math.sqrt on complex, real part

Percentage Accurate: 40.7% → 72.2%

Time: 6.8s

Alternatives: 7

Speedup: 1.7×

Specification

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: 40.7% 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: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{-im\_m}{re} \cdot im\_m} \cdot 0.5\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im\_m} + 2, re, im\_m \cdot 2\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im\_m \cdot im\_m\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{im\_m}{re}, \frac{im\_m}{re}, 4\right) \cdot re} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3e+19)
   (* (sqrt (* (/ (- im_m) re) im_m)) 0.5)
   (if (<= re 9.5e-51)
     (* (sqrt (fma (+ (/ re im_m) 2.0) re (* im_m 2.0))) 0.5)
     (if (<= re 4.7e+32)
       (* (sqrt (* (+ (sqrt (fma re re (* im_m im_m))) re) 2.0)) 0.5)
       (* (sqrt (* (fma (/ im_m re) (/ im_m re) 4.0) re)) 0.5)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3e+19) {
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= 9.5e-51) {
		tmp = sqrt(fma(((re / im_m) + 2.0), re, (im_m * 2.0))) * 0.5;
	} else if (re <= 4.7e+32) {
		tmp = sqrt(((sqrt(fma(re, re, (im_m * im_m))) + re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt((fma((im_m / re), (im_m / re), 4.0) * re)) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3e+19)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im_m) / re) * im_m)) * 0.5);
	elseif (re <= 9.5e-51)
		tmp = Float64(sqrt(fma(Float64(Float64(re / im_m) + 2.0), re, Float64(im_m * 2.0))) * 0.5);
	elseif (re <= 4.7e+32)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(re, re, Float64(im_m * im_m))) + re) * 2.0)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(fma(Float64(im_m / re), Float64(im_m / re), 4.0) * re)) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3e+19], N[(N[Sqrt[N[(N[((-im$95$m) / re), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 9.5e-51], N[(N[Sqrt[N[(N[(N[(re / im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * re + N[(im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 4.7e+32], N[(N[Sqrt[N[(N[(N[Sqrt[N[(re * re + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(im$95$m / re), $MachinePrecision] * N[(im$95$m / re), $MachinePrecision] + 4.0), $MachinePrecision] * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3e19

   1. Initial program 8.6%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -3e19 < re < 9.4999999999999998e-51

   1. Initial program 59.8%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 50.8

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

   5. Applied rewrites 50.8%

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

   if 9.4999999999999998e-51 < re < 4.70000000000000023e32

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 4.70000000000000023e32 < re

   1. Initial program 36.1%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(4 + \frac{{im}^{2}}{{re}^{2}}\right) \cdot re}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(4 + \frac{{im}^{2}}{{re}^{2}}\right) \cdot re}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\frac{{im}^{2}}{{re}^{2}} + 4\right)} \cdot re} \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(\frac{\color{blue}{im \cdot im}}{{re}^{2}} + 4\right) \cdot re} \]

      5. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(\frac{im \cdot im}{\color{blue}{re \cdot re}} + 4\right) \cdot re} \]

      6. times-frac N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(\color{blue}{\frac{im}{re} \cdot \frac{im}{re}} + 4\right) \cdot re} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 81.6

         \[\leadsto 0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{im}{re}, \color{blue}{\frac{im}{re}}, 4\right) \cdot re} \]

   5. Applied rewrites 81.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right) \cdot re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} + 2, re, im \cdot 2\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right) \cdot re} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{-im\_m}{re} \cdot im\_m} \cdot 0.5\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im\_m} + 2, re, im\_m \cdot 2\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im\_m \cdot im\_m\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3e+19)
   (* (sqrt (* (/ (- im_m) re) im_m)) 0.5)
   (if (<= re 9.5e-51)
     (* (sqrt (fma (+ (/ re im_m) 2.0) re (* im_m 2.0))) 0.5)
     (if (<= re 4.7e+32)
       (* (sqrt (* (+ (sqrt (fma re re (* im_m im_m))) re) 2.0)) 0.5)
       (sqrt re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3e+19) {
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= 9.5e-51) {
		tmp = sqrt(fma(((re / im_m) + 2.0), re, (im_m * 2.0))) * 0.5;
	} else if (re <= 4.7e+32) {
		tmp = sqrt(((sqrt(fma(re, re, (im_m * im_m))) + re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3e+19)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im_m) / re) * im_m)) * 0.5);
	elseif (re <= 9.5e-51)
		tmp = Float64(sqrt(fma(Float64(Float64(re / im_m) + 2.0), re, Float64(im_m * 2.0))) * 0.5);
	elseif (re <= 4.7e+32)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(re, re, Float64(im_m * im_m))) + re) * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3e+19], N[(N[Sqrt[N[(N[((-im$95$m) / re), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 9.5e-51], N[(N[Sqrt[N[(N[(N[(re / im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * re + N[(im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 4.7e+32], N[(N[Sqrt[N[(N[(N[Sqrt[N[(re * re + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3e19

   1. Initial program 8.6%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -3e19 < re < 9.4999999999999998e-51

   1. Initial program 59.8%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 50.8

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

   5. Applied rewrites 50.8%

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

   if 9.4999999999999998e-51 < re < 4.70000000000000023e32

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 4.70000000000000023e32 < re

   1. Initial program 36.1%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 81.5

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 81.5%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} + 2, re, im \cdot 2\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{-im\_m}{re} \cdot im\_m} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im\_m} + 2, re, im\_m \cdot 2\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3e+19)
   (* (sqrt (* (/ (- im_m) re) im_m)) 0.5)
   (if (<= re 2.05e-24)
     (* (sqrt (fma (+ (/ re im_m) 2.0) re (* im_m 2.0))) 0.5)
     (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3e+19) {
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= 2.05e-24) {
		tmp = sqrt(fma(((re / im_m) + 2.0), re, (im_m * 2.0))) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3e+19)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im_m) / re) * im_m)) * 0.5);
	elseif (re <= 2.05e-24)
		tmp = Float64(sqrt(fma(Float64(Float64(re / im_m) + 2.0), re, Float64(im_m * 2.0))) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3e+19], N[(N[Sqrt[N[(N[((-im$95$m) / re), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 2.05e-24], N[(N[Sqrt[N[(N[(N[(re / im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * re + N[(im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3e19

   1. Initial program 8.6%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -3e19 < re < 2.05000000000000007e-24

   1. Initial program 60.8%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 50.4

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

   5. Applied rewrites 50.4%

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

   if 2.05000000000000007e-24 < re

   1. Initial program 45.7%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 78.1

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 78.1%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} + 2, re, im \cdot 2\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{-im\_m}{re} \cdot im\_m} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{\left(im\_m + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3e+19)
   (* (sqrt (* (/ (- im_m) re) im_m)) 0.5)
   (if (<= re 2.05e-24) (* (sqrt (* (+ im_m re) 2.0)) 0.5) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3e+19) {
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= 2.05e-24) {
		tmp = sqrt(((im_m + re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-3d+19)) then
        tmp = sqrt(((-im_m / re) * im_m)) * 0.5d0
    else if (re <= 2.05d-24) then
        tmp = sqrt(((im_m + re) * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -3e+19) {
		tmp = Math.sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= 2.05e-24) {
		tmp = Math.sqrt(((im_m + re) * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3e+19:
		tmp = math.sqrt(((-im_m / re) * im_m)) * 0.5
	elif re <= 2.05e-24:
		tmp = math.sqrt(((im_m + re) * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3e+19)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im_m) / re) * im_m)) * 0.5);
	elseif (re <= 2.05e-24)
		tmp = Float64(sqrt(Float64(Float64(im_m + re) * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -3e+19)
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	elseif (re <= 2.05e-24)
		tmp = sqrt(((im_m + re) * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3e+19], N[(N[Sqrt[N[(N[((-im$95$m) / re), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 2.05e-24], N[(N[Sqrt[N[(N[(im$95$m + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3e19

   1. Initial program 8.6%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -3e19 < re < 2.05000000000000007e-24

   1. Initial program 60.8%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 50.7

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

   5. Applied rewrites 50.7%

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

   if 2.05000000000000007e-24 < re

   1. Initial program 45.7%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 78.1

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 78.1%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+229}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) + re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{im\_m \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.2e+229)
   (* (sqrt (* (+ (- re) re) 2.0)) 0.5)
   (if (<= re 2.05e-24) (* (sqrt (* im_m 2.0)) 0.5) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.2e+229) {
		tmp = sqrt(((-re + re) * 2.0)) * 0.5;
	} else if (re <= 2.05e-24) {
		tmp = sqrt((im_m * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-3.2d+229)) then
        tmp = sqrt(((-re + re) * 2.0d0)) * 0.5d0
    else if (re <= 2.05d-24) then
        tmp = sqrt((im_m * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -3.2e+229) {
		tmp = Math.sqrt(((-re + re) * 2.0)) * 0.5;
	} else if (re <= 2.05e-24) {
		tmp = Math.sqrt((im_m * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3.2e+229:
		tmp = math.sqrt(((-re + re) * 2.0)) * 0.5
	elif re <= 2.05e-24:
		tmp = math.sqrt((im_m * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.2e+229)
		tmp = Float64(sqrt(Float64(Float64(Float64(-re) + re) * 2.0)) * 0.5);
	elseif (re <= 2.05e-24)
		tmp = Float64(sqrt(Float64(im_m * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -3.2e+229)
		tmp = sqrt(((-re + re) * 2.0)) * 0.5;
	elseif (re <= 2.05e-24)
		tmp = sqrt((im_m * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.2e+229], N[(N[Sqrt[N[(N[((-re) + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 2.05e-24], N[(N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.1999999999999998e229

   1. Initial program 2.3%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.0

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

   5. Applied rewrites 46.0%

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

   if -3.1999999999999998e229 < re < 2.05000000000000007e-24

   1. Initial program 46.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. lower-*.f64 39.2

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

   5. Applied rewrites 39.2%

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

   if 2.05000000000000007e-24 < re

   1. Initial program 45.7%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 78.1

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 78.1%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+229}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) + re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 2.05 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{im\_m \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 2.05e-24) (* (sqrt (* im_m 2.0)) 0.5) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 2.05e-24) {
		tmp = sqrt((im_m * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 2.05d-24) then
        tmp = sqrt((im_m * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 2.05e-24) {
		tmp = Math.sqrt((im_m * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 2.05e-24:
		tmp = math.sqrt((im_m * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 2.05e-24)
		tmp = Float64(sqrt(Float64(im_m * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 2.05e-24)
		tmp = sqrt((im_m * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 2.05e-24], N[(N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.05000000000000007e-24

   1. Initial program 41.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. lower-*.f64 35.1

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

   5. Applied rewrites 35.1%

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

   if 2.05000000000000007e-24 < re

   1. Initial program 45.7%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 78.1

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 78.1%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.05 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sqrt{re} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (sqrt re))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return sqrt(re);
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = sqrt(re)
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return Math.sqrt(re);
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return math.sqrt(re)

Julia

im_m = abs(im)
function code(re, im_m)
	return sqrt(re)
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = sqrt(re);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 42.6%

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

3. Taylor expanded in re around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

      \[\leadsto \color{blue}{\sqrt{re}} \]

   7. lower-sqrt.f64 22.2

      \[\leadsto \color{blue}{\sqrt{re}} \]

5. Applied rewrites 22.2%

   \[\leadsto \color{blue}{\sqrt{re}} \]
6. Add Preprocessing

Developer Target 1: 47.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t\_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t\_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024304 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< re 0) (* 1/2 (* (sqrt 2) (sqrt (/ (* im im) (- (modulus re im) re))))) (* 1/2 (sqrt (* 2 (+ (modulus re im) re))))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))