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

Percentage Accurate: 41.4% → 90.0%

Time: 8.8s

Alternatives: 7

Speedup: 1.7×

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: 41.4% 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: 90.0% accurate, 0.3× 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 10.3%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(im, im, re \cdot re\right)}} - re\right)} \cdot \frac{1}{2} \]

      9. *-lowering-*.f64 10.3

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

   4. Applied egg-rr 10.3%

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

   5. Taylor expanded in im around 0

      \[\leadsto \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \cdot \frac{1}{2} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 50.4

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

   7. Simplified 50.4%

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

      1. sqrt-div N/A

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

      2. sqrt-prod N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*l/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 91.3

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

   9. Applied egg-rr 91.3%

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

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

   1. Initial program 48.6%

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

      1. accelerator-lowering-hypot.f64 92.6

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

   4. Applied egg-rr 92.6%

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

Alternative 2: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.05 \cdot 10^{+124}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -5.2 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 0.000235:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.05e+124)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re -5.2e-45)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma im im (* re re))) re))))
     (if (<= re 0.000235)
       (* 0.5 (sqrt (* 2.0 (- im re))))
       (/ (* im 0.5) (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.05e+124) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= -5.2e-45) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(im, im, (re * re))) - re)));
	} else if (re <= 0.000235) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.05e+124)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= -5.2e-45)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(im, im, Float64(re * re))) - re))));
	elseif (re <= 0.000235)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.05e+124], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -5.2e-45], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.000235], 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 4 regimes

2. if re < -2.05000000000000001e124

   1. Initial program 15.9%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. *-lowering-*.f64 86.7

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

   5. Simplified 86.7%

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

   if -2.05000000000000001e124 < re < -5.19999999999999973e-45

   1. Initial program 89.3%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(im, im, re \cdot re\right)}} - re\right)} \cdot \frac{1}{2} \]

      9. *-lowering-*.f64 89.3

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

   4. Applied egg-rr 89.3%

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

   if -5.19999999999999973e-45 < re < 2.34999999999999993e-4

   1. Initial program 51.3%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 79.8%

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

   if 2.34999999999999993e-4 < re

   1. Initial program 15.5%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(im, im, re \cdot re\right)}} - re\right)} \cdot \frac{1}{2} \]

      9. *-lowering-*.f64 15.5

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

   4. Applied egg-rr 15.5%

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

   5. Taylor expanded in im around 0

      \[\leadsto \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \cdot \frac{1}{2} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 55.1

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

   7. Simplified 55.1%

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

      1. sqrt-div N/A

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

      2. sqrt-prod N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*l/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 79.7

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

   9. Applied egg-rr 79.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.05 \cdot 10^{+124}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -5.2 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 0.000235:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 0.000118:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3e+16)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 0.000118)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3e+16) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 0.000118) {
		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 <= (-3d+16)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 0.000118d0) 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 <= -3e+16) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 0.000118) {
		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 <= -3e+16:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 0.000118:
		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 <= -3e+16)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 0.000118)
		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 <= -3e+16)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 0.000118)
		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, -3e+16], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.000118], 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 < -3e16

   1. Initial program 42.2%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. *-lowering-*.f64 80.3

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

   5. Simplified 80.3%

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

   if -3e16 < re < 1.18e-4

   1. Initial program 55.1%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 77.6%

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

   if 1.18e-4 < re

   1. Initial program 15.5%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(im, im, re \cdot re\right)}} - re\right)} \cdot \frac{1}{2} \]

      9. *-lowering-*.f64 15.5

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

   4. Applied egg-rr 15.5%

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

   5. Taylor expanded in im around 0

      \[\leadsto \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \cdot \frac{1}{2} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 55.1

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

   7. Simplified 55.1%

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

      1. sqrt-div N/A

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

      2. sqrt-prod N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*l/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 79.7

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

   9. Applied egg-rr 79.7%

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

Alternative 4: 76.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.12 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 0.00013:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.12e+15)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 0.00013)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.12e+15) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 0.00013) {
		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 <= (-1.12d+15)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 0.00013d0) 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 <= -1.12e+15) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 0.00013) {
		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 <= -1.12e+15:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 0.00013:
		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 <= -1.12e+15)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 0.00013)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	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 <= -1.12e+15)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 0.00013)
		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, -1.12e+15], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.00013], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.12e15

   1. Initial program 42.2%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. *-lowering-*.f64 80.3

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

   5. Simplified 80.3%

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

   if -1.12e15 < re < 1.29999999999999989e-4

   1. Initial program 55.1%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 77.6%

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

   if 1.29999999999999989e-4 < re

   1. Initial program 15.5%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(im, im, re \cdot re\right)}} - re\right)} \cdot \frac{1}{2} \]

      9. *-lowering-*.f64 15.5

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

   4. Applied egg-rr 15.5%

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

   5. Taylor expanded in im around 0

      \[\leadsto \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \cdot \frac{1}{2} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 55.1

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

   7. Simplified 55.1%

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

      1. div-inv N/A

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

      2. sqrt-prod N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sqrt-unprod N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

   9. Applied egg-rr 79.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.12 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 0.00013:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.2e-44)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.95e+151) (* 0.5 (sqrt (* im 2.0))) 0.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.2e-44) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.95e+151) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5.2d-44)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.95d+151) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5.2e-44) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.95e+151) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.2e-44:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.95e+151:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.2e-44)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.95e+151)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5.2e-44)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.95e+151)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.2e-44], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.95e+151], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if re < -5.1999999999999996e-44

   1. Initial program 50.2%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. *-lowering-*.f64 75.4

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

   5. Simplified 75.4%

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

   if -5.1999999999999996e-44 < re < 1.94999999999999988e151

   1. Initial program 44.8%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. *-lowering-*.f64 69.8

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

   5. Simplified 69.8%

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

   if 1.94999999999999988e151 < re

   1. Initial program 5.9%

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

   3. Taylor expanded in re around inf

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

   5. Simplified 36.2%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. +-inverses N/A

         \[\leadsto \frac{1}{2} \cdot {\left(\color{blue}{0} \cdot 2\right)}^{\frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot {\color{blue}{0}}^{\frac{1}{2}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{0} \]

      6. metadata-eval 36.2

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 36.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 53.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.25 \cdot 10^{-240}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.25e-240) 0.0 (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.25e-240) {
		tmp = 0.0;
	} 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 (im <= 1.25d-240) then
        tmp = 0.0d0
    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 (im <= 1.25e-240) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.25e-240:
		tmp = 0.0
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.25e-240)
		tmp = 0.0;
	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 (im <= 1.25e-240)
		tmp = 0.0;
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.25e-240], 0.0, N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.2500000000000001e-240

   1. Initial program 43.1%

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

   3. Taylor expanded in re around inf

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

   5. Simplified 29.6%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. +-inverses N/A

         \[\leadsto \frac{1}{2} \cdot {\left(\color{blue}{0} \cdot 2\right)}^{\frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot {\color{blue}{0}}^{\frac{1}{2}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{0} \]

      6. metadata-eval 29.6

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 29.6%

      \[\leadsto \color{blue}{0} \]

   if 1.2500000000000001e-240 < im

   1. Initial program 41.9%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. *-lowering-*.f64 57.4

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

   5. Simplified 57.4%

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

Alternative 7: 6.1% accurate, 47.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 42.0%

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

3. Taylor expanded in re around inf

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

5. Simplified 7.6%

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

   1. pow1/2 N/A

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

   2. *-commutative N/A

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

   3. +-inverses N/A

      \[\leadsto \frac{1}{2} \cdot {\left(\color{blue}{0} \cdot 2\right)}^{\frac{1}{2}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot {\color{blue}{0}}^{\frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{0} \]

   6. metadata-eval 7.6

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 7.6%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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)))))