Average Error: 38.6 → 18.3
Time: 10.5s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -2.0399582988562903 \cdot 10^{+105}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq -5.497046773567483 \cdot 10^{-226}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \frac{\left|im\right| \cdot \sqrt{\sqrt{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{elif}\;re \leq 8.65010257179912 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 3.0513166345511567 \cdot 10^{-168}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;re \leq 1.6419980517252478 \cdot 10^{+115}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -2.0399582988562903 \cdot 10^{+105}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{re \cdot -2}}\\

\mathbf{elif}\;re \leq -5.497046773567483 \cdot 10^{-226}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \frac{\left|im\right| \cdot \sqrt{\sqrt{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\

\mathbf{elif}\;re \leq 8.65010257179912 \cdot 10^{-262}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \leq 3.0513166345511567 \cdot 10^{-168}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\

\mathbf{elif}\;re \leq 1.6419980517252478 \cdot 10^{+115}:\\
\;\;\;\;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 -2.0399582988562903e+105)
   (* 0.5 (/ (* (sqrt 2.0) (fabs im)) (sqrt (* re -2.0))))
   (if (<= re -5.497046773567483e-226)
     (*
      0.5
      (*
       (sqrt (sqrt 2.0))
       (/
        (* (fabs im) (sqrt (sqrt 2.0)))
        (sqrt (- (sqrt (+ (* re re) (* im im))) re)))))
     (if (<= re 8.65010257179912e-262)
       (* 0.5 (sqrt (* 2.0 (+ re im))))
       (if (<= re 3.0513166345511567e-168)
         (* 0.5 (sqrt (* 2.0 (- re im))))
         (if (<= re 1.6419980517252478e+115)
           (* 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 <= -2.0399582988562903e+105) {
		tmp = 0.5 * ((sqrt(2.0) * fabs(im)) / sqrt(re * -2.0));
	} else if (re <= -5.497046773567483e-226) {
		tmp = 0.5 * (sqrt(sqrt(2.0)) * ((fabs(im) * sqrt(sqrt(2.0))) / sqrt(sqrt((re * re) + (im * im)) - re)));
	} else if (re <= 8.65010257179912e-262) {
		tmp = 0.5 * sqrt(2.0 * (re + im));
	} else if (re <= 3.0513166345511567e-168) {
		tmp = 0.5 * sqrt(2.0 * (re - im));
	} else if (re <= 1.6419980517252478e+115) {
		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

Original38.6
Target33.5
Herbie18.3
\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 6 regimes

  2. if re < -2.03995829885629026e105

    1. Initial program 60.9

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

    3. Applied flip-+_binary64_243960.9

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

    4. Applied associate-*r/_binary64_240760.9

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

    5. Applied sqrt-div_binary64_248260.9

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

    6. Simplified44.2

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

    8. Applied sqrt-prod_binary64_248144.3

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

    9. Simplified42.0

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

    10. Taylor expanded around -inf 10.5

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

    11. Simplified10.5

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

    if -2.03995829885629026e105 < re < -5.49704677356748332e-226

    1. Initial program 39.8

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

    3. Applied flip-+_binary64_243939.6

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

    4. Applied associate-*r/_binary64_240739.7

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

    5. Applied sqrt-div_binary64_248239.8

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

    6. Simplified29.0

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

    8. Applied sqrt-prod_binary64_248129.0

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

    9. Simplified18.0

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

    11. Applied add-sqr-sqrt_binary64_248718.1

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

    12. Applied associate-*l*_binary64_240618.1

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

    13. Simplified18.1

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

    15. Applied *-un-lft-identity_binary64_246518.1

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

    16. Applied times-frac_binary64_247118.1

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

    if -5.49704677356748332e-226 < re < 8.65010257179912007e-262

    1. Initial program 30.0

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

    2. Taylor expanded around 0 33.8

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

    3. Simplified33.8

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

    if 8.65010257179912007e-262 < re < 3.05131663455115675e-168

    1. Initial program 31.5

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

    2. Taylor expanded around -inf 35.2

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

    if 3.05131663455115675e-168 < re < 1.6419980517252478e115

    1. Initial program 16.4

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

    if 1.6419980517252478e115 < re

    1. Initial program 54.4

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

    2. Taylor expanded around 0 11.4

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

    3. Simplified10.4

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

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.0399582988562903 \cdot 10^{+105}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq -5.497046773567483 \cdot 10^{-226}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \frac{\left|im\right| \cdot \sqrt{\sqrt{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{elif}\;re \leq 8.65010257179912 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 3.0513166345511567 \cdot 10^{-168}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;re \leq 1.6419980517252478 \cdot 10^{+115}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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)))))