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

Percentage Accurate: 41.8% → 88.5%

Time: 9.3s

Alternatives: 6

Speedup: 2.0×

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: 41.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.00037:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.00037)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.00037) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.00037) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.00037:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.00037)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, 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 <= 0.00037)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.00037], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.6999999999999999e-4

   1. Initial program 49.3%

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

      1. sub-neg 49.3%

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

      2. sqr-neg 49.3%

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

      3. sub-neg 49.3%

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

      4. sqr-neg 49.3%

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

      5. hypot-define 94.3%

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

   3. Simplified 94.3%

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

   if 3.6999999999999999e-4 < re

   1. Initial program 8.7%

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

   3. Taylor expanded in re around inf 39.3%

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

      1. sqrt-div 50.4%

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

      2. sqrt-pow1 82.9%

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

      3. metadata-eval 82.9%

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

      4. pow1 82.9%

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

      5. div-inv 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. associate-*r/ 82.9%

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

      2. *-rgt-identity 82.9%

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

   7. Simplified 82.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.00037:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{re \cdot -2} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.2e+32)
   (* 0.5 (* (sqrt (* re -2.0)) (sqrt 2.0)))
   (if (<= re 9.5e-6)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.2e+32) {
		tmp = 0.5 * (sqrt((re * -2.0)) * sqrt(2.0));
	} else if (re <= 9.5e-6) {
		tmp = 0.5 * sqrt((2.0 * (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 <= (-2.2d+32)) then
        tmp = 0.5d0 * (sqrt((re * (-2.0d0))) * sqrt(2.0d0))
    else if (re <= 9.5d-6) then
        tmp = 0.5d0 * sqrt((2.0d0 * (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 <= -2.2e+32) {
		tmp = 0.5 * (Math.sqrt((re * -2.0)) * Math.sqrt(2.0));
	} else if (re <= 9.5e-6) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.2e+32:
		tmp = 0.5 * (math.sqrt((re * -2.0)) * math.sqrt(2.0))
	elif re <= 9.5e-6:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.2e+32)
		tmp = Float64(0.5 * Float64(sqrt(Float64(re * -2.0)) * sqrt(2.0)));
	elseif (re <= 9.5e-6)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 <= -2.2e+32)
		tmp = 0.5 * (sqrt((re * -2.0)) * sqrt(2.0));
	elseif (re <= 9.5e-6)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.2e+32], N[(0.5 * N[(N[Sqrt[N[(re * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.5e-6], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.20000000000000001e32

   1. Initial program 38.3%

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

      1. sqrt-prod 38.0%

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

      2. *-commutative 38.0%

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

      3. hypot-define 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in re around -inf 79.1%

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

      1. *-commutative 79.1%

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

   7. Simplified 79.1%

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

   if -2.20000000000000001e32 < re < 9.5000000000000005e-6

   1. Initial program 54.1%

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

   3. Taylor expanded in re around 0 79.9%

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

   if 9.5000000000000005e-6 < re

   1. Initial program 8.7%

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

   3. Taylor expanded in re around inf 39.3%

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

      1. sqrt-div 50.4%

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

      2. sqrt-pow1 82.9%

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

      3. metadata-eval 82.9%

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

      4. pow1 82.9%

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

      5. div-inv 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. associate-*r/ 82.9%

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

      2. *-rgt-identity 82.9%

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

   7. Simplified 82.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{re \cdot -2} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -5.1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -2.2 \cdot 10^{-126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -1.25 \cdot 10^{-148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.006:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (* 0.5 (sqrt (* 2.0 im)))))
   (if (<= re -5.1e+16)
     t_0
     (if (<= re -2.2e-126)
       t_1
       (if (<= re -1.25e-148)
         t_0
         (if (<= re 0.006) t_1 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((2.0 * im));
	double tmp;
	if (re <= -5.1e+16) {
		tmp = t_0;
	} else if (re <= -2.2e-126) {
		tmp = t_1;
	} else if (re <= -1.25e-148) {
		tmp = t_0;
	} else if (re <= 0.006) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((2.0d0 * im))
    if (re <= (-5.1d+16)) then
        tmp = t_0
    else if (re <= (-2.2d-126)) then
        tmp = t_1
    else if (re <= (-1.25d-148)) then
        tmp = t_0
    else if (re <= 0.006d0) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((2.0 * im));
	double tmp;
	if (re <= -5.1e+16) {
		tmp = t_0;
	} else if (re <= -2.2e-126) {
		tmp = t_1;
	} else if (re <= -1.25e-148) {
		tmp = t_0;
	} else if (re <= 0.006) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((2.0 * im))
	tmp = 0
	if re <= -5.1e+16:
		tmp = t_0
	elif re <= -2.2e-126:
		tmp = t_1
	elif re <= -1.25e-148:
		tmp = t_0
	elif re <= 0.006:
		tmp = t_1
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * im)))
	tmp = 0.0
	if (re <= -5.1e+16)
		tmp = t_0;
	elseif (re <= -2.2e-126)
		tmp = t_1;
	elseif (re <= -1.25e-148)
		tmp = t_0;
	elseif (re <= 0.006)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * sqrt((2.0 * im));
	tmp = 0.0;
	if (re <= -5.1e+16)
		tmp = t_0;
	elseif (re <= -2.2e-126)
		tmp = t_1;
	elseif (re <= -1.25e-148)
		tmp = t_0;
	elseif (re <= 0.006)
		tmp = t_1;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -5.1e+16], t$95$0, If[LessEqual[re, -2.2e-126], t$95$1, If[LessEqual[re, -1.25e-148], t$95$0, If[LessEqual[re, 0.006], t$95$1, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.1e16 or -2.20000000000000014e-126 < re < -1.25e-148

   1. Initial program 45.3%

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

   3. Taylor expanded in re around -inf 77.6%

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

      1. *-commutative 77.6%

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

   5. Simplified 77.6%

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

   if -5.1e16 < re < -2.20000000000000014e-126 or -1.25e-148 < re < 0.0060000000000000001

   1. Initial program 51.6%

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

   3. Taylor expanded in re around 0 82.7%

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

      1. *-commutative 82.7%

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

   5. Simplified 82.7%

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

   if 0.0060000000000000001 < re

   1. Initial program 8.7%

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

   3. Taylor expanded in re around inf 39.3%

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

      1. sqrt-div 50.4%

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

      2. sqrt-pow1 82.9%

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

      3. metadata-eval 82.9%

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

      4. pow1 82.9%

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

      5. div-inv 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. associate-*r/ 82.9%

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

      2. *-rgt-identity 82.9%

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

   7. Simplified 82.9%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.1 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -2.2 \cdot 10^{-126}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq -1.25 \cdot 10^{-148}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 0.006:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.12 \cdot 10^{+16} \lor \neg \left(re \leq -2.2 \cdot 10^{-126}\right) \land re \leq -1.25 \cdot 10^{-148}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -1.12e+16) (and (not (<= re -2.2e-126)) (<= re -1.25e-148)))
   (* 0.5 (sqrt (* re -4.0)))
   (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -1.12e+16) || (!(re <= -2.2e-126) && (re <= -1.25e-148))) {
		tmp = 0.5 * sqrt((re * -4.0));
	} 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 ((re <= (-1.12d+16)) .or. (.not. (re <= (-2.2d-126))) .and. (re <= (-1.25d-148))) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    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 ((re <= -1.12e+16) || (!(re <= -2.2e-126) && (re <= -1.25e-148))) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -1.12e+16) or (not (re <= -2.2e-126) and (re <= -1.25e-148)):
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -1.12e+16) || (!(re <= -2.2e-126) && (re <= -1.25e-148)))
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	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 ((re <= -1.12e+16) || (~((re <= -2.2e-126)) && (re <= -1.25e-148)))
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -1.12e+16], And[N[Not[LessEqual[re, -2.2e-126]], $MachinePrecision], LessEqual[re, -1.25e-148]]], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.12e16 or -2.20000000000000014e-126 < re < -1.25e-148

   1. Initial program 45.3%

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

   3. Taylor expanded in re around -inf 77.6%

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

      1. *-commutative 77.6%

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

   5. Simplified 77.6%

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

   if -1.12e16 < re < -2.20000000000000014e-126 or -1.25e-148 < re

   1. Initial program 36.6%

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

   3. Taylor expanded in re around 0 61.7%

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

      1. *-commutative 61.7%

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

   5. Simplified 61.7%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.12 \cdot 10^{+16} \lor \neg \left(re \leq -2.2 \cdot 10^{-126}\right) \land re \leq -1.25 \cdot 10^{-148}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.95 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 0.00155:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.95e+32)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 0.00155)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.95e+32) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 0.00155) {
		tmp = 0.5 * sqrt((2.0 * (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 <= (-2.95d+32)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 0.00155d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (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 <= -2.95e+32) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 0.00155) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.95e+32:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 0.00155:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.95e+32)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 0.00155)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 <= -2.95e+32)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 0.00155)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.95e+32], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.00155], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.94999999999999983e32

   1. Initial program 38.3%

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

   3. Taylor expanded in re around -inf 78.2%

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

      1. *-commutative 78.2%

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

   5. Simplified 78.2%

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

   if -2.94999999999999983e32 < re < 0.00154999999999999995

   1. Initial program 54.1%

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

   3. Taylor expanded in re around 0 79.9%

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

   if 0.00154999999999999995 < re

   1. Initial program 8.7%

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

   3. Taylor expanded in re around inf 39.3%

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

      1. sqrt-div 50.4%

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

      2. sqrt-pow1 82.9%

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

      3. metadata-eval 82.9%

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

      4. pow1 82.9%

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

      5. div-inv 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. associate-*r/ 82.9%

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

      2. *-rgt-identity 82.9%

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

   7. Simplified 82.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.95 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 0.00155:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in re around 0 52.3%

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

   1. *-commutative 52.3%

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

5. Simplified 52.3%

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

6. Final simplification 52.3%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(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)))))