Average Error: 38.5 → 11.6
Time: 4.9s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -9.46019038849945 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -9.46019038849945e+108)
   (* 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 (re <= -9.46019038849945e+108) {
		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;
}

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}
↓
public static double code(double re, double im) {
	double tmp;
	if (re <= -9.46019038849945e+108) {
		tmp = 0.5 * Math.sqrt((2.0 * (-0.5 * (Math.pow(im, 2.0) / re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))
↓
def code(re, im):
	tmp = 0
	if re <= -9.46019038849945e+108:
		tmp = 0.5 * math.sqrt((2.0 * (-0.5 * (math.pow(im, 2.0) / re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end
↓
function code(re, im)
	tmp = 0.0
	if (re <= -9.46019038849945e+108)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(-0.5 * Float64((im ^ 2.0) / re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end
↓
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9.46019038849945e+108)
		tmp = 0.5 * sqrt((2.0 * (-0.5 * ((im ^ 2.0) / re))));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[re_, im_] := If[LessEqual[re, -9.46019038849945e+108], N[(0.5 * N[Sqrt[N[(2.0 * N[(-0.5 * N[(N[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -9.46019038849945 \cdot 10^{+108}:\\
\;\;\;\;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}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.5
Target33.3
Herbie11.6
\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if re < -9.4601903884994502e108

    1. Initial program 61.6

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

    2. Simplified39.6

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

    3. Taylor expanded in re around -inf 31.3

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

    if -9.4601903884994502e108 < re

    1. Initial program 34.0

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

    2. Simplified7.8

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.46019038849945 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(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)))))