math.sqrt on complex, real part

Percentage Accurate: 42.3% → 85.3%

Time: 4.7s

Alternatives: 9

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: 42.3% 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: 85.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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 * ((im / (re / im)) * -0.5)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision] * -0.5), $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 (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 12.1%

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

      1. +-commutative 12.1%

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

      2. hypot-def 12.1%

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

   3. Simplified 12.1%

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

   4. Taylor expanded in re around -inf 63.4%

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

      1. *-commutative 63.4%

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

      2. unpow2 63.4%

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

      3. associate-/l* 64.8%

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

   6. Simplified 64.8%

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

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

   1. Initial program 45.1%

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

      1. +-commutative 45.1%

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

      2. hypot-def 90.7%

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

   3. Simplified 90.7%

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

4. Final simplification 87.7%

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

Alternative 2: 52.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.65 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq -6.8 \cdot 10^{-256}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{re \cdot re}{im} + 2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.65e+61)
   (* 0.5 (sqrt (* 2.0 (* (/ im (/ re im)) -0.5))))
   (if (<= re -6.8e-256)
     (* 0.5 (sqrt (+ (/ (* re re) im) (* 2.0 (+ re im)))))
     (if (<= re 7.5e-79)
       (* 0.5 (sqrt (* 2.0 (- re im))))
       (* 0.5 (* 2.0 (sqrt re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.65e+61) {
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	} else if (re <= -6.8e-256) {
		tmp = 0.5 * sqrt((((re * re) / im) + (2.0 * (re + im))));
	} else if (re <= 7.5e-79) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.65d+61)) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((im / (re / im)) * (-0.5d0))))
    else if (re <= (-6.8d-256)) then
        tmp = 0.5d0 * sqrt((((re * re) / im) + (2.0d0 * (re + im))))
    else if (re <= 7.5d-79) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.65e+61) {
		tmp = 0.5 * Math.sqrt((2.0 * ((im / (re / im)) * -0.5)));
	} else if (re <= -6.8e-256) {
		tmp = 0.5 * Math.sqrt((((re * re) / im) + (2.0 * (re + im))));
	} else if (re <= 7.5e-79) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.65e+61:
		tmp = 0.5 * math.sqrt((2.0 * ((im / (re / im)) * -0.5)))
	elif re <= -6.8e-256:
		tmp = 0.5 * math.sqrt((((re * re) / im) + (2.0 * (re + im))))
	elif re <= 7.5e-79:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.65e+61)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im / Float64(re / im)) * -0.5))));
	elseif (re <= -6.8e-256)
		tmp = Float64(0.5 * sqrt(Float64(Float64(Float64(re * re) / im) + Float64(2.0 * Float64(re + im)))));
	elseif (re <= 7.5e-79)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.65e+61)
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	elseif (re <= -6.8e-256)
		tmp = 0.5 * sqrt((((re * re) / im) + (2.0 * (re + im))));
	elseif (re <= 7.5e-79)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.65e+61], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -6.8e-256], N[(0.5 * N[Sqrt[N[(N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision] + N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.5e-79], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.64999999999999987e61

   1. Initial program 10.6%

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

      1. +-commutative 10.6%

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

      2. hypot-def 30.8%

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

   3. Simplified 30.8%

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

   4. Taylor expanded in re around -inf 54.6%

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

      1. *-commutative 54.6%

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

      2. unpow2 54.6%

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

      3. associate-/l* 59.2%

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

   6. Simplified 59.2%

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

   if -4.64999999999999987e61 < re < -6.8000000000000001e-256

   1. Initial program 45.5%

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

      1. +-commutative 45.5%

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

      2. hypot-def 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in re around 0 52.6%

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

      1. unpow2 52.6%

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

      2. distribute-lft-out 52.6%

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

   6. Simplified 52.6%

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

   if -6.8000000000000001e-256 < re < 7.49999999999999969e-79

   1. Initial program 47.9%

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

      1. +-commutative 47.9%

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

      2. hypot-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in im around -inf 56.3%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + -1 \cdot im\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 56.3%

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

      2. sub-neg 56.3%

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

   6. Simplified 56.3%

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

   if 7.49999999999999969e-79 < re

   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 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in im around 0 74.5%

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

      1. unpow2 74.5%

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

      2. rem-square-sqrt 75.9%

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

   6. Simplified 75.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot \sqrt{re}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.65 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq -6.8 \cdot 10^{-256}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{re \cdot re}{im} + 2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 3: 59.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.4 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \left(-im\right)}{re}}\\ \mathbf{elif}\;im \leq -1.16 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- re im))))))
   (if (<= im -1.1e+93)
     t_0
     (if (<= im -1.4e+65)
       (* 0.5 (sqrt (/ (* im (- im)) re)))
       (if (<= im -1.16e-54)
         t_0
         (if (<= im 1.55e-120)
           (* 0.5 (* 2.0 (sqrt re)))
           (* 0.5 (sqrt (* 2.0 (+ re im))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (re - im)));
	double tmp;
	if (im <= -1.1e+93) {
		tmp = t_0;
	} else if (im <= -1.4e+65) {
		tmp = 0.5 * sqrt(((im * -im) / re));
	} else if (im <= -1.16e-54) {
		tmp = t_0;
	} else if (im <= 1.55e-120) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	}
	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 = 0.5d0 * sqrt((2.0d0 * (re - im)))
    if (im <= (-1.1d+93)) then
        tmp = t_0
    else if (im <= (-1.4d+65)) then
        tmp = 0.5d0 * sqrt(((im * -im) / re))
    else if (im <= (-1.16d-54)) then
        tmp = t_0
    else if (im <= 1.55d-120) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (re - im)));
	double tmp;
	if (im <= -1.1e+93) {
		tmp = t_0;
	} else if (im <= -1.4e+65) {
		tmp = 0.5 * Math.sqrt(((im * -im) / re));
	} else if (im <= -1.16e-54) {
		tmp = t_0;
	} else if (im <= 1.55e-120) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (re - im)))
	tmp = 0
	if im <= -1.1e+93:
		tmp = t_0
	elif im <= -1.4e+65:
		tmp = 0.5 * math.sqrt(((im * -im) / re))
	elif im <= -1.16e-54:
		tmp = t_0
	elif im <= 1.55e-120:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))))
	tmp = 0.0
	if (im <= -1.1e+93)
		tmp = t_0;
	elseif (im <= -1.4e+65)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * Float64(-im)) / re)));
	elseif (im <= -1.16e-54)
		tmp = t_0;
	elseif (im <= 1.55e-120)
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (re - im)));
	tmp = 0.0;
	if (im <= -1.1e+93)
		tmp = t_0;
	elseif (im <= -1.4e+65)
		tmp = 0.5 * sqrt(((im * -im) / re));
	elseif (im <= -1.16e-54)
		tmp = t_0;
	elseif (im <= 1.55e-120)
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.1e+93], t$95$0, If[LessEqual[im, -1.4e+65], N[(0.5 * N[Sqrt[N[(N[(im * (-im)), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -1.16e-54], t$95$0, If[LessEqual[im, 1.55e-120], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.10000000000000011e93 or -1.3999999999999999e65 < im < -1.16e-54

   1. Initial program 39.3%

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

      1. +-commutative 39.3%

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

      2. hypot-def 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in im around -inf 73.3%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + -1 \cdot im\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 73.3%

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

      2. sub-neg 73.3%

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

   6. Simplified 73.3%

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

   if -1.10000000000000011e93 < im < -1.3999999999999999e65

   1. Initial program 3.2%

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

      1. +-commutative 3.2%

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

      2. hypot-def 19.4%

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

   3. Simplified 19.4%

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

   4. Taylor expanded in re around -inf 83.3%

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

      1. associate-*r/ 83.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{-1 \cdot {im}^{2}}{re}}} \]

      2. neg-mul-1 83.3%

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

      3. unpow2 83.3%

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

      4. distribute-rgt-neg-in 83.3%

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

   6. Simplified 83.3%

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

   if -1.16e-54 < im < 1.5500000000000001e-120

   1. Initial program 39.0%

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

      1. +-commutative 39.0%

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

      2. hypot-def 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in im around 0 47.9%

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

      1. unpow2 47.9%

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

      2. rem-square-sqrt 48.8%

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

   6. Simplified 48.8%

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

   if 1.5500000000000001e-120 < im

   1. Initial program 47.6%

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

      1. +-commutative 47.6%

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

      2. hypot-def 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in re around 0 70.6%

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

      1. distribute-lft-out 70.6%

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

      2. +-commutative 70.6%

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

      3. *-commutative 70.6%

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

      4. +-commutative 70.6%

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

   6. Simplified 70.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right) \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -1.4 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \left(-im\right)}{re}}\\ \mathbf{elif}\;im \leq -1.16 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4: 52.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.65 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq -4 \cdot 10^{-255}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.65e+61)
   (* 0.5 (sqrt (* 2.0 (* (/ im (/ re im)) -0.5))))
   (if (<= re -4e-255)
     (* 0.5 (sqrt (* 2.0 im)))
     (if (<= re 6.2e-78)
       (* 0.5 (sqrt (* 2.0 (- re im))))
       (* 0.5 (* 2.0 (sqrt re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.65e+61) {
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	} else if (re <= -4e-255) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else if (re <= 6.2e-78) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.65d+61)) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((im / (re / im)) * (-0.5d0))))
    else if (re <= (-4d-255)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else if (re <= 6.2d-78) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.65e+61) {
		tmp = 0.5 * Math.sqrt((2.0 * ((im / (re / im)) * -0.5)));
	} else if (re <= -4e-255) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else if (re <= 6.2e-78) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.65e+61:
		tmp = 0.5 * math.sqrt((2.0 * ((im / (re / im)) * -0.5)))
	elif re <= -4e-255:
		tmp = 0.5 * math.sqrt((2.0 * im))
	elif re <= 6.2e-78:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.65e+61)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im / Float64(re / im)) * -0.5))));
	elseif (re <= -4e-255)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	elseif (re <= 6.2e-78)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.65e+61)
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	elseif (re <= -4e-255)
		tmp = 0.5 * sqrt((2.0 * im));
	elseif (re <= 6.2e-78)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.65e+61], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -4e-255], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.2e-78], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.64999999999999987e61

   1. Initial program 10.6%

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

      1. +-commutative 10.6%

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

      2. hypot-def 30.8%

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

   3. Simplified 30.8%

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

   4. Taylor expanded in re around -inf 54.6%

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

      1. *-commutative 54.6%

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

      2. unpow2 54.6%

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

      3. associate-/l* 59.2%

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

   6. Simplified 59.2%

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

   if -4.64999999999999987e61 < re < -4e-255

   1. Initial program 45.5%

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

      1. +-commutative 45.5%

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

      2. hypot-def 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in re around 0 52.4%

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

      1. *-commutative 52.4%

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

   6. Simplified 52.4%

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

   if -4e-255 < re < 6.20000000000000035e-78

   1. Initial program 47.9%

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

      1. +-commutative 47.9%

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

      2. hypot-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in im around -inf 56.3%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + -1 \cdot im\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 56.3%

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

      2. sub-neg 56.3%

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

   6. Simplified 56.3%

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

   if 6.20000000000000035e-78 < re

   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 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in im around 0 74.5%

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

      1. unpow2 74.5%

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

      2. rem-square-sqrt 75.9%

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

   6. Simplified 75.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot \sqrt{re}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.65 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq -4 \cdot 10^{-255}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 5: 60.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{-121}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.1e-55)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im 3.6e-121)
     (* 0.5 (* 2.0 (sqrt re)))
     (* 0.5 (sqrt (* 2.0 (+ re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.1e-55) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= 3.6e-121) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.1d-55)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= 3.6d-121) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.1e-55) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= 3.6e-121) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.1e-55:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= 3.6e-121:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.1e-55)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= 3.6e-121)
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.1e-55)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= 3.6e-121)
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.1e-55], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.6e-121], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.1e-55

   1. Initial program 36.4%

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

      1. +-commutative 36.4%

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

      2. hypot-def 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in im around -inf 67.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + -1 \cdot im\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 67.6%

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

      2. sub-neg 67.6%

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

   6. Simplified 67.6%

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

   if -1.1e-55 < im < 3.59999999999999984e-121

   1. Initial program 39.0%

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

      1. +-commutative 39.0%

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

      2. hypot-def 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in im around 0 47.9%

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

      1. unpow2 47.9%

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

      2. rem-square-sqrt 48.8%

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

   6. Simplified 48.8%

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

   if 3.59999999999999984e-121 < im

   1. Initial program 47.6%

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

      1. +-commutative 47.6%

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

      2. hypot-def 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in re around 0 70.6%

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

      1. distribute-lft-out 70.6%

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

      2. +-commutative 70.6%

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

      3. *-commutative 70.6%

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

      4. +-commutative 70.6%

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

   6. Simplified 70.6%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{-121}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6: 59.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.32e-55)
   (* 0.5 (sqrt (* im -2.0)))
   (if (<= im 2.8e-120) (* 0.5 (* 2.0 (sqrt re))) (* 0.5 (sqrt (* 2.0 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.32e-55) {
		tmp = 0.5 * sqrt((im * -2.0));
	} else if (im <= 2.8e-120) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.32d-55)) then
        tmp = 0.5d0 * sqrt((im * (-2.0d0)))
    else if (im <= 2.8d-120) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.32e-55) {
		tmp = 0.5 * Math.sqrt((im * -2.0));
	} else if (im <= 2.8e-120) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.32e-55:
		tmp = 0.5 * math.sqrt((im * -2.0))
	elif im <= 2.8e-120:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.32e-55)
		tmp = Float64(0.5 * sqrt(Float64(im * -2.0)));
	elseif (im <= 2.8e-120)
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.32e-55)
		tmp = 0.5 * sqrt((im * -2.0));
	elseif (im <= 2.8e-120)
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.32e-55], N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.8e-120], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.31999999999999993e-55

   1. Initial program 36.4%

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

      1. +-commutative 36.4%

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

      2. hypot-def 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in im around -inf 66.6%

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

      1. *-commutative 66.6%

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

   6. Simplified 66.6%

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

   if -1.31999999999999993e-55 < im < 2.79999999999999994e-120

   1. Initial program 39.0%

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

      1. +-commutative 39.0%

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

      2. hypot-def 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in im around 0 47.9%

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

      1. unpow2 47.9%

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

      2. rem-square-sqrt 48.8%

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

   6. Simplified 48.8%

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

   if 2.79999999999999994e-120 < im

   1. Initial program 47.6%

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

      1. +-commutative 47.6%

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

      2. hypot-def 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in re around 0 69.3%

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

      1. *-commutative 69.3%

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

   6. Simplified 69.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 7: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8.5 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 4 \cdot 10^{-121}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -8.5e-55)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im 4e-121) (* 0.5 (* 2.0 (sqrt re))) (* 0.5 (sqrt (* 2.0 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -8.5e-55) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= 4e-121) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-8.5d-55)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= 4d-121) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -8.5e-55) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= 4e-121) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -8.5e-55:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= 4e-121:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -8.5e-55)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= 4e-121)
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -8.5e-55)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= 4e-121)
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -8.5e-55], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4e-121], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.49999999999999968e-55

   1. Initial program 36.4%

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

      1. +-commutative 36.4%

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

      2. hypot-def 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in im around -inf 67.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + -1 \cdot im\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 67.6%

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

      2. sub-neg 67.6%

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

   6. Simplified 67.6%

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

   if -8.49999999999999968e-55 < im < 3.9999999999999999e-121

   1. Initial program 39.0%

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

      1. +-commutative 39.0%

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

      2. hypot-def 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in im around 0 47.9%

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

      1. unpow2 47.9%

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

      2. rem-square-sqrt 48.8%

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

   6. Simplified 48.8%

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

   if 3.9999999999999999e-121 < im

   1. Initial program 47.6%

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

      1. +-commutative 47.6%

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

      2. hypot-def 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in re around 0 69.3%

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

      1. *-commutative 69.3%

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

   6. Simplified 69.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.5 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 4 \cdot 10^{-121}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 8: 53.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{-311}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -4e-311) (* 0.5 (sqrt (* im -2.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -4e-311) {
		tmp = 0.5 * sqrt((im * -2.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-4d-311)) then
        tmp = 0.5d0 * sqrt((im * (-2.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -4e-311) {
		tmp = 0.5 * Math.sqrt((im * -2.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -4e-311:
		tmp = 0.5 * math.sqrt((im * -2.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -4e-311)
		tmp = Float64(0.5 * sqrt(Float64(im * -2.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -4e-311)
		tmp = 0.5 * sqrt((im * -2.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -4e-311], N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.99999999999979e-311

   1. Initial program 36.7%

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

      1. +-commutative 36.7%

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

      2. hypot-def 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in im around -inf 52.1%

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

      1. *-commutative 52.1%

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

   6. Simplified 52.1%

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

   if -3.99999999999979e-311 < im

   1. Initial program 45.7%

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

      1. +-commutative 45.7%

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

      2. hypot-def 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in re around 0 55.4%

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

      1. *-commutative 55.4%

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

   6. Simplified 55.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{-311}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 9: 26.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.3%

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

   1. +-commutative 41.3%

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

   2. hypot-def 81.5%

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

3. Simplified 81.5%

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

4. Taylor expanded in im around -inf 25.4%

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

   1. *-commutative 25.4%

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

6. Simplified 25.4%

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

7. Final simplification 25.4%

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

Developer target: 48.9% 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 2023189 
(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)))))