Average Error: 37.7 → 20.9
Time: 4.2s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.306963558091192 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq 1.0928277540347782 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|im\right| \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.306963558091192 \cdot 10^{+18}:\\
\;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{re \cdot -2}}\\

\mathbf{elif}\;re \leq 1.0928277540347782 \cdot 10^{-101}:\\
\;\;\;\;0.5 \cdot \left(\left(\left|im\right| \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\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 -1.306963558091192e+18)
   (* 0.5 (/ (* (fabs im) (sqrt 2.0)) (sqrt (* re -2.0))))
   (if (<= re 1.0928277540347782e-101)
     (*
      0.5
      (*
       (* (fabs im) (sqrt 2.0))
       (/ 1.0 (sqrt (- (sqrt (+ (* re re) (* im im))) 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 <= -1.306963558091192e+18)) {
		tmp = ((double) (0.5 * (((double) (((double) fabs(im)) * ((double) sqrt(2.0)))) / ((double) sqrt(((double) (re * -2.0)))))));
	} else {
		double tmp_1;
		if ((re <= 1.0928277540347782e-101)) {
			tmp_1 = ((double) (0.5 * ((double) (((double) (((double) fabs(im)) * ((double) sqrt(2.0)))) * (1.0 / ((double) sqrt(((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)))))))));
		} else {
			tmp_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + re))))))));
		}
		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

Original37.7
Target32.9
Herbie20.9
\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -1306963558091192060

    1. Initial program Error: 57.7 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: 57.7 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: 57.8 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: 57.8 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: 40.1 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: 40.1 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: 34.8 bits

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

    10. Taylor expanded around -inf Error: 14.8 bits

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

    11. SimplifiedError: 14.8 bits

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

    if -1306963558091192060 < re < 1.09282775403477823e-101

    1. Initial program Error: 30.2 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: 32.2 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: 32.2 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: 32.6 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: 30.2 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: 30.2 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: 24.4 bits

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

    11. Applied div-invError: 24.4 bits

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

    if 1.09282775403477823e-101 < re

    1. Initial program Error: 33.6 bits

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

    2. Taylor expanded around inf Error: 20.5 bits

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

  4. Final simplificationError: 20.9 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.306963558091192 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;re \leq 1.0928277540347782 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|im\right| \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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)))))