
   (*
    0.5
    (*
     (* (sqrt (fabs im)) (sqrt (sqrt 2.0)))
     (/ (sqrt (sqrt 2.0)) (/ (sqrt (- (- re) re)) (sqrt (fabs im))))))
   (if (<= re 6.993600331950945e-261)
     (*
      0.5
      (*
       (sqrt (* (fabs im) (sqrt 2.0)))
       (/
        (sqrt (sqrt 2.0))
        (/ (sqrt (- (sqrt (+ (* re re) (* im im))) re)) (sqrt (fabs im))))))
     (if (<= re 9.000466420424405e-202)
       (* 0.5 (sqrt (* 2.0 (+ re (+ im (* 0.5 (/ (* re re) im)))))))
       (if (or (<= re 1.0073950938736855e-164)
               (not (<= re 2.632424878302857e+113)))
         (* 0.5 (sqrt (* 2.0 (* re 2.0))))
         (* 0.5 (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im))))))))))))
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 <= -1.3360753823747522e+154) {
		tmp = 0.5 * ((sqrt(fabs(im)) * sqrt(sqrt(2.0))) * (sqrt(sqrt(2.0)) / (sqrt(-re - re) / sqrt(fabs(im)))));
	} else if (re <= 6.993600331950945e-261) {
		tmp = 0.5 * (sqrt(fabs(im) * sqrt(2.0)) * (sqrt(sqrt(2.0)) / (sqrt(sqrt((re * re) + (im * im)) - re) / sqrt(fabs(im)))));
	} else if (re <= 9.000466420424405e-202) {
		tmp = 0.5 * sqrt(2.0 * (re + (im + (0.5 * ((re * re) / im)))));
	} else if ((re <= 1.0073950938736855e-164) || !(re <= 2.632424878302857e+113)) {
		tmp = 0.5 * sqrt(2.0 * (re * 2.0));
	} else {
		tmp = 0.5 * sqrt(2.0 * (re + sqrt((re * re) + (im * im))));
	}
	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

Original38.3
Target33.2
Herbie18.1
\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -1.3360753823747522e154

    1. Initial program 64.0

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

    3. Applied flip-+_binary64_414464.0

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

    4. Applied associate-*r/_binary64_411264.0

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

    5. Applied sqrt-div_binary64_418764.0

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

    6. Simplified48.6

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

    8. Applied sqrt-prod_binary64_418648.6

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

    9. Applied associate-/l*_binary64_411548.6

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

    10. Simplified48.2

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

    12. Applied add-sqr-sqrt_binary64_419248.2

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

    13. Applied *-un-lft-identity_binary64_417048.2

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

    14. Applied sqrt-prod_binary64_418648.2

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

    15. Applied times-frac_binary64_417648.2

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

    16. Applied add-sqr-sqrt_binary64_419248.2

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

    17. Applied times-frac_binary64_417648.2

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

    18. Simplified48.2

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

    19. Taylor expanded around -inf 7.3

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

    if -1.3360753823747522e154 < re < 6.993600331950945e-261

    1. Initial program 39.4

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

    3. Applied flip-+_binary64_414439.3

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

    4. Applied associate-*r/_binary64_411239.3

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

    5. Applied sqrt-div_binary64_418739.5

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

    6. Simplified30.2

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

    8. Applied sqrt-prod_binary64_418630.2

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

    9. Applied associate-/l*_binary64_411530.2

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

    10. Simplified21.2

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

    12. Applied add-sqr-sqrt_binary64_419221.3

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

    13. Applied *-un-lft-identity_binary64_417021.3

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

    14. Applied sqrt-prod_binary64_418621.3

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

    15. Applied times-frac_binary64_417621.3

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

    16. Applied add-sqr-sqrt_binary64_419221.3

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

    17. Applied times-frac_binary64_417621.1

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

    18. Simplified21.1

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

    20. Applied sqrt-unprod_binary64_419021.0

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

    if 6.993600331950945e-261 < re < 9.00046642042440462e-202

    1. Initial program 29.7

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

    2. Taylor expanded around 0 33.2

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

    3. Simplified33.2

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

    if 9.00046642042440462e-202 < re < 1.00739509387369e-164 or 2.63242487830285688e113 < re

    1. Initial program 49.0

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

    2. Taylor expanded around inf 17.8

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

    3. Simplified17.8

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

    if 1.00739509387369e-164 < re < 2.63242487830285688e113

    1. Initial program 15.5

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

    3. Applied pow1_binary64_423115.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt{re \cdot re + im \cdot im} + re\right)}^{1}}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.3360753823747522 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{\left|im\right|} \cdot \sqrt{\sqrt{2}}\right) \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\sqrt{\left(-re\right) - re}}{\sqrt{\left|im\right|}}}\right)\\ \mathbf{elif}\;re \leq 6.993600331950945 \cdot 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\left|im\right| \cdot \sqrt{2}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\sqrt{\left|im\right|}}}\right)\\ \mathbf{elif}\;re \leq 9.000466420424405 \cdot 10^{-202}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)\right)}\\ \mathbf{elif}\;re \leq 1.0073950938736855 \cdot 10^{-164} \lor \neg \left(re \leq 2.632424878302857 \cdot 10^{+113}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021075 
(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)))))