math.sqrt on complex, real part

Percentage Accurate: 41.3% → 89.9%

Time: 3.7s

Alternatives: 6

Speedup: 1.8×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    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: 89.9% accurate, 0.5× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* 2.0 (+ (sqrt (+ (* re re) (* im_m im_m))) re)) 0.0)
   (* 0.5 (* (* im_m (sqrt (* -0.5 (/ 1.0 re)))) (sqrt 2.0)))
   (* 0.5 (sqrt (* 2.0 (+ (hypot re im_m) re))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(im$95$m * N[Sqrt[N[(-0.5 * N[(1.0 / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)) < 0.0

   1. Initial program 7.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 7.0%

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

   4. Taylor expanded in re around -inf

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.5

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

   6. Applied rewrites 88.5%

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

   if 0.0 < (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))

   1. Initial program 47.4%

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

      1. lift-sqrt.f64 N/A

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

      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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-hypot.f64 90.2

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

   3. Applied rewrites 90.2%

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

Alternative 2: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \left(\left(im\_m \cdot \sqrt{-0.5 \cdot \frac{1}{re}}\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{-95}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im\_m + re\right)}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(im\_m, im\_m, re \cdot re\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.32e-102)
   (* 0.5 (* (* im_m (sqrt (* -0.5 (/ 1.0 re)))) (sqrt 2.0)))
   (if (<= re 2.5e-95)
     (* 0.5 (sqrt (* 2.0 (+ im_m re))))
     (if (<= re 4.6e+82)
       (* (sqrt (* (+ (sqrt (fma im_m im_m (* re re))) re) 2.0)) 0.5)
       (sqrt re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.32e-102) {
		tmp = 0.5 * ((im_m * sqrt((-0.5 * (1.0 / re)))) * sqrt(2.0));
	} else if (re <= 2.5e-95) {
		tmp = 0.5 * sqrt((2.0 * (im_m + re)));
	} else if (re <= 4.6e+82) {
		tmp = sqrt(((sqrt(fma(im_m, im_m, (re * re))) + 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 <= -1.32e-102)
		tmp = Float64(0.5 * Float64(Float64(im_m * sqrt(Float64(-0.5 * Float64(1.0 / re)))) * sqrt(2.0)));
	elseif (re <= 2.5e-95)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im_m + re))));
	elseif (re <= 4.6e+82)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(im_m, im_m, Float64(re * re))) + 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, -1.32e-102], N[(0.5 * N[(N[(im$95$m * N[Sqrt[N[(-0.5 * N[(1.0 / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.5e-95], N[(0.5 * N[Sqrt[N[(2.0 * N[(im$95$m + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.6e+82], N[(N[Sqrt[N[(N[(N[Sqrt[N[(im$95$m * im$95$m + N[(re * re), $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 < -1.31999999999999996e-102

   1. Initial program 18.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 18.3%

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

   4. Taylor expanded in re around -inf

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 66.5

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

   6. Applied rewrites 66.5%

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

   if -1.31999999999999996e-102 < re < 2.4999999999999999e-95

   1. Initial program 55.9%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 84.7%

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

   if 2.4999999999999999e-95 < re < 4.59999999999999976e82

   1. Initial program 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 75.3%

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

   if 4.59999999999999976e82 < re

   1. Initial program 27.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 27.6%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 27.5

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

   5. Applied rewrites 27.5%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\sqrt{re}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 84.0

         \[\leadsto \sqrt{re} \]

   8. Applied rewrites 84.0%

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

Alternative 3: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \left(\left(im\_m \cdot \sqrt{-0.5 \cdot \frac{1}{re}}\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{+63}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im\_m + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.32e-102)
   (* 0.5 (* (* im_m (sqrt (* -0.5 (/ 1.0 re)))) (sqrt 2.0)))
   (if (<= re 2.9e+63) (* 0.5 (sqrt (* 2.0 (+ im_m re)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.32e-102) {
		tmp = 0.5 * ((im_m * sqrt((-0.5 * (1.0 / re)))) * sqrt(2.0));
	} else if (re <= 2.9e+63) {
		tmp = 0.5 * sqrt((2.0 * (im_m + re)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-1.32d-102)) then
        tmp = 0.5d0 * ((im_m * sqrt(((-0.5d0) * (1.0d0 / re)))) * sqrt(2.0d0))
    else if (re <= 2.9d+63) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im_m + re)))
    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 <= -1.32e-102) {
		tmp = 0.5 * ((im_m * Math.sqrt((-0.5 * (1.0 / re)))) * Math.sqrt(2.0));
	} else if (re <= 2.9e+63) {
		tmp = 0.5 * Math.sqrt((2.0 * (im_m + re)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -1.32e-102:
		tmp = 0.5 * ((im_m * math.sqrt((-0.5 * (1.0 / re)))) * math.sqrt(2.0))
	elif re <= 2.9e+63:
		tmp = 0.5 * math.sqrt((2.0 * (im_m + re)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.32e-102)
		tmp = Float64(0.5 * Float64(Float64(im_m * sqrt(Float64(-0.5 * Float64(1.0 / re)))) * sqrt(2.0)));
	elseif (re <= 2.9e+63)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im_m + re))));
	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 <= -1.32e-102)
		tmp = 0.5 * ((im_m * sqrt((-0.5 * (1.0 / re)))) * sqrt(2.0));
	elseif (re <= 2.9e+63)
		tmp = 0.5 * sqrt((2.0 * (im_m + re)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.32e-102], N[(0.5 * N[(N[(im$95$m * N[Sqrt[N[(-0.5 * N[(1.0 / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.9e+63], N[(0.5 * N[Sqrt[N[(2.0 * N[(im$95$m + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.31999999999999996e-102

   1. Initial program 18.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 18.3%

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

   4. Taylor expanded in re around -inf

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 66.5

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

   6. Applied rewrites 66.5%

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

   if -1.31999999999999996e-102 < re < 2.8999999999999999e63

   1. Initial program 61.2%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 77.9%

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

   if 2.8999999999999999e63 < re

   1. Initial program 31.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 31.6%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 31.5

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\sqrt{re}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 81.8

         \[\leadsto \sqrt{re} \]

   8. Applied rewrites 81.8%

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

Alternative 4: 68.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5.6 \cdot 10^{-85}:\\ \;\;\;\;0.5 \cdot \sqrt{-im\_m \cdot \frac{im\_m}{re}}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{+63}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im\_m + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5.6e-85)
   (* 0.5 (sqrt (- (* im_m (/ im_m re)))))
   (if (<= re 2.9e+63) (* 0.5 (sqrt (* 2.0 (+ im_m re)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5.6e-85) {
		tmp = 0.5 * sqrt(-(im_m * (im_m / re)));
	} else if (re <= 2.9e+63) {
		tmp = 0.5 * sqrt((2.0 * (im_m + re)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-5.6d-85)) then
        tmp = 0.5d0 * sqrt(-(im_m * (im_m / re)))
    else if (re <= 2.9d+63) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im_m + re)))
    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 <= -5.6e-85) {
		tmp = 0.5 * Math.sqrt(-(im_m * (im_m / re)));
	} else if (re <= 2.9e+63) {
		tmp = 0.5 * Math.sqrt((2.0 * (im_m + re)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -5.6e-85:
		tmp = 0.5 * math.sqrt(-(im_m * (im_m / re)))
	elif re <= 2.9e+63:
		tmp = 0.5 * math.sqrt((2.0 * (im_m + re)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -5.6e-85)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(im_m * Float64(im_m / re)))));
	elseif (re <= 2.9e+63)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im_m + re))));
	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 <= -5.6e-85)
		tmp = 0.5 * sqrt(-(im_m * (im_m / re)));
	elseif (re <= 2.9e+63)
		tmp = 0.5 * sqrt((2.0 * (im_m + re)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -5.6e-85], N[(0.5 * N[Sqrt[(-N[(im$95$m * N[(im$95$m / re), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.9e+63], N[(0.5 * N[Sqrt[N[(2.0 * N[(im$95$m + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.60000000000000033e-85

   1. Initial program 17.1%

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

   2. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 40.1

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

   4. Applied rewrites 40.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 45.5

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

   6. Applied rewrites 45.5%

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

   if -5.60000000000000033e-85 < re < 2.8999999999999999e63

   1. Initial program 60.6%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 77.3%

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

   if 2.8999999999999999e63 < re

   1. Initial program 31.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 31.6%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 31.5

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\sqrt{re}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 81.8

         \[\leadsto \sqrt{re} \]

   8. Applied rewrites 81.8%

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

Alternative 5: 64.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 6.6 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m + im\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 6.6e-67) (* 0.5 (sqrt (+ im_m im_m))) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 6.6e-67) {
		tmp = 0.5 * sqrt((im_m + im_m));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 6.6d-67) then
        tmp = 0.5d0 * sqrt((im_m + im_m))
    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 <= 6.6e-67) {
		tmp = 0.5 * Math.sqrt((im_m + im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 6.6e-67:
		tmp = 0.5 * math.sqrt((im_m + im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 6.6e-67)
		tmp = Float64(0.5 * sqrt(Float64(im_m + im_m)));
	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 <= 6.6e-67)
		tmp = 0.5 * sqrt((im_m + im_m));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 6.6e-67], N[(0.5 * N[Sqrt[N[(im$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.6000000000000003e-67

   1. Initial program 38.6%

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

   2. Taylor expanded in re around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

   if 6.6000000000000003e-67 < re

   1. Initial program 47.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 47.3%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\sqrt{re}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 72.2

         \[\leadsto \sqrt{re} \]

   8. Applied rewrites 72.2%

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

Alternative 6: 25.8% accurate, 7.9× 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 =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im_m)
use fmin_fmax_functions
    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 41.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. *-commutative N/A

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

   9. sqrt-prod N/A

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

   10. pow1/2 N/A

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

   11. lower-*.f64 N/A

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

3. Applied rewrites 41.0%

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

   1. lift-sqrt.f64 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. lift-sqrt.f64 40.9

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

5. Applied rewrites 40.9%

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

6. Taylor expanded in re around inf

   \[\leadsto \color{blue}{\sqrt{re}} \]
7. Step-by-step derivation

   1. lower-sqrt.f64 25.8

      \[\leadsto \sqrt{re} \]

8. Applied rewrites 25.8%

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

Developer Target 1: 48.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    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 2025106 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform c (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)))))