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

Percentage Accurate: 41.9% → 90.7%

Time: 10.6s

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: 41.9% 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: 90.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $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 12.4%

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

   2. Taylor expanded in re around inf 34.8%

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

      1. unpow2 34.8%

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

   4. Simplified 34.8%

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

      1. expm1-log1p-u 34.8%

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

      2. expm1-udef 12.4%

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

      3. sqrt-div 12.4%

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

      4. sqrt-prod 12.4%

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

      5. add-sqr-sqrt 12.4%

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

   6. Applied egg-rr 12.4%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

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

   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. hypot-def 90.0%

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

   3. Simplified 90.0%

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

4. Final simplification 91.1%

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

Alternative 2: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-104} \lor \neg \left(re \leq 3.9 \cdot 10^{-70}\right) \land \left(re \leq 3 \cdot 10^{+66} \lor \neg \left(re \leq 2.05 \cdot 10^{+97}\right) \land re \leq 2 \cdot 10^{+113}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.5e+32)
   (* 0.5 (sqrt (* re -4.0)))
   (if (or (<= re 5.5e-104)
           (and (not (<= re 3.9e-70))
                (or (<= re 3e+66)
                    (and (not (<= re 2.05e+97)) (<= re 2e+113)))))
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.5e+32) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if ((re <= 5.5e-104) || (!(re <= 3.9e-70) && ((re <= 3e+66) || (!(re <= 2.05e+97) && (re <= 2e+113))))) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= (-5.5d+32)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if ((re <= 5.5d-104) .or. (.not. (re <= 3.9d-70)) .and. (re <= 3d+66) .or. (.not. (re <= 2.05d+97)) .and. (re <= 2d+113)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= -5.5e+32) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if ((re <= 5.5e-104) || (!(re <= 3.9e-70) && ((re <= 3e+66) || (!(re <= 2.05e+97) && (re <= 2e+113))))) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.5e+32:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif (re <= 5.5e-104) or (not (re <= 3.9e-70) and ((re <= 3e+66) or (not (re <= 2.05e+97) and (re <= 2e+113)))):
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.5e+32)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif ((re <= 5.5e-104) || (!(re <= 3.9e-70) && ((re <= 3e+66) || (!(re <= 2.05e+97) && (re <= 2e+113)))))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= -5.5e+32)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif ((re <= 5.5e-104) || (~((re <= 3.9e-70)) && ((re <= 3e+66) || (~((re <= 2.05e+97)) && (re <= 2e+113)))))
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.5e+32], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 5.5e-104], And[N[Not[LessEqual[re, 3.9e-70]], $MachinePrecision], Or[LessEqual[re, 3e+66], And[N[Not[LessEqual[re, 2.05e+97]], $MachinePrecision], LessEqual[re, 2e+113]]]]], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.49999999999999984e32

   1. Initial program 38.1%

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

   2. Taylor expanded in re around -inf 82.9%

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

      1. *-commutative 82.9%

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

   4. Simplified 82.9%

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

   if -5.49999999999999984e32 < re < 5.4999999999999998e-104 or 3.90000000000000019e-70 < re < 3.00000000000000002e66 or 2.04999999999999994e97 < re < 2e113

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0 79.1%

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

      1. *-commutative 79.1%

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

   4. Simplified 79.1%

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

   if 5.4999999999999998e-104 < re < 3.90000000000000019e-70 or 3.00000000000000002e66 < re < 2.04999999999999994e97 or 2e113 < re

   1. Initial program 11.8%

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

   2. Taylor expanded in re around inf 42.8%

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

      1. unpow2 42.8%

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

   4. Simplified 42.8%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-udef 21.7%

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

      3. sqrt-div 21.7%

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

      4. sqrt-prod 29.1%

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

      5. add-sqr-sqrt 29.1%

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

   6. Applied egg-rr 29.1%

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

      1. expm1-def 86.2%

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

      2. expm1-log1p 87.0%

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

   8. Simplified 87.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-104} \lor \neg \left(re \leq 3.9 \cdot 10^{-70}\right) \land \left(re \leq 3 \cdot 10^{+66} \lor \neg \left(re \leq 2.05 \cdot 10^{+97}\right) \land re \leq 2 \cdot 10^{+113}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -2 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ im (sqrt re)))) (t_1 (* 0.5 (sqrt (* 2.0 im)))))
   (if (<= re -2e+32)
     (* 0.5 (sqrt (* re -4.0)))
     (if (<= re 5.5e-104)
       t_1
       (if (<= re 5.2e-68)
         t_0
         (if (<= re 1.25e+69)
           t_1
           (if (<= re 3.2e+97)
             (* 0.5 (* im (pow re -0.5)))
             (if (<= re 2e+113) t_1 t_0))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im / sqrt(re));
	double t_1 = 0.5 * sqrt((2.0 * im));
	double tmp;
	if (re <= -2e+32) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5.5e-104) {
		tmp = t_1;
	} else if (re <= 5.2e-68) {
		tmp = t_0;
	} else if (re <= 1.25e+69) {
		tmp = t_1;
	} else if (re <= 3.2e+97) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else if (re <= 2e+113) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 * (im / sqrt(re))
    t_1 = 0.5d0 * sqrt((2.0d0 * im))
    if (re <= (-2d+32)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5.5d-104) then
        tmp = t_1
    else if (re <= 5.2d-68) then
        tmp = t_0
    else if (re <= 1.25d+69) then
        tmp = t_1
    else if (re <= 3.2d+97) then
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    else if (re <= 2d+113) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im / Math.sqrt(re));
	double t_1 = 0.5 * Math.sqrt((2.0 * im));
	double tmp;
	if (re <= -2e+32) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5.5e-104) {
		tmp = t_1;
	} else if (re <= 5.2e-68) {
		tmp = t_0;
	} else if (re <= 1.25e+69) {
		tmp = t_1;
	} else if (re <= 3.2e+97) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else if (re <= 2e+113) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im / math.sqrt(re))
	t_1 = 0.5 * math.sqrt((2.0 * im))
	tmp = 0
	if re <= -2e+32:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5.5e-104:
		tmp = t_1
	elif re <= 5.2e-68:
		tmp = t_0
	elif re <= 1.25e+69:
		tmp = t_1
	elif re <= 3.2e+97:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	elif re <= 2e+113:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im / sqrt(re)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * im)))
	tmp = 0.0
	if (re <= -2e+32)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5.5e-104)
		tmp = t_1;
	elseif (re <= 5.2e-68)
		tmp = t_0;
	elseif (re <= 1.25e+69)
		tmp = t_1;
	elseif (re <= 3.2e+97)
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	elseif (re <= 2e+113)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im / sqrt(re));
	t_1 = 0.5 * sqrt((2.0 * im));
	tmp = 0.0;
	if (re <= -2e+32)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5.5e-104)
		tmp = t_1;
	elseif (re <= 5.2e-68)
		tmp = t_0;
	elseif (re <= 1.25e+69)
		tmp = t_1;
	elseif (re <= 3.2e+97)
		tmp = 0.5 * (im * (re ^ -0.5));
	elseif (re <= 2e+113)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2e+32], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.5e-104], t$95$1, If[LessEqual[re, 5.2e-68], t$95$0, If[LessEqual[re, 1.25e+69], t$95$1, If[LessEqual[re, 3.2e+97], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2e+113], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.00000000000000011e32

   1. Initial program 38.1%

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

   2. Taylor expanded in re around -inf 82.9%

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

      1. *-commutative 82.9%

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

   4. Simplified 82.9%

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

   if -2.00000000000000011e32 < re < 5.4999999999999998e-104 or 5.1999999999999996e-68 < re < 1.25000000000000009e69 or 3.20000000000000016e97 < re < 2e113

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0 79.1%

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

      1. *-commutative 79.1%

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

   4. Simplified 79.1%

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

   if 5.4999999999999998e-104 < re < 5.1999999999999996e-68 or 2e113 < re

   1. Initial program 10.3%

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

   2. Taylor expanded in re around inf 44.7%

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

      1. unpow2 44.7%

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

   4. Simplified 44.7%

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

      1. expm1-log1p-u 44.4%

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

      2. expm1-udef 21.3%

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

      3. sqrt-div 21.3%

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

      4. sqrt-prod 29.5%

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

      5. add-sqr-sqrt 29.5%

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

   6. Applied egg-rr 29.5%

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

      1. expm1-def 84.7%

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

      2. expm1-log1p 85.6%

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

   8. Simplified 85.6%

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

   if 1.25000000000000009e69 < re < 3.20000000000000016e97

   1. Initial program 25.4%

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

   2. Taylor expanded in re around inf 25.4%

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

      1. unpow2 25.4%

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

   4. Simplified 25.4%

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

      1. div-inv 25.4%

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

      2. sqrt-prod 25.4%

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

      3. sqrt-prod 99.5%

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

      4. add-sqr-sqrt 99.7%

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

      5. inv-pow 99.7%

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

      6. sqrt-pow1 99.7%

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

      7. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-104}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{-68}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 75.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -6.1 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ im (sqrt re)))) (t_1 (* 0.5 (sqrt (* 2.0 im)))))
   (if (<= re -6.1e+34)
     (* 0.5 (sqrt (* re -4.0)))
     (if (<= re 2.35e-104)
       (* 0.5 (sqrt (* 2.0 (- im re))))
       (if (<= re 1.05e-69)
         t_0
         (if (<= re 8e+68)
           t_1
           (if (<= re 2.4e+97)
             (* 0.5 (* im (pow re -0.5)))
             (if (<= re 8e+114) t_1 t_0))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im / sqrt(re));
	double t_1 = 0.5 * sqrt((2.0 * im));
	double tmp;
	if (re <= -6.1e+34) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.35e-104) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 1.05e-69) {
		tmp = t_0;
	} else if (re <= 8e+68) {
		tmp = t_1;
	} else if (re <= 2.4e+97) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else if (re <= 8e+114) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 * (im / sqrt(re))
    t_1 = 0.5d0 * sqrt((2.0d0 * im))
    if (re <= (-6.1d+34)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.35d-104) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 1.05d-69) then
        tmp = t_0
    else if (re <= 8d+68) then
        tmp = t_1
    else if (re <= 2.4d+97) then
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    else if (re <= 8d+114) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im / Math.sqrt(re));
	double t_1 = 0.5 * Math.sqrt((2.0 * im));
	double tmp;
	if (re <= -6.1e+34) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.35e-104) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 1.05e-69) {
		tmp = t_0;
	} else if (re <= 8e+68) {
		tmp = t_1;
	} else if (re <= 2.4e+97) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else if (re <= 8e+114) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im / math.sqrt(re))
	t_1 = 0.5 * math.sqrt((2.0 * im))
	tmp = 0
	if re <= -6.1e+34:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.35e-104:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 1.05e-69:
		tmp = t_0
	elif re <= 8e+68:
		tmp = t_1
	elif re <= 2.4e+97:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	elif re <= 8e+114:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im / sqrt(re)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * im)))
	tmp = 0.0
	if (re <= -6.1e+34)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.35e-104)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 1.05e-69)
		tmp = t_0;
	elseif (re <= 8e+68)
		tmp = t_1;
	elseif (re <= 2.4e+97)
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	elseif (re <= 8e+114)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im / sqrt(re));
	t_1 = 0.5 * sqrt((2.0 * im));
	tmp = 0.0;
	if (re <= -6.1e+34)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.35e-104)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 1.05e-69)
		tmp = t_0;
	elseif (re <= 8e+68)
		tmp = t_1;
	elseif (re <= 2.4e+97)
		tmp = 0.5 * (im * (re ^ -0.5));
	elseif (re <= 8e+114)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -6.1e+34], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.35e-104], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.05e-69], t$95$0, If[LessEqual[re, 8e+68], t$95$1, If[LessEqual[re, 2.4e+97], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8e+114], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -6.09999999999999996e34

   1. Initial program 36.1%

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

   2. Taylor expanded in re around -inf 83.8%

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

      1. *-commutative 83.8%

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

   4. Simplified 83.8%

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

   if -6.09999999999999996e34 < re < 2.35e-104

   1. Initial program 59.4%

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

   2. Taylor expanded in re around 0 81.5%

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

   if 2.35e-104 < re < 1.05e-69 or 8e114 < re

   1. Initial program 10.3%

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

   2. Taylor expanded in re around inf 44.7%

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

      1. unpow2 44.7%

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

   4. Simplified 44.7%

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

      1. expm1-log1p-u 44.4%

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

      2. expm1-udef 21.3%

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

      3. sqrt-div 21.3%

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

      4. sqrt-prod 29.5%

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

      5. add-sqr-sqrt 29.5%

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

   6. Applied egg-rr 29.5%

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

      1. expm1-def 84.7%

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

      2. expm1-log1p 85.6%

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

   8. Simplified 85.6%

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

   if 1.05e-69 < re < 7.99999999999999962e68 or 2.4e97 < re < 8e114

   1. Initial program 40.2%

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

   2. Taylor expanded in re around 0 76.5%

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

      1. *-commutative 76.5%

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

   4. Simplified 76.5%

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

   if 7.99999999999999962e68 < re < 2.4e97

   1. Initial program 25.4%

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

   2. Taylor expanded in re around inf 25.4%

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

      1. unpow2 25.4%

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

   4. Simplified 25.4%

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

      1. div-inv 25.4%

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

      2. sqrt-prod 25.4%

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

      3. sqrt-prod 99.5%

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

      4. add-sqr-sqrt 99.7%

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

      5. inv-pow 99.7%

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

      6. sqrt-pow1 99.7%

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

      7. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.1 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 64.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e+31) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e+31) {
		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 <= (-4.5d+31)) 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 <= -4.5e+31) {
		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 <= -4.5e+31:
		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 <= -4.5e+31)
		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 <= -4.5e+31)
		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[LessEqual[re, -4.5e+31], 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 < -4.4999999999999996e31

   1. Initial program 38.1%

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

   2. Taylor expanded in re around -inf 82.9%

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

      1. *-commutative 82.9%

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

   4. Simplified 82.9%

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

   if -4.4999999999999996e31 < re

   1. Initial program 43.0%

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

   2. Taylor expanded in re around 0 63.1%

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

      1. *-commutative 63.1%

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

   4. Simplified 63.1%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 6: 52.9% 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 41.8%

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

2. Taylor expanded in re around 0 53.0%

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

   1. *-commutative 53.0%

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

4. Simplified 53.0%

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

5. Final simplification 53.0%

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

Reproduce

herbie shell --seed 2023257 
(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)))))