Average Error: 38.2 → 9.9
Time: 6.7s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\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 (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (* 0.5 (sqrt (* 2.0 (* -0.5 (/ (pow im 2.0) re)))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re 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 (sqrt(2.0 * (re + sqrt((re * re) + (im * im)))) <= 0.0) {
		tmp = 0.5 * sqrt(2.0 * (-0.5 * (pow(im, 2.0) / re)));
	} else {
		tmp = 0.5 * sqrt(2.0 * (re + hypot(re, 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

Original38.2
Target33.2
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

    1. Initial program 57.6

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

    2. Simplified57.6

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

    3. Taylor expanded in re around -inf 29.9

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

    if 0.0 < (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

    1. Initial program 35.6

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

    2. Simplified7.1

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021224 
(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)))))