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

Percentage Accurate: 41.3% → 90.0%

Time: 8.2s

Alternatives: 5

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: 41.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.3%

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

   2. Taylor expanded in im around 0 91.6%

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

      1. associate-*r* 91.6%

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

      2. associate-*r* 92.0%

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

   4. Simplified 92.0%

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

      1. expm1-log1p-u 92.0%

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

      2. expm1-udef 13.5%

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

   6. Applied egg-rr 13.5%

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

      1. expm1-def 92.8%

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

      2. expm1-log1p 92.8%

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

      3. associate-*r/ 92.9%

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

      4. *-commutative 92.9%

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

   8. Simplified 92.9%

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

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

   1. Initial program 47.3%

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

      1. hypot-udef 89.7%

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

      2. add-sqr-sqrt 89.0%

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

      3. sqrt-unprod 89.7%

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

      4. *-commutative 89.7%

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

      5. *-commutative 89.7%

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

      6. swap-sqr 89.7%

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

      7. add-sqr-sqrt 89.7%

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

      8. metadata-eval 89.7%

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

   3. Applied egg-rr 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*r* 89.7%

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

      3. metadata-eval 89.7%

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

   5. Simplified 89.7%

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

4. Final simplification 90.2%

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

Alternative 2: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 0.000185:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.2e-45)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 0.000185)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.2e-45) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 0.000185) {
		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 <= (-3.2d-45)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 0.000185d0) 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 <= -3.2e-45) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 0.000185) {
		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 <= -3.2e-45:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 0.000185:
		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 <= -3.2e-45)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 0.000185)
		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 <= -3.2e-45)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 0.000185)
		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, -3.2e-45], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.000185], 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 < -3.20000000000000007e-45

   1. Initial program 42.8%

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

   2. Taylor expanded in re around -inf 74.4%

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

      1. *-commutative 74.4%

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

   4. Simplified 74.4%

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

   if -3.20000000000000007e-45 < re < 1.85e-4

   1. Initial program 57.1%

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

   2. Taylor expanded in re around 0 77.7%

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

   if 1.85e-4 < re

   1. Initial program 12.7%

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

   2. Taylor expanded in im around 0 77.5%

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

      1. associate-*r* 77.5%

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

      2. associate-*r* 78.0%

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

   4. Simplified 78.0%

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

      1. expm1-log1p-u 77.4%

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

      2. expm1-udef 28.2%

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

   6. Applied egg-rr 28.2%

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

      1. expm1-def 77.8%

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

      2. expm1-log1p 78.6%

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

      3. associate-*r/ 78.6%

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

      4. *-commutative 78.6%

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

   8. Simplified 78.6%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 0.000185:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 52.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e-310)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (/ (* im 0.5) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.5 * sqrt((2.0 * (re * -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 <= (-5d-310)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-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 <= -5e-310) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-310:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -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 <= -5e-310)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-310], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 51.2%

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

   2. Taylor expanded in re around -inf 53.6%

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

      1. *-commutative 53.6%

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

   4. Simplified 53.6%

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

   if -4.999999999999985e-310 < re

   1. Initial program 30.5%

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

   2. Taylor expanded in im around 0 56.2%

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

      1. associate-*r* 56.2%

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

      2. associate-*r* 56.5%

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

   4. Simplified 56.5%

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

      1. expm1-log1p-u 56.2%

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

      2. expm1-udef 17.8%

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

   6. Applied egg-rr 17.8%

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

      1. expm1-def 56.5%

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

      2. expm1-log1p 57.0%

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

      3. associate-*r/ 57.0%

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

      4. *-commutative 57.0%

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

   8. Simplified 57.0%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 26.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{\sqrt{re}}{im}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 0.5 (/ (sqrt re) im)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

2. Taylor expanded in re around inf 13.7%

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

   1. associate-*r/ 13.7%

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

4. Simplified 13.7%

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

   1. associate-*r/ 13.7%

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

   2. sqrt-div 18.1%

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

   3. rem-square-sqrt 18.0%

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

   4. *-commutative 18.0%

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

   5. unpow2 18.0%

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

   6. rem-square-sqrt 17.9%

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

   7. swap-sqr 17.9%

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

   8. swap-sqr 17.9%

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

   9. *-commutative 17.9%

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

   10. *-commutative 17.9%

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

   11. sqrt-unprod 26.4%

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

   12. add-sqr-sqrt 26.5%

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

   13. un-div-inv 26.5%

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

   14. metadata-eval 26.5%

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

   15. sqrt-div 26.5%

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

6. Applied egg-rr 26.7%

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

   1. associate-*r* 26.7%

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

   2. metadata-eval 26.7%

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

   3. sqrt-pow1 26.7%

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

   4. inv-pow 26.7%

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

   5. associate-*l* 26.7%

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

   6. sqrt-div 26.7%

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

   7. metadata-eval 26.7%

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

   8. div-inv 26.7%

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

   9. associate-*r/ 26.7%

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

   10. associate-/l* 25.9%

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

8. Applied egg-rr 25.9%

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

9. Final simplification 25.9%

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

Alternative 5: 26.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

2. Taylor expanded in im around 0 26.4%

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

   1. associate-*r* 26.4%

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

   2. associate-*r* 26.5%

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

4. Simplified 26.5%

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

   1. expm1-log1p-u 26.3%

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

   2. expm1-udef 8.3%

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

6. Applied egg-rr 8.3%

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

   1. expm1-def 26.5%

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

   2. expm1-log1p 26.7%

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

   3. associate-*r/ 26.7%

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

   4. *-commutative 26.7%

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

8. Simplified 26.7%

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

9. Final simplification 26.7%

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

Reproduce

herbie shell --seed 2023321 
(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)))))