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

Percentage Accurate: 42.1% → 90.6%

Time: 7.4s

Alternatives: 7

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: 42.1% 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: 90.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = (im * 0.5) / 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(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = (im * 0.5) / 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(((re * re) + (im * im))) - re) <= 0.0:
		tmp = (im * 0.5) / 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 (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp = Float64(Float64(im * 0.5) / 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(((re * re) + (im * im))) - re) <= 0.0)
		tmp = (im * 0.5) / 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[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $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 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 4.1%

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

   2. Taylor expanded in im around 0 90.9%

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

      1. *-commutative 90.9%

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

   4. Simplified 90.9%

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

      1. associate-*r* 90.9%

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

      2. sqrt-div 90.9%

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

      3. metadata-eval 90.9%

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

      4. un-div-inv 90.9%

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

      5. sqrt-unprod 92.3%

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

      6. metadata-eval 92.3%

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

      7. metadata-eval 92.3%

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

      8. *-rgt-identity 92.3%

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

      9. *-commutative 92.3%

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

   6. Applied egg-rr 92.3%

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

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

   1. Initial program 47.1%

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

      1. sub-neg 47.1%

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

      2. sqr-neg 47.1%

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

      3. sub-neg 47.1%

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

      4. sqr-neg 47.1%

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

      5. hypot-def 93.3%

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

   3. Simplified 93.3%

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

4. Final simplification 93.2%

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

Alternative 2: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.44:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.44)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 5e-39)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.44) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 5e-39) {
		tmp = 0.5 * sqrt((2.0 * (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 <= (-0.44d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 5d-39) then
        tmp = 0.5d0 * sqrt((2.0d0 * (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 <= -0.44) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 5e-39) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.44:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 5e-39:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.44)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 5e-39)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 <= -0.44)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 5e-39)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.44], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e-39], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.440000000000000002

   1. Initial program 32.2%

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

   2. Taylor expanded in re around -inf 79.0%

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

      1. *-commutative 79.0%

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

   4. Simplified 79.0%

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

   if -0.440000000000000002 < re < 4.9999999999999998e-39

   1. Initial program 56.0%

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

   2. Taylor expanded in re around 0 80.0%

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

   if 4.9999999999999998e-39 < re

   1. Initial program 16.7%

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

   2. Taylor expanded in im around 0 69.7%

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

      1. *-commutative 69.7%

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

   4. Simplified 69.7%

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

      1. associate-*r* 69.7%

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

      2. sqrt-div 69.6%

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

      3. metadata-eval 69.6%

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

      4. un-div-inv 69.6%

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

      5. sqrt-unprod 70.5%

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

      6. metadata-eval 70.5%

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

      7. metadata-eval 70.5%

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

      8. *-rgt-identity 70.5%

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

      9. *-commutative 70.5%

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

   6. Applied egg-rr 70.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.44:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 76.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.078:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1750000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.078)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1750000000.0)
     (* 0.5 (sqrt (* im 2.0)))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.078) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1750000000.0) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= (-0.078d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1750000000.0d0) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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 <= -0.078) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1750000000.0) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.078:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1750000000.0:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.078)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1750000000.0)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	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 <= -0.078)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1750000000.0)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.078], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1750000000.0], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0779999999999999999

   1. Initial program 32.2%

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

   2. Taylor expanded in re around -inf 79.0%

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

      1. *-commutative 79.0%

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

   4. Simplified 79.0%

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

   if -0.0779999999999999999 < re < 1.75e9

   1. Initial program 53.7%

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

   2. Taylor expanded in re around 0 75.8%

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

      1. expm1-log1p-u 71.9%

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

      2. expm1-udef 57.9%

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

      3. sqrt-unprod 57.9%

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

   4. Applied egg-rr 57.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im \cdot 2}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 72.0%

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

      2. expm1-log1p 76.2%

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

   6. Simplified 76.2%

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

   if 1.75e9 < 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 im around 0 75.2%

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

      1. *-commutative 75.2%

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

   4. Simplified 75.2%

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

      1. associate-*r* 75.2%

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

      2. sqrt-div 75.1%

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

      3. metadata-eval 75.1%

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

      4. un-div-inv 75.1%

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

      5. sqrt-unprod 76.1%

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

      6. metadata-eval 76.1%

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

      7. metadata-eval 76.1%

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

      8. *-rgt-identity 76.1%

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

      9. *-commutative 76.1%

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

   6. Applied egg-rr 76.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.078:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1750000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 17000000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 17000000000.0) (* 0.5 (sqrt (* im 2.0))) (* im (/ 0.5 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 17000000000.0) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= 17000000000.0d0) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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 <= 17000000000.0) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 17000000000.0:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 17000000000.0)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 17000000000.0)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 17000000000.0], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.7e10

   1. Initial program 47.1%

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

   2. Taylor expanded in re around 0 60.2%

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

      1. expm1-log1p-u 57.0%

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

      2. expm1-udef 47.0%

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

      3. sqrt-unprod 47.0%

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

   4. Applied egg-rr 47.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im \cdot 2}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 57.1%

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

      2. expm1-log1p 60.6%

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

   6. Simplified 60.6%

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

   if 1.7e10 < 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 im around 0 75.2%

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

      1. *-commutative 75.2%

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

   4. Simplified 75.2%

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

      1. expm1-log1p-u 74.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      2. expm1-udef 26.5%

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

      3. associate-*r* 26.5%

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

      4. sqrt-div 26.5%

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

      5. metadata-eval 26.5%

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

      6. un-div-inv 26.5%

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

      7. sqrt-unprod 26.5%

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

      8. metadata-eval 26.5%

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

      9. metadata-eval 26.5%

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

      10. *-rgt-identity 26.5%

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

      11. *-commutative 26.5%

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

   6. Applied egg-rr 26.5%

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

      1. expm1-def 75.5%

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

      2. expm1-log1p 76.1%

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

      3. *-commutative 76.1%

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

      4. associate-/l* 76.0%

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

   8. Simplified 76.0%

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

      1. associate-/r/ 76.0%

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

   10. Applied egg-rr 76.0%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 17000000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 63.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 320000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 320000000.0) (* 0.5 (sqrt (* im 2.0))) (/ 0.5 (/ (sqrt re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 320000000.0) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 / (sqrt(re) / im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 320000000.0d0) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 / (sqrt(re) / im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 320000000.0) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 / (Math.sqrt(re) / im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 320000000.0:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 / (math.sqrt(re) / im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 320000000.0)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 / Float64(sqrt(re) / im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 320000000.0)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 / (sqrt(re) / im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 320000000.0], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[Sqrt[re], $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.2e8

   1. Initial program 47.1%

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

   2. Taylor expanded in re around 0 60.2%

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

      1. expm1-log1p-u 57.0%

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

      2. expm1-udef 47.0%

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

      3. sqrt-unprod 47.0%

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

   4. Applied egg-rr 47.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im \cdot 2}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 57.1%

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

      2. expm1-log1p 60.6%

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

   6. Simplified 60.6%

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

   if 3.2e8 < 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 im around 0 75.2%

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

      1. *-commutative 75.2%

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

   4. Simplified 75.2%

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

      1. expm1-log1p-u 74.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      2. expm1-udef 26.5%

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

      3. associate-*r* 26.5%

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

      4. sqrt-div 26.5%

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

      5. metadata-eval 26.5%

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

      6. un-div-inv 26.5%

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

      7. sqrt-unprod 26.5%

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

      8. metadata-eval 26.5%

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

      9. metadata-eval 26.5%

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

      10. *-rgt-identity 26.5%

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

      11. *-commutative 26.5%

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

   6. Applied egg-rr 26.5%

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

      1. expm1-def 75.5%

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

      2. expm1-log1p 76.1%

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

      3. *-commutative 76.1%

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

      4. associate-/l* 76.0%

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

   8. Simplified 76.0%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 320000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \end{array} \]

Alternative 6: 63.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 430000000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 430000000000.0)
   (* 0.5 (sqrt (* im 2.0)))
   (/ (* im 0.5) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 430000000000.0) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= 430000000000.0d0) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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 <= 430000000000.0) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 430000000000.0:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 430000000000.0)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	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 <= 430000000000.0)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 430000000000.0], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.3e11

   1. Initial program 47.1%

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

   2. Taylor expanded in re around 0 60.2%

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

      1. expm1-log1p-u 57.0%

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

      2. expm1-udef 47.0%

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

      3. sqrt-unprod 47.0%

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

   4. Applied egg-rr 47.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im \cdot 2}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 57.1%

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

      2. expm1-log1p 60.6%

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

   6. Simplified 60.6%

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

   if 4.3e11 < 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 im around 0 75.2%

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

      1. *-commutative 75.2%

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

   4. Simplified 75.2%

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

      1. associate-*r* 75.2%

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

      2. sqrt-div 75.1%

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

      3. metadata-eval 75.1%

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

      4. un-div-inv 75.1%

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

      5. sqrt-unprod 76.1%

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

      6. metadata-eval 76.1%

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

      7. metadata-eval 76.1%

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

      8. *-rgt-identity 76.1%

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

      9. *-commutative 76.1%

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

   6. Applied egg-rr 76.1%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 430000000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 7: 52.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.0%

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

2. Taylor expanded in re around 0 54.7%

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

   1. expm1-log1p-u 51.8%

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

   2. expm1-udef 45.1%

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

   3. sqrt-unprod 45.1%

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

4. Applied egg-rr 45.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im \cdot 2}\right)} - 1\right)} \]
5. Step-by-step derivation

   1. expm1-def 51.9%

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

   2. expm1-log1p 55.0%

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

6. Simplified 55.0%

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

7. Final simplification 55.0%

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

Reproduce

herbie shell --seed 2023330 
(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)))))