Average Error: 38.5 → 24.6
Time: 4.1s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -9.234847622570144 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{re \cdot -2}}\\ \mathbf{elif}\;re \leq -9.044521588158757 \cdot 10^{-202}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{im \cdot im + re \cdot re} - re}\right)}\\ \mathbf{elif}\;re \leq 3.499018610539718 \cdot 10^{-195}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 5.816453865787388 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + e^{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -9.234847622570144 \cdot 10^{+165}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{re \cdot -2}}\\

\mathbf{elif}\;re \leq -9.044521588158757 \cdot 10^{-202}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{im \cdot im + re \cdot re} - re}\right)}\\

\mathbf{elif}\;re \leq 3.499018610539718 \cdot 10^{-195}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \leq 5.816453865787388 \cdot 10^{+70}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + e^{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}\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 -9.234847622570144e+165)
   (* 0.5 (sqrt (* 2.0 (/ (* im im) (* re -2.0)))))
   (if (<= re -9.044521588158757e-202)
     (* 0.5 (sqrt (* 2.0 (* im (/ im (- (sqrt (+ (* im im) (* re re))) re))))))
     (if (<= re 3.499018610539718e-195)
       (* 0.5 (sqrt (* 2.0 (+ re im))))
       (if (<= re 5.816453865787388e+70)
         (*
          0.5
          (sqrt (* 2.0 (+ re (exp (log (sqrt (+ (* im im) (* re re)))))))))
         (* 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 <= -9.234847622570144e+165)) {
		tmp = ((double) (0.5 * ((double) sqrt(((double) (2.0 * (((double) (im * im)) / ((double) (re * -2.0)))))))));
	} else {
		double tmp_1;
		if ((re <= -9.044521588158757e-202)) {
			tmp_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im * (im / ((double) (((double) sqrt(((double) (((double) (im * im)) + ((double) (re * re)))))) - re)))))))))));
		} else {
			double tmp_2;
			if ((re <= 3.499018610539718e-195)) {
				tmp_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + im))))))));
			} else {
				double tmp_3;
				if ((re <= 5.816453865787388e+70)) {
					tmp_3 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + ((double) exp(((double) log(((double) sqrt(((double) (((double) (im * im)) + ((double) (re * re))))))))))))))))));
				} else {
					tmp_3 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + re))))))));
				}
				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

Original38.5
Target33.3
Herbie24.6
\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -9.23484762257014383e165

    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. SimplifiedError: 49.7 bits

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

    5. Taylor expanded around -inf Error: 29.6 bits

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

    6. SimplifiedError: 29.6 bits

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

    if -9.23484762257014383e165 < re < -9.04452158815875729e-202

    1. Initial program Error: 44.2 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: 44.1 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. SimplifiedError: 32.1 bits

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

    6. Applied *-un-lft-identityError: 32.1 bits

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

    7. Applied times-fracError: 29.6 bits

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

    8. SimplifiedError: 29.6 bits

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

    if -9.04452158815875729e-202 < re < 3.49901861053971786e-195

    1. Initial program Error: 28.3 bits

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

    2. Taylor expanded around 0 Error: 32.7 bits

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

    if 3.49901861053971786e-195 < re < 5.81645386578738842e70

    1. Initial program Error: 17.7 bits

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

    3. Applied add-exp-logError: 20.0 bits

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

    if 5.81645386578738842e70 < re

    1. Initial program Error: 47.5 bits

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

    2. Taylor expanded around inf Error: 11.8 bits

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

  4. Final simplificationError: 24.6 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.234847622570144 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{re \cdot -2}}\\ \mathbf{elif}\;re \leq -9.044521588158757 \cdot 10^{-202}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{im \cdot im + re \cdot re} - re}\right)}\\ \mathbf{elif}\;re \leq 3.499018610539718 \cdot 10^{-195}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 5.816453865787388 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + e^{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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)))))