Average Error: 38.6 → 0.4
Time: 4.3s
Precision: binary64
\[im > 0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq 4.982407308203682 \cdot 10^{-234}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{2}}{\sqrt{re + \mathsf{hypot}\left(im, re\right)}}\\ \end{array} \]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq 4.982407308203682 \cdot 10^{-234}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{2}}{\sqrt{re + \mathsf{hypot}\left(im, 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 4.982407308203682e-234)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (/ (* im (sqrt 2.0)) (sqrt (+ re (hypot im 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 <= 4.982407308203682e-234) {
		tmp = 0.5 * sqrt(2.0 * (hypot(re, im) - re));
	} else {
		tmp = 0.5 * ((im * sqrt(2.0)) / sqrt(re + hypot(im, 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

Derivation

  1. Split input into 2 regimes

  2. if re < 4.98240730820368227e-234

    1. Initial program 31.1

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

    2. Simplified0.3

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

    if 4.98240730820368227e-234 < re

    1. Initial program 48.0

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

    2. Applied flip--_binary6448.0

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

    3. Applied associate-*r/_binary6448.0

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

    4. Applied sqrt-div_binary6448.0

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

    5. Simplified35.2

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}} \]

    6. Simplified26.7

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\color{blue}{\sqrt{re + \mathsf{hypot}\left(im, re\right)}}} \]

    7. Taylor expanded in im around 0 0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.982407308203682 \cdot 10^{-234}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{2}}{\sqrt{re + \mathsf{hypot}\left(im, re\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))