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

Percentage Accurate: 41.5% → 88.0%

Time: 9.8s

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: 41.5% 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: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.7 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.7e-37)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (/ (* 0.5 im) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.7e-37) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.7e-37) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.7e-37:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.7e-37)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.7e-37)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.7e-37], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.7e-37

   1. Initial program 50.7%

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right)\right) \]

      5. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right), re\right)\right)\right)\right) \]

      6. hypot-lowering-hypot.f64 97.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right)\right) \]

   3. Simplified 97.0%

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

   if 3.7e-37 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{{im}^{2}}{re}\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right) \]

      3. *-lowering-*.f64 40.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right) \]

   5. Simplified 40.4%

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

      1. sqrt-div N/A

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

      2. associate-*r/ N/A

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

      3. pow1/2 N/A

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

      4. unpow-prod-down N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left({im}^{\frac{1}{2}} \cdot {im}^{\frac{1}{2}}\right)\right), \color{blue}{\left(\sqrt{re}\right)}\right) \]

      6. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {im}^{\left(\frac{1}{2} + \frac{1}{2}\right)}\right), \left(\sqrt{re}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {im}^{1}\right), \left(\sqrt{re}\right)\right) \]

      8. unpow1 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot im\right), \left(\sqrt{re}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, im\right), \left(\sqrt{\color{blue}{re}}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, im\right), \mathsf{sqrt.f64}\left(re\right)\right) \]

   7. Applied egg-rr 76.2%

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

Alternative 2: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{re}{-0.25} - \frac{im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 7.8 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.75e-31)
   (* 0.5 (sqrt (- (/ re -0.25) (/ im (/ re im)))))
   (if (<= re 7.8e-26)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.75e-31) {
		tmp = 0.5 * sqrt(((re / -0.25) - (im / (re / im))));
	} else if (re <= 7.8e-26) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.75d-31)) then
        tmp = 0.5d0 * sqrt(((re / (-0.25d0)) - (im / (re / im))))
    else if (re <= 7.8d-26) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (0.5d0 * im) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.75e-31) {
		tmp = 0.5 * Math.sqrt(((re / -0.25) - (im / (re / im))));
	} else if (re <= 7.8e-26) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.75e-31:
		tmp = 0.5 * math.sqrt(((re / -0.25) - (im / (re / im))))
	elif re <= 7.8e-26:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.75e-31)
		tmp = Float64(0.5 * sqrt(Float64(Float64(re / -0.25) - Float64(im / Float64(re / im)))));
	elseif (re <= 7.8e-26)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.75e-31)
		tmp = 0.5 * sqrt(((re / -0.25) - (im / (re / im))));
	elseif (re <= 7.8e-26)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.75e-31], N[(0.5 * N[Sqrt[N[(N[(re / -0.25), $MachinePrecision] - N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.8e-26], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.74999999999999993e-31

   1. Initial program 42.3%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(re \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(re \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\left(4 + \frac{{im}^{2}}{{re}^{2}}\right) \cdot re\right)\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\left(4 + \frac{{im}^{2}}{{re}^{2}}\right) \cdot \left(\mathsf{neg}\left(re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(4 + \frac{{im}^{2}}{{re}^{2}}\right), \left(\mathsf{neg}\left(re\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{{im}^{2}}{{re}^{2}} + 4\right), \left(\mathsf{neg}\left(re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{{im}^{2}}{{re}^{2}}\right), 4\right), \left(\mathsf{neg}\left(re\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({im}^{2}\right), \left({re}^{2}\right)\right), 4\right), \left(\mathsf{neg}\left(re\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(im \cdot im\right), \left({re}^{2}\right)\right), 4\right), \left(\mathsf{neg}\left(re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({re}^{2}\right)\right), 4\right), \left(\mathsf{neg}\left(re\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(re \cdot re\right)\right), 4\right), \left(\mathsf{neg}\left(re\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(re, re\right)\right), 4\right), \left(\mathsf{neg}\left(re\right)\right)\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(re, re\right)\right), 4\right), \left(0 - re\right)\right)\right)\right) \]

      13. --lowering--.f64 74.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(re, re\right)\right), 4\right), \mathsf{\_.f64}\left(0, re\right)\right)\right)\right) \]

   5. Simplified 74.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(\frac{im \cdot im}{re \cdot re} + 4\right) \cdot \left(0 - re\right)}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re + -1 \cdot \frac{{im}^{2}}{re}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4 + -1 \cdot \frac{{im}^{2}}{re}\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot \left(-2 - 2\right) + -1 \cdot \frac{{im}^{2}}{re}\right)\right)\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\left(-2 \cdot re - 2 \cdot re\right) + -1 \cdot \frac{{im}^{2}}{re}\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\left(-2 \cdot re - 2 \cdot re\right) + \left(\mathsf{neg}\left(\frac{{im}^{2}}{re}\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\left(-2 \cdot re - 2 \cdot re\right) - \frac{{im}^{2}}{re}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(-2 \cdot re - 2 \cdot re\right), \left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(re \cdot \left(-2 - 2\right)\right), \left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(re \cdot -4\right), \left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, -4\right), \left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, -4\right), \mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, -4\right), \mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right)\right) \]

      12. *-lowering-*.f64 75.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right)\right) \]

   8. Simplified 75.1%

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{re \cdot -4 - \frac{im \cdot im}{re}}\right), \color{blue}{\frac{1}{2}}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(re \cdot -4 - \frac{im \cdot im}{re}\right)\right), \frac{1}{2}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(re \cdot \frac{1}{\frac{-1}{4}} - \frac{im \cdot im}{re}\right)\right), \frac{1}{2}\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{re}{\frac{-1}{4}} - \frac{im \cdot im}{re}\right)\right), \frac{1}{2}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(\frac{re}{\frac{-1}{4}}\right), \left(\frac{im \cdot im}{re}\right)\right)\right), \frac{1}{2}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(re, \frac{-1}{4}\right), \left(\frac{im \cdot im}{re}\right)\right)\right), \frac{1}{2}\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(re, \frac{-1}{4}\right), \left(\frac{im}{re} \cdot im\right)\right)\right), \frac{1}{2}\right) \]

      9. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(re, \frac{-1}{4}\right), \left(\frac{im}{\frac{re}{im}}\right)\right)\right), \frac{1}{2}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(re, \frac{-1}{4}\right), \mathsf{/.f64}\left(im, \left(\frac{re}{im}\right)\right)\right)\right), \frac{1}{2}\right) \]

      11. /-lowering-/.f64 81.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(re, \frac{-1}{4}\right), \mathsf{/.f64}\left(im, \mathsf{/.f64}\left(re, im\right)\right)\right)\right), \frac{1}{2}\right) \]

   10. Applied egg-rr 81.8%

       \[\leadsto \color{blue}{\sqrt{\frac{re}{-0.25} - \frac{im}{\frac{re}{im}}} \cdot 0.5} \]

   if -1.74999999999999993e-31 < re < 7.79999999999999973e-26

   1. Initial program 55.9%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\color{blue}{im}, re\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 87.8%

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

   if 7.79999999999999973e-26 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{{im}^{2}}{re}\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right) \]

      3. *-lowering-*.f64 40.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right) \]

   5. Simplified 40.4%

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

      1. sqrt-div N/A

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

      2. associate-*r/ N/A

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

      3. pow1/2 N/A

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

      4. unpow-prod-down N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left({im}^{\frac{1}{2}} \cdot {im}^{\frac{1}{2}}\right)\right), \color{blue}{\left(\sqrt{re}\right)}\right) \]

      6. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {im}^{\left(\frac{1}{2} + \frac{1}{2}\right)}\right), \left(\sqrt{re}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {im}^{1}\right), \left(\sqrt{re}\right)\right) \]

      8. unpow1 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot im\right), \left(\sqrt{re}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, im\right), \left(\sqrt{\color{blue}{re}}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, im\right), \mathsf{sqrt.f64}\left(re\right)\right) \]

   7. Applied egg-rr 76.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{re}{-0.25} - \frac{im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 7.8 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.65 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.65e-31)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.3e-35)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.65e-31) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.3e-35) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.65d-31)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.3d-35) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (0.5d0 * im) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.65e-31) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.3e-35) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.65e-31:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.3e-35:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.65e-31)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.3e-35)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.65e-31)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.3e-35)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.65e-31], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.3e-35], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.65e-31

   1. Initial program 42.3%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 81.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 81.7%

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

   if -1.65e-31 < re < 1.30000000000000002e-35

   1. Initial program 55.9%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\color{blue}{im}, re\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 87.8%

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

   if 1.30000000000000002e-35 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{{im}^{2}}{re}\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right) \]

      3. *-lowering-*.f64 40.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right) \]

   5. Simplified 40.4%

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

      1. sqrt-div N/A

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

      2. associate-*r/ N/A

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

      3. pow1/2 N/A

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

      4. unpow-prod-down N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left({im}^{\frac{1}{2}} \cdot {im}^{\frac{1}{2}}\right)\right), \color{blue}{\left(\sqrt{re}\right)}\right) \]

      6. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {im}^{\left(\frac{1}{2} + \frac{1}{2}\right)}\right), \left(\sqrt{re}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {im}^{1}\right), \left(\sqrt{re}\right)\right) \]

      8. unpow1 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot im\right), \left(\sqrt{re}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, im\right), \left(\sqrt{\color{blue}{re}}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, im\right), \mathsf{sqrt.f64}\left(re\right)\right) \]

   7. Applied egg-rr 76.2%

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

Alternative 4: 74.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3e-84)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 5.5e-40) (* 0.5 (sqrt (* 2.0 im))) (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3e-84) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5.5e-40) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3d-84)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5.5d-40) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = (0.5d0 * im) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3e-84) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5.5e-40) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3e-84:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5.5e-40:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3e-84)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5.5e-40)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3e-84)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5.5e-40)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3e-84], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.5e-40], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.0000000000000001e-84

   1. Initial program 45.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 79.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 79.6%

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

   if -3.0000000000000001e-84 < re < 5.50000000000000002e-40

   1. Initial program 54.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 88.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 88.4%

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

   if 5.50000000000000002e-40 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{{im}^{2}}{re}\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right) \]

      3. *-lowering-*.f64 40.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right) \]

   5. Simplified 40.4%

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

      1. sqrt-div N/A

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

      2. associate-*r/ N/A

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

      3. pow1/2 N/A

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

      4. unpow-prod-down N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left({im}^{\frac{1}{2}} \cdot {im}^{\frac{1}{2}}\right)\right), \color{blue}{\left(\sqrt{re}\right)}\right) \]

      6. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {im}^{\left(\frac{1}{2} + \frac{1}{2}\right)}\right), \left(\sqrt{re}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {im}^{1}\right), \left(\sqrt{re}\right)\right) \]

      8. unpow1 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot im\right), \left(\sqrt{re}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, im\right), \left(\sqrt{\color{blue}{re}}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, im\right), \mathsf{sqrt.f64}\left(re\right)\right) \]

   7. Applied egg-rr 76.2%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.65 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.65e-84)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.7e-37) (* 0.5 (sqrt (* 2.0 im))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.65e-84) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.7e-37) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= (-1.65d-84)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.7d-37) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= -1.65e-84) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.7e-37) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.65e-84:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.7e-37:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.65e-84)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.7e-37)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= -1.65e-84)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.7e-37)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.65e-84], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.7e-37], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.64999999999999992e-84

   1. Initial program 45.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 79.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 79.6%

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

   if -1.64999999999999992e-84 < re < 2.70000000000000016e-37

   1. Initial program 54.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 88.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 88.4%

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

   if 2.70000000000000016e-37 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{{im}^{2}}{re}\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right) \]

      3. *-lowering-*.f64 40.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right) \]

   5. Simplified 40.4%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{\frac{1}{2}}{\sqrt{re}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{re}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 76.0%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(re\right)\right)\right) \]

   7. Applied egg-rr 76.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.65 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.8e-84) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-84) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.8d-84)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-84) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.8e-84:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.8e-84)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.8e-84)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.8e-84], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.80000000000000035e-84

   1. Initial program 45.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 79.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 79.6%

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

   if -4.80000000000000035e-84 < re

   1. Initial program 38.7%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 66.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 66.8%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

3. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

   2. *-lowering-*.f64 54.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

5. Simplified 54.2%

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

6. Final simplification 54.2%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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)))))