math.sqrt on complex, imaginary part, im greater than 0 branch

Percentage Accurate: 40.9% → 90.3%

Time: 9.0s

Alternatives: 6

Speedup: 1.9×

Specification

\[im > 0\]

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: 40.9% 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: 90.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $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 7.6%

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

   3. Taylor expanded in re around inf 39.9%

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

      1. sqrt-div 50.3%

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

      2. sqrt-pow1 92.3%

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

      3. metadata-eval 92.3%

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

      4. pow1 92.3%

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

      5. div-inv 91.9%

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

   5. Applied egg-rr 91.9%

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

      1. associate-*r/ 92.3%

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

      2. *-rgt-identity 92.3%

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

   7. Simplified 92.3%

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

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

   1. Initial program 45.0%

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

      1. pow1 45.0%

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

   4. Applied egg-rr 91.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 91.4%

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}} \]

      2. *-commutative 91.4%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

      3. associate-*r* 91.4%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      4. metadata-eval 91.4%

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

   6. Simplified 91.4%

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

Alternative 2: 74.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9e-71)
   (sqrt (- re))
   (if (<= re 1.65e-83) (sqrt (* 0.5 (- im re))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9e-71) {
		tmp = sqrt(-re);
	} else if (re <= 1.65e-83) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) / 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 <= (-9d-71)) then
        tmp = sqrt(-re)
    else if (re <= 1.65d-83) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9e-71) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.65e-83) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9e-71:
		tmp = math.sqrt(-re)
	elif re <= 1.65e-83:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9e-71)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.65e-83)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9e-71)
		tmp = sqrt(-re);
	elseif (re <= 1.65e-83)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9e-71], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.65e-83], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -9.0000000000000004e-71

   1. Initial program 44.2%

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

   3. Taylor expanded in re around -inf 81.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 81.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 81.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 81.4%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{re \cdot -4}} \cdot \sqrt{0.5 \cdot \sqrt{re \cdot -4}}} \]

      2. sqrt-unprod 81.8%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{re \cdot -4}\right) \cdot \left(0.5 \cdot \sqrt{re \cdot -4}\right)}} \]

      3. *-commutative 81.8%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot -4} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{re \cdot -4}\right)} \]

      4. *-commutative 81.8%

         \[\leadsto \sqrt{\left(\sqrt{re \cdot -4} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{re \cdot -4} \cdot 0.5\right)}} \]

      5. swap-sqr 81.8%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot -4} \cdot \sqrt{re \cdot -4}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 81.8%

         \[\leadsto \sqrt{\color{blue}{\left(re \cdot -4\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. metadata-eval 81.8%

         \[\leadsto \sqrt{\left(re \cdot -4\right) \cdot \color{blue}{0.25}} \]

   7. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\sqrt{\left(re \cdot -4\right) \cdot 0.25}} \]
   8. Step-by-step derivation

      1. associate-*l* 81.8%

         \[\leadsto \sqrt{\color{blue}{re \cdot \left(-4 \cdot 0.25\right)}} \]

      2. metadata-eval 81.8%

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

      3. *-commutative 81.8%

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

      4. mul-1-neg 81.8%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 81.8%

      \[\leadsto \color{blue}{\sqrt{-re}} \]

   if -9.0000000000000004e-71 < re < 1.65e-83

   1. Initial program 51.5%

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

      1. pow1 51.5%

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

   4. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 87.6%

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}} \]

      2. *-commutative 87.6%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

      3. associate-*r* 87.6%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      4. metadata-eval 87.6%

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

   6. Simplified 87.6%

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

   7. Taylor expanded in re around 0 75.6%

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

      1. neg-mul-1 75.6%

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

      2. unsub-neg 75.6%

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

   9. Simplified 75.6%

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

   if 1.65e-83 < re

   1. Initial program 17.2%

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

   3. Taylor expanded in re around inf 42.2%

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

      1. sqrt-div 53.8%

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

      2. sqrt-pow1 75.6%

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

      3. metadata-eval 75.6%

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

      4. pow1 75.6%

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

      5. clear-num 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. un-div-inv 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. associate-/r/ 75.5%

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

      2. associate-*l/ 75.7%

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

      3. *-commutative 75.7%

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

   9. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.4 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 7.4 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.4e-71)
   (sqrt (- re))
   (if (<= re 7.4e-82) (sqrt (* 0.5 (- im re))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.4e-71) {
		tmp = sqrt(-re);
	} else if (re <= 7.4e-82) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = 0.5 * (im / 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 <= (-1.4d-71)) then
        tmp = sqrt(-re)
    else if (re <= 7.4d-82) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.4e-71) {
		tmp = Math.sqrt(-re);
	} else if (re <= 7.4e-82) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.4e-71:
		tmp = math.sqrt(-re)
	elif re <= 7.4e-82:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.4e-71)
		tmp = sqrt(Float64(-re));
	elseif (re <= 7.4e-82)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.4e-71)
		tmp = sqrt(-re);
	elseif (re <= 7.4e-82)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.4e-71], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 7.4e-82], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.4e-71

   1. Initial program 44.2%

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

   3. Taylor expanded in re around -inf 81.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 81.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 81.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 81.4%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{re \cdot -4}} \cdot \sqrt{0.5 \cdot \sqrt{re \cdot -4}}} \]

      2. sqrt-unprod 81.8%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{re \cdot -4}\right) \cdot \left(0.5 \cdot \sqrt{re \cdot -4}\right)}} \]

      3. *-commutative 81.8%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot -4} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{re \cdot -4}\right)} \]

      4. *-commutative 81.8%

         \[\leadsto \sqrt{\left(\sqrt{re \cdot -4} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{re \cdot -4} \cdot 0.5\right)}} \]

      5. swap-sqr 81.8%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot -4} \cdot \sqrt{re \cdot -4}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 81.8%

         \[\leadsto \sqrt{\color{blue}{\left(re \cdot -4\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. metadata-eval 81.8%

         \[\leadsto \sqrt{\left(re \cdot -4\right) \cdot \color{blue}{0.25}} \]

   7. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\sqrt{\left(re \cdot -4\right) \cdot 0.25}} \]
   8. Step-by-step derivation

      1. associate-*l* 81.8%

         \[\leadsto \sqrt{\color{blue}{re \cdot \left(-4 \cdot 0.25\right)}} \]

      2. metadata-eval 81.8%

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

      3. *-commutative 81.8%

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

      4. mul-1-neg 81.8%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 81.8%

      \[\leadsto \color{blue}{\sqrt{-re}} \]

   if -1.4e-71 < re < 7.4000000000000002e-82

   1. Initial program 51.5%

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

      1. pow1 51.5%

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

   4. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 87.6%

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}} \]

      2. *-commutative 87.6%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

      3. associate-*r* 87.6%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      4. metadata-eval 87.6%

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

   6. Simplified 87.6%

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

   7. Taylor expanded in re around 0 75.6%

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

      1. neg-mul-1 75.6%

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

      2. unsub-neg 75.6%

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

   9. Simplified 75.6%

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

   if 7.4000000000000002e-82 < re

   1. Initial program 17.2%

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

   3. Taylor expanded in re around inf 42.2%

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

      1. sqrt-div 53.8%

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

      2. sqrt-pow1 75.6%

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

      3. metadata-eval 75.6%

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

      4. pow1 75.6%

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

      5. div-inv 75.5%

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

   5. Applied egg-rr 75.5%

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

      1. associate-*r/ 75.6%

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

      2. *-rgt-identity 75.6%

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

   7. Simplified 75.6%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 61.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.3e-133)
   (sqrt (- re))
   (if (<= re 6.6e+146) (sqrt (* im 0.5)) 0.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.3e-133) {
		tmp = sqrt(-re);
	} else if (re <= 6.6e+146) {
		tmp = sqrt((im * 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.3d-133)) then
        tmp = sqrt(-re)
    else if (re <= 6.6d+146) then
        tmp = sqrt((im * 0.5d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.3e-133) {
		tmp = Math.sqrt(-re);
	} else if (re <= 6.6e+146) {
		tmp = Math.sqrt((im * 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.3e-133:
		tmp = math.sqrt(-re)
	elif re <= 6.6e+146:
		tmp = math.sqrt((im * 0.5))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.3e-133)
		tmp = sqrt(Float64(-re));
	elseif (re <= 6.6e+146)
		tmp = sqrt(Float64(im * 0.5));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.3e-133)
		tmp = sqrt(-re);
	elseif (re <= 6.6e+146)
		tmp = sqrt((im * 0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.3e-133], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 6.6e+146], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if re < -3.30000000000000009e-133

   1. Initial program 50.0%

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

   3. Taylor expanded in re around -inf 78.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 78.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 78.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 77.6%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{re \cdot -4}} \cdot \sqrt{0.5 \cdot \sqrt{re \cdot -4}}} \]

      2. sqrt-unprod 78.0%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{re \cdot -4}\right) \cdot \left(0.5 \cdot \sqrt{re \cdot -4}\right)}} \]

      3. *-commutative 78.0%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot -4} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{re \cdot -4}\right)} \]

      4. *-commutative 78.0%

         \[\leadsto \sqrt{\left(\sqrt{re \cdot -4} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{re \cdot -4} \cdot 0.5\right)}} \]

      5. swap-sqr 78.0%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot -4} \cdot \sqrt{re \cdot -4}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 78.0%

         \[\leadsto \sqrt{\color{blue}{\left(re \cdot -4\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. metadata-eval 78.0%

         \[\leadsto \sqrt{\left(re \cdot -4\right) \cdot \color{blue}{0.25}} \]

   7. Applied egg-rr 78.0%

      \[\leadsto \color{blue}{\sqrt{\left(re \cdot -4\right) \cdot 0.25}} \]
   8. Step-by-step derivation

      1. associate-*l* 78.0%

         \[\leadsto \sqrt{\color{blue}{re \cdot \left(-4 \cdot 0.25\right)}} \]

      2. metadata-eval 78.0%

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

      3. *-commutative 78.0%

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

      4. mul-1-neg 78.0%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 78.0%

      \[\leadsto \color{blue}{\sqrt{-re}} \]

   if -3.30000000000000009e-133 < re < 6.60000000000000032e146

   1. Initial program 38.7%

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

      1. pow1 38.7%

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

   4. Applied egg-rr 71.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 71.1%

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}} \]

      2. *-commutative 71.1%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

      3. associate-*r* 71.1%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      4. metadata-eval 71.1%

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

   6. Simplified 71.1%

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

   7. Taylor expanded in re around 0 64.4%

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

   if 6.60000000000000032e146 < re

   1. Initial program 2.7%

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

   3. Taylor expanded in re around inf 35.0%

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

   4. Taylor expanded in re around 0 35.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 30.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -5e-310) (sqrt (- re)) 0.0))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = sqrt(-re);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5d-310)) then
        tmp = sqrt(-re)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-310:
		tmp = math.sqrt(-re)
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = sqrt(Float64(-re));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e-310)
		tmp = sqrt(-re);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-310], N[Sqrt[(-re)], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 49.7%

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

   3. Taylor expanded in re around -inf 64.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 64.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 64.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 64.0%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{re \cdot -4}} \cdot \sqrt{0.5 \cdot \sqrt{re \cdot -4}}} \]

      2. sqrt-unprod 64.4%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{re \cdot -4}\right) \cdot \left(0.5 \cdot \sqrt{re \cdot -4}\right)}} \]

      3. *-commutative 64.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot -4} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{re \cdot -4}\right)} \]

      4. *-commutative 64.4%

         \[\leadsto \sqrt{\left(\sqrt{re \cdot -4} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{re \cdot -4} \cdot 0.5\right)}} \]

      5. swap-sqr 64.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot -4} \cdot \sqrt{re \cdot -4}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 64.4%

         \[\leadsto \sqrt{\color{blue}{\left(re \cdot -4\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. metadata-eval 64.4%

         \[\leadsto \sqrt{\left(re \cdot -4\right) \cdot \color{blue}{0.25}} \]

   7. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{\sqrt{\left(re \cdot -4\right) \cdot 0.25}} \]
   8. Step-by-step derivation

      1. associate-*l* 64.4%

         \[\leadsto \sqrt{\color{blue}{re \cdot \left(-4 \cdot 0.25\right)}} \]

      2. metadata-eval 64.4%

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

      3. *-commutative 64.4%

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

      4. mul-1-neg 64.4%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 64.4%

      \[\leadsto \color{blue}{\sqrt{-re}} \]

   if -4.999999999999985e-310 < re

   1. Initial program 27.1%

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

   3. Taylor expanded in re around inf 13.0%

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

   4. Taylor expanded in re around 0 13.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 6.3% accurate, 213.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 37.9%

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

3. Taylor expanded in re around inf 8.1%

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

4. Taylor expanded in re around 0 8.1%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))