math.sqrt on complex, real part

Percentage Accurate: 41.8% → 89.5%

Time: 16.2s

Alternatives: 8

Speedup: 2.0×

Specification

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

Wolfram

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]

Initial Program: 41.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

Wolfram

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]

Alternative 1: 89.5% accurate, 0.7× speedup

Math

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

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im im)))) 0.0)
   (* 0.5 (* (sqrt im) (sqrt (/ (- im) re))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if ((re + sqrt(((re * re) + (im * im)))) <= 0.0) {
		tmp = 0.5 * (sqrt(im) * sqrt((-im / re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if ((re + Math.sqrt(((re * re) + (im * im)))) <= 0.0) {
		tmp = 0.5 * (Math.sqrt(im) * Math.sqrt((-im / re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if (re + math.sqrt(((re * re) + (im * im)))) <= 0.0:
		tmp = 0.5 * (math.sqrt(im) * math.sqrt((-im / re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[Sqrt[im], $MachinePrecision] * N[Sqrt[N[((-im) / re), $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]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 9.5%

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

      2. hypot-def 19.3%

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

   3. Simplified 19.3%

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

   4. Taylor expanded in re around -inf 42.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 42.2%

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

      2. unpow2 42.2%

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

      3. associate-/l* 52.4%

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

   6. Simplified 52.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 52.2%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)\right)} \]

      2. expm1-udef 14.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)} - 1\right)} \]

      3. *-commutative 14.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right) \cdot 2}}\right)} - 1\right) \]

      4. associate-*l* 14.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{im}{\frac{re}{im}} \cdot \left(-0.5 \cdot 2\right)}}\right)} - 1\right) \]

      5. associate-/r/ 14.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      6. *-commutative 14.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      7. metadata-eval 14.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-1}}\right)} - 1\right) \]

   8. Applied egg-rr 14.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 52.3%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)\right)} \]

      2. expm1-log1p 52.5%

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

      3. associate-*l* 52.5%

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

      4. associate-*l/ 52.5%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \color{blue}{\frac{im \cdot -1}{re}}} \]

      5. *-commutative 52.5%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \frac{\color{blue}{-1 \cdot im}}{re}} \]

      6. neg-mul-1 52.5%

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

   10. Simplified 52.5%

       \[\leadsto 0.5 \cdot \color{blue}{\sqrt{im \cdot \frac{-im}{re}}} \]
   11. Step-by-step derivation

       1. distribute-frac-neg 52.5%

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

       2. distribute-rgt-neg-out 52.5%

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

       3. mul-1-neg 52.5%

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

       4. *-commutative 52.5%

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

       5. associate-*l* 52.5%

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

       6. sqrt-prod 50.2%

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

   12. Applied egg-rr 50.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im} \cdot \sqrt{\frac{im}{re} \cdot -1}\right)} \]
   13. Step-by-step derivation

       1. *-commutative 50.2%

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

       2. associate-*r/ 50.2%

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

       3. neg-mul-1 50.2%

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

   14. Simplified 50.2%

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

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

   1. Initial program 46.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 46.6%

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

      2. hypot-def 90.3%

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

   3. Simplified 90.3%

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

4. Final simplification 82.6%

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

Alternative 2: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -2.8e+77)
   (* 0.5 (sqrt (/ (- im) (/ re im))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -2.8e+77) {
		tmp = 0.5 * sqrt((-im / (re / im)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -2.8e+77) {
		tmp = 0.5 * Math.sqrt((-im / (re / im)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -2.8e+77:
		tmp = 0.5 * math.sqrt((-im / (re / im)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -2.8e+77)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) / Float64(re / im))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.8e+77)
		tmp = 0.5 * sqrt((-im / (re / im)));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -2.8e+77], N[(0.5 * N[Sqrt[N[((-im) / N[(re / im), $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]]

Derivation

1. Split input into 2 regimes

2. if re < -2.8e77

   1. Initial program 8.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 8.9%

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

      2. hypot-def 35.6%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in re around -inf 59.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 59.2%

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

      2. unpow2 59.2%

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

      3. associate-/l* 63.8%

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

   6. Simplified 63.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 63.3%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)\right)} \]

      2. expm1-udef 32.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)} - 1\right)} \]

      3. *-commutative 32.5%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right) \cdot 2}}\right)} - 1\right) \]

      4. associate-*l* 32.5%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{im}{\frac{re}{im}} \cdot \left(-0.5 \cdot 2\right)}}\right)} - 1\right) \]

      5. associate-/r/ 32.5%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      6. *-commutative 32.5%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      7. metadata-eval 32.5%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-1}}\right)} - 1\right) \]

   8. Applied egg-rr 32.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 63.2%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)\right)} \]

      2. expm1-log1p 63.7%

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

      3. associate-*l* 63.7%

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

      4. associate-*l/ 63.7%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \color{blue}{\frac{im \cdot -1}{re}}} \]

      5. *-commutative 63.7%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \frac{\color{blue}{-1 \cdot im}}{re}} \]

      6. neg-mul-1 63.7%

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

   10. Simplified 63.7%

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

   11. Taylor expanded in im around 0 59.2%

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   12. Step-by-step derivation

       1. mul-1-neg 59.2%

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

       2. unpow2 59.2%

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

       3. associate-/l* 63.8%

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

       4. distribute-neg-frac 63.8%

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

   13. Simplified 63.8%

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

   if -2.8e77 < re

   1. Initial program 46.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 46.7%

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

      2. hypot-def 86.5%

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

   3. Simplified 86.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3: 70.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{re \cdot re}{im} + 2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -4.8e+53)
   (* 0.5 (sqrt (/ (- im) (/ re im))))
   (if (<= re 1.12e-44)
     (* 0.5 (sqrt (+ (/ (* re re) im) (* 2.0 (+ re im)))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -4.8e+53) {
		tmp = 0.5 * sqrt((-im / (re / im)));
	} else if (re <= 1.12e-44) {
		tmp = 0.5 * sqrt((((re * re) / im) + (2.0 * (re + im))));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.8d+53)) then
        tmp = 0.5d0 * sqrt((-im / (re / im)))
    else if (re <= 1.12d-44) then
        tmp = 0.5d0 * sqrt((((re * re) / im) + (2.0d0 * (re + im))))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -4.8e+53) {
		tmp = 0.5 * Math.sqrt((-im / (re / im)));
	} else if (re <= 1.12e-44) {
		tmp = 0.5 * Math.sqrt((((re * re) / im) + (2.0 * (re + im))));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -4.8e+53:
		tmp = 0.5 * math.sqrt((-im / (re / im)))
	elif re <= 1.12e-44:
		tmp = 0.5 * math.sqrt((((re * re) / im) + (2.0 * (re + im))))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -4.8e+53)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) / Float64(re / im))));
	elseif (re <= 1.12e-44)
		tmp = Float64(0.5 * sqrt(Float64(Float64(Float64(re * re) / im) + Float64(2.0 * Float64(re + im)))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.8e+53)
		tmp = 0.5 * sqrt((-im / (re / im)));
	elseif (re <= 1.12e-44)
		tmp = 0.5 * sqrt((((re * re) / im) + (2.0 * (re + im))));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -4.8e+53], N[(0.5 * N[Sqrt[N[((-im) / N[(re / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.12e-44], N[(0.5 * N[Sqrt[N[(N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision] + N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.8e53

   1. Initial program 12.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 12.0%

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

      2. hypot-def 38.1%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in re around -inf 57.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 57.8%

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

      2. unpow2 57.8%

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

      3. associate-/l* 61.9%

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

   6. Simplified 61.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 61.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)\right)} \]

      2. expm1-udef 31.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)} - 1\right)} \]

      3. *-commutative 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right) \cdot 2}}\right)} - 1\right) \]

      4. associate-*l* 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{im}{\frac{re}{im}} \cdot \left(-0.5 \cdot 2\right)}}\right)} - 1\right) \]

      5. associate-/r/ 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      6. *-commutative 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      7. metadata-eval 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-1}}\right)} - 1\right) \]

   8. Applied egg-rr 31.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 61.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)\right)} \]

      2. expm1-log1p 61.9%

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

      3. associate-*l* 61.9%

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

      4. associate-*l/ 61.9%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \color{blue}{\frac{im \cdot -1}{re}}} \]

      5. *-commutative 61.9%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \frac{\color{blue}{-1 \cdot im}}{re}} \]

      6. neg-mul-1 61.9%

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

   10. Simplified 61.9%

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

   11. Taylor expanded in im around 0 57.8%

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   12. Step-by-step derivation

       1. mul-1-neg 57.8%

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

       2. unpow2 57.8%

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

       3. associate-/l* 61.9%

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

       4. distribute-neg-frac 61.9%

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

   13. Simplified 61.9%

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

   if -4.8e53 < re < 1.1200000000000001e-44

   1. Initial program 49.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 49.5%

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

      2. hypot-def 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in re around 0 36.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{re}^{2}}{im} + \left(2 \cdot im + 2 \cdot re\right)}} \]
   5. Step-by-step derivation

      1. unpow2 36.1%

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

      2. distribute-lft-out 36.1%

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

   6. Simplified 36.1%

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

   if 1.1200000000000001e-44 < re

   1. Initial program 41.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 41.4%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 76.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]
   5. Step-by-step derivation

      1. unpow2 76.2%

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

      2. rem-square-sqrt 77.7%

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

   6. Simplified 77.7%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{re \cdot re}{im} + 2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 4: 70.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -4.2e+47)
   (* 0.5 (sqrt (* im (/ (- im) re))))
   (if (<= re 4.1e-45) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (* 2.0 (sqrt re))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -4.2e+47) {
		tmp = 0.5 * sqrt((im * (-im / re)));
	} else if (re <= 4.1e-45) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.2d+47)) then
        tmp = 0.5d0 * sqrt((im * (-im / re)))
    else if (re <= 4.1d-45) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -4.2e+47) {
		tmp = 0.5 * Math.sqrt((im * (-im / re)));
	} else if (re <= 4.1e-45) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -4.2e+47:
		tmp = 0.5 * math.sqrt((im * (-im / re)))
	elif re <= 4.1e-45:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -4.2e+47)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(Float64(-im) / re))));
	elseif (re <= 4.1e-45)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.2e+47)
		tmp = 0.5 * sqrt((im * (-im / re)));
	elseif (re <= 4.1e-45)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -4.2e+47], N[(0.5 * N[Sqrt[N[(im * N[((-im) / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.1e-45], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.2e47

   1. Initial program 12.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 12.0%

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

      2. hypot-def 38.1%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in re around -inf 57.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 57.8%

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

      2. unpow2 57.8%

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

      3. associate-/l* 61.9%

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

   6. Simplified 61.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 61.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)\right)} \]

      2. expm1-udef 31.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)} - 1\right)} \]

      3. *-commutative 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right) \cdot 2}}\right)} - 1\right) \]

      4. associate-*l* 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{im}{\frac{re}{im}} \cdot \left(-0.5 \cdot 2\right)}}\right)} - 1\right) \]

      5. associate-/r/ 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      6. *-commutative 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      7. metadata-eval 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-1}}\right)} - 1\right) \]

   8. Applied egg-rr 31.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 61.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)\right)} \]

      2. expm1-log1p 61.9%

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

      3. associate-*l* 61.9%

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

      4. associate-*l/ 61.9%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \color{blue}{\frac{im \cdot -1}{re}}} \]

      5. *-commutative 61.9%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \frac{\color{blue}{-1 \cdot im}}{re}} \]

      6. neg-mul-1 61.9%

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

   10. Simplified 61.9%

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

   if -4.2e47 < re < 4.0999999999999999e-45

   1. Initial program 49.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 49.5%

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

      2. hypot-def 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in re around 0 35.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 35.9%

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

   6. Simplified 35.9%

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

   if 4.0999999999999999e-45 < re

   1. Initial program 41.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 41.4%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 76.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]
   5. Step-by-step derivation

      1. unpow2 76.2%

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

      2. rem-square-sqrt 77.7%

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

   6. Simplified 77.7%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 5: 70.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -1.35e+39)
   (* 0.5 (sqrt (/ (- im) (/ re im))))
   (if (<= re 6.6e-44) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (* 2.0 (sqrt re))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -1.35e+39) {
		tmp = 0.5 * sqrt((-im / (re / im)));
	} else if (re <= 6.6e-44) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.35d+39)) then
        tmp = 0.5d0 * sqrt((-im / (re / im)))
    else if (re <= 6.6d-44) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -1.35e+39) {
		tmp = 0.5 * Math.sqrt((-im / (re / im)));
	} else if (re <= 6.6e-44) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -1.35e+39:
		tmp = 0.5 * math.sqrt((-im / (re / im)))
	elif re <= 6.6e-44:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -1.35e+39)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) / Float64(re / im))));
	elseif (re <= 6.6e-44)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.35e+39)
		tmp = 0.5 * sqrt((-im / (re / im)));
	elseif (re <= 6.6e-44)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -1.35e+39], N[(0.5 * N[Sqrt[N[((-im) / N[(re / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.6e-44], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.35000000000000002e39

   1. Initial program 12.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 12.0%

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

      2. hypot-def 38.1%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in re around -inf 57.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 57.8%

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

      2. unpow2 57.8%

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

      3. associate-/l* 61.9%

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

   6. Simplified 61.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 61.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)\right)} \]

      2. expm1-udef 31.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)} - 1\right)} \]

      3. *-commutative 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right) \cdot 2}}\right)} - 1\right) \]

      4. associate-*l* 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{im}{\frac{re}{im}} \cdot \left(-0.5 \cdot 2\right)}}\right)} - 1\right) \]

      5. associate-/r/ 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      6. *-commutative 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      7. metadata-eval 31.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-1}}\right)} - 1\right) \]

   8. Applied egg-rr 31.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 61.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)\right)} \]

      2. expm1-log1p 61.9%

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

      3. associate-*l* 61.9%

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

      4. associate-*l/ 61.9%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \color{blue}{\frac{im \cdot -1}{re}}} \]

      5. *-commutative 61.9%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \frac{\color{blue}{-1 \cdot im}}{re}} \]

      6. neg-mul-1 61.9%

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

   10. Simplified 61.9%

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

   11. Taylor expanded in im around 0 57.8%

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   12. Step-by-step derivation

       1. mul-1-neg 57.8%

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

       2. unpow2 57.8%

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

       3. associate-/l* 61.9%

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

       4. distribute-neg-frac 61.9%

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

   13. Simplified 61.9%

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

   if -1.35000000000000002e39 < re < 6.60000000000000011e-44

   1. Initial program 49.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 49.5%

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

      2. hypot-def 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in re around 0 35.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 35.9%

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

   6. Simplified 35.9%

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

   if 6.60000000000000011e-44 < re

   1. Initial program 41.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 41.4%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 76.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]
   5. Step-by-step derivation

      1. unpow2 76.2%

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

      2. rem-square-sqrt 77.7%

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

   6. Simplified 77.7%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 6: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -1.65 \cdot 10^{+206}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{re}}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -1.65e+206)
   (* 0.5 (sqrt (/ (* im im) re)))
   (if (<= re 3.1e-45) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (* 2.0 (sqrt re))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -1.65e+206) {
		tmp = 0.5 * sqrt(((im * im) / re));
	} else if (re <= 3.1e-45) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.65d+206)) then
        tmp = 0.5d0 * sqrt(((im * im) / re))
    else if (re <= 3.1d-45) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -1.65e+206) {
		tmp = 0.5 * Math.sqrt(((im * im) / re));
	} else if (re <= 3.1e-45) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -1.65e+206:
		tmp = 0.5 * math.sqrt(((im * im) / re))
	elif re <= 3.1e-45:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -1.65e+206)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * im) / re)));
	elseif (re <= 3.1e-45)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.65e+206)
		tmp = 0.5 * sqrt(((im * im) / re));
	elseif (re <= 3.1e-45)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -1.65e+206], N[(0.5 * N[Sqrt[N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.1e-45], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.64999999999999992e206

   1. Initial program 2.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 2.2%

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

      2. hypot-def 38.5%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in re around -inf 48.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 48.0%

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

      2. unpow2 48.0%

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

      3. associate-/l* 58.2%

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

   6. Simplified 58.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 57.6%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)\right)} \]

      2. expm1-udef 43.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)} - 1\right)} \]

      3. *-commutative 43.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right) \cdot 2}}\right)} - 1\right) \]

      4. associate-*l* 43.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{im}{\frac{re}{im}} \cdot \left(-0.5 \cdot 2\right)}}\right)} - 1\right) \]

      5. associate-/r/ 43.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      6. *-commutative 43.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      7. metadata-eval 43.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-1}}\right)} - 1\right) \]

   8. Applied egg-rr 43.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 57.6%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)\right)} \]

      2. expm1-log1p 58.2%

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

      3. associate-*l* 58.2%

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

      4. associate-*l/ 58.2%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \color{blue}{\frac{im \cdot -1}{re}}} \]

      5. *-commutative 58.2%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \frac{\color{blue}{-1 \cdot im}}{re}} \]

      6. neg-mul-1 58.2%

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

   10. Simplified 58.2%

       \[\leadsto 0.5 \cdot \color{blue}{\sqrt{im \cdot \frac{-im}{re}}} \]
   11. Step-by-step derivation

       1. pow1/2 58.2%

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

       2. add-sqr-sqrt 58.2%

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

       3. sqrt-unprod 38.6%

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

       4. swap-sqr 33.8%

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

       5. distribute-frac-neg 33.8%

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

       6. distribute-frac-neg 33.8%

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

       7. sqr-neg 33.8%

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

       8. swap-sqr 38.6%

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

       9. sqrt-unprod 32.6%

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

       10. add-sqr-sqrt 32.6%

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

       11. clear-num 32.6%

           \[\leadsto 0.5 \cdot {\left(im \cdot \color{blue}{\frac{1}{\frac{re}{im}}}\right)}^{0.5} \]

       12. un-div-inv 32.6%

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

   12. Applied egg-rr 32.6%

       \[\leadsto 0.5 \cdot \color{blue}{{\left(\frac{im}{\frac{re}{im}}\right)}^{0.5}} \]
   13. Step-by-step derivation

       1. unpow1/2 32.6%

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

       2. associate-/l* 32.6%

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

   14. Simplified 32.6%

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

   if -1.64999999999999992e206 < re < 3.1000000000000001e-45

   1. Initial program 43.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 43.2%

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

      2. hypot-def 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in re around 0 32.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 32.2%

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

   6. Simplified 32.2%

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

   if 3.1000000000000001e-45 < re

   1. Initial program 41.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 41.4%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 76.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]
   5. Step-by-step derivation

      1. unpow2 76.2%

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

      2. rem-square-sqrt 77.7%

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

   6. Simplified 77.7%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.65 \cdot 10^{+206}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{re}}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 7: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 2.12 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re 2.12e-44) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (* 2.0 (sqrt re)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= 2.12e-44) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.12d-44) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= 2.12e-44) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= 2.12e-44:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= 2.12e-44)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.12e-44)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, 2.12e-44], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.1199999999999999e-44

   1. Initial program 38.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 38.8%

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

      2. hypot-def 68.5%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in re around 0 29.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 29.4%

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

   6. Simplified 29.4%

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

   if 2.1199999999999999e-44 < re

   1. Initial program 41.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 41.4%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 76.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]
   5. Step-by-step derivation

      1. unpow2 76.2%

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

      2. rem-square-sqrt 77.7%

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

   6. Simplified 77.7%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.12 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 8: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ 0.5 \cdot \sqrt{im \cdot 2} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (* 0.5 (sqrt (* im 2.0))))

C

im = abs(im);
double code(double re, double im) {
	return 0.5 * sqrt((im * 2.0));
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((im * 2.0d0))
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return 0.5 * Math.sqrt((im * 2.0));
}

Python

im = abs(im)
def code(re, im):
	return 0.5 * math.sqrt((im * 2.0))

Julia

im = abs(im)
function code(re, im)
	return Float64(0.5 * sqrt(Float64(im * 2.0)))
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = 0.5 * sqrt((im * 2.0));
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.5%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
2. Step-by-step derivation

   1. +-commutative 39.5%

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

   2. hypot-def 76.7%

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

3. Simplified 76.7%

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

4. Taylor expanded in re around 0 26.6%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
5. Step-by-step derivation

   1. *-commutative 26.6%

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

6. Simplified 26.6%

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

7. Final simplification 26.6%

   \[\leadsto 0.5 \cdot \sqrt{im \cdot 2} \]

Developer target: 48.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023278 
(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)))))