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

Percentage Accurate: 42.4% → 88.6%

Time: 9.4s

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: 42.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{if}\;re \leq 2.2 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 750000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \sqrt{\frac{0.5}{re}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))
   (if (<= re 2.2e-66)
     t_0
     (if (<= re 6.5e-53)
       (* 0.5 (* im (sqrt (/ 1.0 re))))
       (if (<= re 750000000000.0)
         t_0
         (* 0.5 (* (sqrt 2.0) (* im (sqrt (/ 0.5 re))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	double tmp;
	if (re <= 2.2e-66) {
		tmp = t_0;
	} else if (re <= 6.5e-53) {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	} else if (re <= 750000000000.0) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (sqrt(2.0) * (im * sqrt((0.5 / re))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	double tmp;
	if (re <= 2.2e-66) {
		tmp = t_0;
	} else if (re <= 6.5e-53) {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	} else if (re <= 750000000000.0) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (Math.sqrt(2.0) * (im * Math.sqrt((0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	tmp = 0
	if re <= 2.2e-66:
		tmp = t_0
	elif re <= 6.5e-53:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	elif re <= 750000000000.0:
		tmp = t_0
	else:
		tmp = 0.5 * (math.sqrt(2.0) * (im * math.sqrt((0.5 / re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))))
	tmp = 0.0
	if (re <= 2.2e-66)
		tmp = t_0;
	elseif (re <= 6.5e-53)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	elseif (re <= 750000000000.0)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(sqrt(2.0) * Float64(im * sqrt(Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	tmp = 0.0;
	if (re <= 2.2e-66)
		tmp = t_0;
	elseif (re <= 6.5e-53)
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	elseif (re <= 750000000000.0)
		tmp = t_0;
	else
		tmp = 0.5 * (sqrt(2.0) * (im * sqrt((0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 2.2e-66], t$95$0, If[LessEqual[re, 6.5e-53], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 750000000000.0], t$95$0, N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(im * N[Sqrt[N[(0.5 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 2.2000000000000001e-66 or 6.4999999999999997e-53 < re < 7.5e11

   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-def 92.6%

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

   3. Simplified 92.6%

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

   if 2.2000000000000001e-66 < re < 6.4999999999999997e-53

   1. Initial program 4.8%

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

      1. hypot-udef 4.8%

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

      2. expm1-log1p-u 4.8%

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

   3. Applied egg-rr 4.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]
   4. Step-by-step derivation

      1. pow1/2 4.8%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)\right)}^{0.5}} \]

      2. pow-to-exp 4.8%

         \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)\right) \cdot 0.5}} \]

      3. expm1-log1p-u 4.8%

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

      4. *-commutative 4.8%

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

   5. Applied egg-rr 4.8%

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

   6. Taylor expanded in im around 0 100.0%

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

   if 7.5e11 < re

   1. Initial program 8.8%

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

   2. Taylor expanded in im around 0 76.3%

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

      1. associate-*l* 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-*l* 76.3%

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

   4. Simplified 76.3%

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

      1. expm1-log1p-u 76.3%

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

      2. expm1-udef 12.6%

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

      3. sqrt-unprod 12.6%

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

      4. un-div-inv 12.6%

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

   6. Applied egg-rr 12.6%

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

      1. expm1-def 76.9%

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

      2. expm1-log1p 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.2 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 750000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \sqrt{\frac{0.5}{re}}\right)\right)\\ \end{array} \]

Alternative 2: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* 0.5 (* im (sqrt (/ 1.0 re))))
   (* 0.5 (pow (* 2.0 (- (hypot re im) re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * pow((2.0 * (hypot(re, im) - re)), 0.5);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * Math.pow((2.0 * (Math.hypot(re, im) - re)), 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	else:
		tmp = 0.5 * math.pow((2.0 * (math.hypot(re, im) - re)), 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	else
		tmp = Float64(0.5 * (Float64(2.0 * Float64(hypot(re, im) - re)) ^ 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0)
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	else
		tmp = 0.5 * ((2.0 * (hypot(re, im) - re)) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], 0.5], $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 4.4%

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

      1. hypot-udef 10.1%

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

      2. expm1-log1p-u 10.1%

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

   3. Applied egg-rr 10.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]
   4. Step-by-step derivation

      1. pow1/2 10.1%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)\right)}^{0.5}} \]

      2. pow-to-exp 9.7%

         \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)\right) \cdot 0.5}} \]

      3. expm1-log1p-u 9.7%

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

      4. *-commutative 9.7%

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

   5. Applied egg-rr 9.7%

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

   6. Taylor expanded in im around 0 94.8%

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

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

   1. Initial program 40.9%

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

      1. pow1/2 40.9%

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

      2. hypot-udef 85.9%

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

   3. Applied egg-rr 85.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}\\ \end{array} \]

Alternative 3: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.45 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-53} \lor \neg \left(re \leq 1.05 \cdot 10^{+32}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.45e-64)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (if (or (<= re 4.6e-53) (not (<= re 1.05e+32)))
     (* 0.5 (* im (sqrt (/ 1.0 re))))
     (* 0.5 (sqrt (* im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.45e-64) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else if ((re <= 4.6e-53) || !(re <= 1.05e+32)) {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.45e-64) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else if ((re <= 4.6e-53) || !(re <= 1.05e+32)) {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.45e-64:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	elif (re <= 4.6e-53) or not (re <= 1.05e+32):
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.45e-64)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	elseif ((re <= 4.6e-53) || !(re <= 1.05e+32))
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.45e-64)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	elseif ((re <= 4.6e-53) || ~((re <= 1.05e+32)))
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.45e-64], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 4.6e-53], N[Not[LessEqual[re, 1.05e+32]], $MachinePrecision]], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 2.4500000000000001e-64

   1. Initial program 47.0%

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

      1. hypot-def 93.5%

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

   3. Simplified 93.5%

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

   if 2.4500000000000001e-64 < re < 4.6000000000000003e-53 or 1.05e32 < re

   1. Initial program 8.8%

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

      1. hypot-udef 32.9%

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

      2. expm1-log1p-u 31.2%

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

   3. Applied egg-rr 31.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]
   4. Step-by-step derivation

      1. pow1/2 31.2%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)\right)}^{0.5}} \]

      2. pow-to-exp 30.9%

         \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)\right) \cdot 0.5}} \]

      3. expm1-log1p-u 31.0%

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

      4. *-commutative 31.0%

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

   5. Applied egg-rr 31.0%

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

   6. Taylor expanded in im around 0 79.7%

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

   if 4.6000000000000003e-53 < re < 1.05e32

   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 77.2%

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

      1. expm1-log1p-u 72.5%

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

      2. expm1-udef 69.0%

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

      3. *-commutative 69.0%

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

      4. sqrt-unprod 69.0%

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

   4. Applied egg-rr 69.0%

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

      1. expm1-def 72.6%

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

      2. expm1-log1p 77.7%

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

   6. Simplified 77.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.45 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-53} \lor \neg \left(re \leq 1.05 \cdot 10^{+32}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 4: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.6 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.86 \cdot 10^{-65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-53} \lor \neg \left(re \leq 1.4 \cdot 10^{+36}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.6e+70)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1.86e-65)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (or (<= re 5.8e-53) (not (<= re 1.4e+36)))
       (* 0.5 (* im (sqrt (/ 1.0 re))))
       (* 0.5 (sqrt (* im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.6e+70) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.86e-65) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if ((re <= 5.8e-53) || !(re <= 1.4e+36)) {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-9.6d+70)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1.86d-65) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if ((re <= 5.8d-53) .or. (.not. (re <= 1.4d+36))) then
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9.6e+70) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.86e-65) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if ((re <= 5.8e-53) || !(re <= 1.4e+36)) {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.6e+70:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1.86e-65:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif (re <= 5.8e-53) or not (re <= 1.4e+36):
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.6e+70)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1.86e-65)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif ((re <= 5.8e-53) || !(re <= 1.4e+36))
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9.6e+70)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1.86e-65)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif ((re <= 5.8e-53) || ~((re <= 1.4e+36)))
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.6e+70], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.86e-65], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 5.8e-53], N[Not[LessEqual[re, 1.4e+36]], $MachinePrecision]], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -9.59999999999999947e70

   1. Initial program 28.2%

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

   2. Taylor expanded in re around -inf 89.6%

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

      1. *-commutative 89.6%

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

   4. Simplified 89.6%

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

   if -9.59999999999999947e70 < re < 1.86000000000000006e-65

   1. Initial program 54.0%

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

   2. Taylor expanded in re around 0 81.0%

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

   if 1.86000000000000006e-65 < re < 5.7999999999999996e-53 or 1.4e36 < re

   1. Initial program 8.8%

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

      1. hypot-udef 32.9%

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

      2. expm1-log1p-u 31.2%

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

   3. Applied egg-rr 31.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]
   4. Step-by-step derivation

      1. pow1/2 31.2%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)\right)}^{0.5}} \]

      2. pow-to-exp 30.9%

         \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)\right) \cdot 0.5}} \]

      3. expm1-log1p-u 31.0%

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

      4. *-commutative 31.0%

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

   5. Applied egg-rr 31.0%

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

   6. Taylor expanded in im around 0 79.7%

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

   if 5.7999999999999996e-53 < re < 1.4e36

   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 77.2%

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

      1. expm1-log1p-u 72.5%

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

      2. expm1-udef 69.0%

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

      3. *-commutative 69.0%

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

      4. sqrt-unprod 69.0%

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

   4. Applied egg-rr 69.0%

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

      1. expm1-def 72.6%

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

      2. expm1-log1p 77.7%

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

   6. Simplified 77.7%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{im \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.6 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.86 \cdot 10^{-65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-53} \lor \neg \left(re \leq 1.4 \cdot 10^{+36}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.55 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 280000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.55e+70)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 280000000000.0)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (sqrt (/ (* im im) re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.55e+70) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 280000000000.0) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * sqrt(((im * im) / re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.55d+70)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 280000000000.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * sqrt(((im * im) / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.55e+70) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 280000000000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * Math.sqrt(((im * im) / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.55e+70:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 280000000000.0:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * math.sqrt(((im * im) / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.55e+70)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 280000000000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * im) / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.55e+70)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 280000000000.0)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * sqrt(((im * im) / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.55e+70], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 280000000000.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.55000000000000007e70

   1. Initial program 28.2%

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

   2. Taylor expanded in re around -inf 89.6%

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

      1. *-commutative 89.6%

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

   4. Simplified 89.6%

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

   if -2.55000000000000007e70 < re < 2.8e11

   1. Initial program 51.7%

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

   2. Taylor expanded in re around 0 78.2%

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

   if 2.8e11 < re

   1. Initial program 8.8%

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

   2. Taylor expanded in im around 0 76.3%

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

      1. associate-*l* 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-*l* 76.3%

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

   4. Simplified 76.3%

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

      1. expm1-log1p-u 75.7%

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

      2. expm1-udef 22.9%

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

      3. *-commutative 22.9%

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

      4. *-commutative 22.9%

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

      5. associate-*l* 22.9%

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

      6. sqrt-unprod 22.9%

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

      7. un-div-inv 22.9%

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

   6. Applied egg-rr 22.9%

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

      1. expm1-def 76.0%

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

      2. expm1-log1p 76.7%

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

      3. associate-*r* 76.9%

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

      4. *-commutative 76.9%

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

      5. associate-*l* 76.7%

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

   8. Simplified 76.7%

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

      1. add-sqr-sqrt 76.5%

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

      2. sqrt-unprod 56.8%

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

      3. swap-sqr 48.4%

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

      4. sqrt-unprod 48.5%

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

      5. sqrt-unprod 48.5%

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

      6. add-sqr-sqrt 48.6%

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

      7. associate-*l/ 48.6%

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

      8. metadata-eval 48.6%

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

   10. Applied egg-rr 48.6%

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

       1. unpow2 48.6%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{im}^{2}} \cdot \frac{1}{re}} \]

       2. associate-*r/ 48.6%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2} \cdot 1}{re}}} \]

       3. *-rgt-identity 48.6%

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

       4. unpow2 48.6%

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

   12. Simplified 48.6%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.55 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 280000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{re}}\\ \end{array} \]

Alternative 6: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e+71)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e+71) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.6d+71)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.6e+71) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.6e+71:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.6e+71)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.6e+71)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e+71], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.6e71

   1. Initial program 28.2%

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

   2. Taylor expanded in re around -inf 89.6%

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

      1. *-commutative 89.6%

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

   4. Simplified 89.6%

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

   if -3.6e71 < re

   1. Initial program 37.1%

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

   2. Taylor expanded in re around 0 59.0%

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

      1. expm1-log1p-u 55.8%

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

      2. expm1-udef 48.4%

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

      3. *-commutative 48.4%

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

      4. sqrt-unprod 48.4%

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

   4. Applied egg-rr 48.4%

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

      1. expm1-def 55.8%

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

      2. expm1-log1p 59.4%

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

   6. Simplified 59.4%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7: 53.6% 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 35.6%

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

2. Taylor expanded in re around 0 51.7%

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

   1. expm1-log1p-u 48.9%

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

   2. expm1-udef 42.6%

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

   3. *-commutative 42.6%

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

   4. sqrt-unprod 42.6%

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

4. Applied egg-rr 42.6%

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

   1. expm1-def 48.9%

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

   2. expm1-log1p 52.1%

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

6. Simplified 52.1%

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

7. Final simplification 52.1%

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

Reproduce

herbie shell --seed 2023196 
(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)))))