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

Percentage Accurate: 41.2% → 87.5%

Time: 6.8s

Alternatives: 13

Speedup: 2.5×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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: 41.2% 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: 87.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.1 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left({0.5}^{0.25} \cdot {0.5}^{0.25}\right) \cdot 0.5\right) \cdot \left(im \cdot \left({re}^{-0.5} \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.1e-35)
   (* (sqrt (* (- (hypot im re) re) 2.0)) 0.5)
   (*
    (* (* (pow 0.5 0.25) (pow 0.5 0.25)) 0.5)
    (* im (* (pow re -0.5) (sqrt 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.1e-35) {
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = ((pow(0.5, 0.25) * pow(0.5, 0.25)) * 0.5) * (im * (pow(re, -0.5) * sqrt(2.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.1e-35) {
		tmp = Math.sqrt(((Math.hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = ((Math.pow(0.5, 0.25) * Math.pow(0.5, 0.25)) * 0.5) * (im * (Math.pow(re, -0.5) * Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.1e-35:
		tmp = math.sqrt(((math.hypot(im, re) - re) * 2.0)) * 0.5
	else:
		tmp = ((math.pow(0.5, 0.25) * math.pow(0.5, 0.25)) * 0.5) * (im * (math.pow(re, -0.5) * math.sqrt(2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.1e-35)
		tmp = Float64(sqrt(Float64(Float64(hypot(im, re) - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(Float64((0.5 ^ 0.25) * (0.5 ^ 0.25)) * 0.5) * Float64(im * Float64((re ^ -0.5) * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.1e-35)
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	else
		tmp = (((0.5 ^ 0.25) * (0.5 ^ 0.25)) * 0.5) * (im * ((re ^ -0.5) * sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.1e-35], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Power[0.5, 0.25], $MachinePrecision] * N[Power[0.5, 0.25], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * N[(N[Power[re, -0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.1e-35

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 55.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 95.2

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

   4. Applied rewrites 95.2%

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

   if 2.1e-35 < re

   1. Initial program 10.5%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 75.8%

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

   9. Applied rewrites 76.2%

      \[\leadsto \left(\left({0.5}^{0.25} \cdot {0.5}^{0.25}\right) \cdot 0.5\right) \cdot \left(im \cdot \left({re}^{-0.5} \cdot \sqrt{2}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 87.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.1 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{{re}^{-1}} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot e^{\log 2 \cdot 0.5}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.1e-35)
   (* (sqrt (* (- (hypot im re) re) 2.0)) 0.5)
   (*
    (* (* (sqrt (pow re -1.0)) (* (sqrt 0.5) im)) (exp (* (log 2.0) 0.5)))
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.1e-35) {
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = ((sqrt(pow(re, -1.0)) * (sqrt(0.5) * im)) * exp((log(2.0) * 0.5))) * 0.5;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.1e-35) {
		tmp = Math.sqrt(((Math.hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = ((Math.sqrt(Math.pow(re, -1.0)) * (Math.sqrt(0.5) * im)) * Math.exp((Math.log(2.0) * 0.5))) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.1e-35:
		tmp = math.sqrt(((math.hypot(im, re) - re) * 2.0)) * 0.5
	else:
		tmp = ((math.sqrt(math.pow(re, -1.0)) * (math.sqrt(0.5) * im)) * math.exp((math.log(2.0) * 0.5))) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.1e-35)
		tmp = Float64(sqrt(Float64(Float64(hypot(im, re) - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(Float64(sqrt((re ^ -1.0)) * Float64(sqrt(0.5) * im)) * exp(Float64(log(2.0) * 0.5))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.1e-35)
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	else
		tmp = ((sqrt((re ^ -1.0)) * (sqrt(0.5) * im)) * exp((log(2.0) * 0.5))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.1e-35], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Sqrt[N[Power[re, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.1e-35

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 55.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 95.2

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

   4. Applied rewrites 95.2%

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

   if 2.1e-35 < re

   1. Initial program 10.5%

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

      1. 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. *-commutative N/A

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

      3. lower-*.f64 10.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 10.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 38.4

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

   4. Applied rewrites 38.4%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 75.6

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

   7. Applied rewrites 75.6%

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

   9. Applied rewrites 76.1%

      \[\leadsto \left(\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot e^{\log 2 \cdot 0.5}\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.1 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{{re}^{-1}} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot e^{\log 2 \cdot 0.5}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq -9 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, re, im\right) + im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot 0.5\right) \cdot \left(im \cdot \left(\sqrt{{re}^{-1}} \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7e+146)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re -9e-125)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma im im (* re re))) re))))
     (if (<= re 2.1e-35)
       (* (sqrt (+ (fma -2.0 re im) im)) 0.5)
       (* (* (sqrt 0.5) 0.5) (* im (* (sqrt (pow re -1.0)) (sqrt 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7e+146) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= -9e-125) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(im, im, (re * re))) - re)));
	} else if (re <= 2.1e-35) {
		tmp = sqrt((fma(-2.0, re, im) + im)) * 0.5;
	} else {
		tmp = (sqrt(0.5) * 0.5) * (im * (sqrt(pow(re, -1.0)) * sqrt(2.0)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7e+146)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= -9e-125)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(im, im, Float64(re * re))) - re))));
	elseif (re <= 2.1e-35)
		tmp = Float64(sqrt(Float64(fma(-2.0, re, im) + im)) * 0.5);
	else
		tmp = Float64(Float64(sqrt(0.5) * 0.5) * Float64(im * Float64(sqrt((re ^ -1.0)) * sqrt(2.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7e+146], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, -9e-125], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e-35], N[(N[Sqrt[N[(N[(-2.0 * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Sqrt[0.5], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * N[(N[Sqrt[N[Power[re, -1.0], $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.0000000000000002e146

   1. Initial program 4.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 4.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 4.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 84.4

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

   7. Applied rewrites 84.4%

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

   if -7.0000000000000002e146 < re < -9.00000000000000024e-125

   1. Initial program 79.6%

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

      1. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 79.6

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

   4. Applied rewrites 79.6%

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

   if -9.00000000000000024e-125 < re < 2.1e-35

   1. Initial program 50.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.4

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 50.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 90.7

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 84.1%

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

   if 2.1e-35 < re

   1. Initial program 10.5%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 75.7%

       \[\leadsto \left(\sqrt{0.5} \cdot 0.5\right) \cdot \left(im \cdot \left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\sqrt{2}}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq -9 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, re, im\right) + im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot 0.5\right) \cdot \left(im \cdot \left(\sqrt{{re}^{-1}} \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.45 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot 0.5\right) \cdot \left(im \cdot \left(\sqrt{{re}^{-1}} \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.45e-32)
   (* (sqrt (* (- (hypot im re) re) 2.0)) 0.5)
   (* (* (sqrt 0.5) 0.5) (* im (* (sqrt (pow re -1.0)) (sqrt 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.45e-32) {
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = (sqrt(0.5) * 0.5) * (im * (sqrt(pow(re, -1.0)) * sqrt(2.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.45e-32) {
		tmp = Math.sqrt(((Math.hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = (Math.sqrt(0.5) * 0.5) * (im * (Math.sqrt(Math.pow(re, -1.0)) * Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.45e-32:
		tmp = math.sqrt(((math.hypot(im, re) - re) * 2.0)) * 0.5
	else:
		tmp = (math.sqrt(0.5) * 0.5) * (im * (math.sqrt(math.pow(re, -1.0)) * math.sqrt(2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.45e-32)
		tmp = Float64(sqrt(Float64(Float64(hypot(im, re) - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(sqrt(0.5) * 0.5) * Float64(im * Float64(sqrt((re ^ -1.0)) * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.45e-32)
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	else
		tmp = (sqrt(0.5) * 0.5) * (im * (sqrt((re ^ -1.0)) * sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.45e-32], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Sqrt[0.5], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * N[(N[Sqrt[N[Power[re, -1.0], $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.4499999999999999e-32

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 55.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 95.2

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

   4. Applied rewrites 95.2%

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

   if 2.4499999999999999e-32 < re

   1. Initial program 10.5%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 75.7%

       \[\leadsto \left(\sqrt{0.5} \cdot 0.5\right) \cdot \left(im \cdot \left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\sqrt{2}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.45 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot 0.5\right) \cdot \left(im \cdot \left(\sqrt{{re}^{-1}} \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\frac{im \cdot im}{re}} \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+73}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))))
   (if (<= t_0 5e-83)
     (* (sqrt (/ (* im im) re)) 0.5)
     (if (<= t_0 2e+73)
       (* 0.5 (sqrt (* 2.0 (- (sqrt (fma im im (* re re))) re))))
       (* 0.5 (sqrt (* 2.0 (- im re))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	double tmp;
	if (t_0 <= 5e-83) {
		tmp = sqrt(((im * im) / re)) * 0.5;
	} else if (t_0 <= 2e+73) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(im, im, (re * re))) - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
	tmp = 0.0
	if (t_0 <= 5e-83)
		tmp = Float64(sqrt(Float64(Float64(im * im) / re)) * 0.5);
	elseif (t_0 <= 2e+73)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(im, im, Float64(re * re))) - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 5e-83], N[(N[Sqrt[N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e+73], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 5e-83

   1. Initial program 12.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 12.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 12.7

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 26.3

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

   4. Applied rewrites 26.3%

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

   5. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 49.0

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

   7. Applied rewrites 49.0%

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

   if 5e-83 < (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 1.99999999999999997e73

   1. Initial program 98.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if 1.99999999999999997e73 < (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))

   1. Initial program 6.7%

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

   3. Taylor expanded in re around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 59.5

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

   5. Applied rewrites 59.5%

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

Alternative 6: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.45 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot 0.5\right) \cdot \left(im \cdot \left({re}^{-0.5} \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.45e-32)
   (* (sqrt (* (- (hypot im re) re) 2.0)) 0.5)
   (* (* (sqrt 0.5) 0.5) (* im (* (pow re -0.5) (sqrt 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.45e-32) {
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = (sqrt(0.5) * 0.5) * (im * (pow(re, -0.5) * sqrt(2.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.45e-32) {
		tmp = Math.sqrt(((Math.hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = (Math.sqrt(0.5) * 0.5) * (im * (Math.pow(re, -0.5) * Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.45e-32:
		tmp = math.sqrt(((math.hypot(im, re) - re) * 2.0)) * 0.5
	else:
		tmp = (math.sqrt(0.5) * 0.5) * (im * (math.pow(re, -0.5) * math.sqrt(2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.45e-32)
		tmp = Float64(sqrt(Float64(Float64(hypot(im, re) - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(sqrt(0.5) * 0.5) * Float64(im * Float64((re ^ -0.5) * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.45e-32)
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	else
		tmp = (sqrt(0.5) * 0.5) * (im * ((re ^ -0.5) * sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.45e-32], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Sqrt[0.5], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * N[(N[Power[re, -0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.4499999999999999e-32

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 55.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 95.2

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

   4. Applied rewrites 95.2%

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

   if 2.4499999999999999e-32 < re

   1. Initial program 10.5%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 75.8%

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

Alternative 7: 78.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq -9 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, re, im\right) + im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot im\right) \cdot \sqrt{0.5}}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7e+146)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re -9e-125)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma im im (* re re))) re))))
     (if (<= re 2.1e-35)
       (* (sqrt (+ (fma -2.0 re im) im)) 0.5)
       (* (/ (* (* (sqrt 2.0) im) (sqrt 0.5)) (sqrt re)) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7e+146) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= -9e-125) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(im, im, (re * re))) - re)));
	} else if (re <= 2.1e-35) {
		tmp = sqrt((fma(-2.0, re, im) + im)) * 0.5;
	} else {
		tmp = (((sqrt(2.0) * im) * sqrt(0.5)) / sqrt(re)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7e+146)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= -9e-125)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(im, im, Float64(re * re))) - re))));
	elseif (re <= 2.1e-35)
		tmp = Float64(sqrt(Float64(fma(-2.0, re, im) + im)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * im) * sqrt(0.5)) / sqrt(re)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7e+146], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, -9e-125], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e-35], N[(N[Sqrt[N[(N[(-2.0 * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * im), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.0000000000000002e146

   1. Initial program 4.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 4.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 4.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 84.4

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

   7. Applied rewrites 84.4%

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

   if -7.0000000000000002e146 < re < -9.00000000000000024e-125

   1. Initial program 79.6%

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

      1. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 79.6

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

   4. Applied rewrites 79.6%

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

   if -9.00000000000000024e-125 < re < 2.1e-35

   1. Initial program 50.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.4

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 50.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 90.7

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 84.1%

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

   if 2.1e-35 < re

   1. Initial program 10.5%

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

      1. 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. *-commutative N/A

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

      3. lower-*.f64 10.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 10.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 38.4

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

   4. Applied rewrites 38.4%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 75.6

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

   7. Applied rewrites 75.6%

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

   9. Applied rewrites 75.6%

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

   11. Applied rewrites 75.6%

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

Alternative 8: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq -9 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, re, im\right) + im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \frac{\sqrt{0.5} \cdot im}{\sqrt{re}}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7e+146)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re -9e-125)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma im im (* re re))) re))))
     (if (<= re 2.1e-35)
       (* (sqrt (+ (fma -2.0 re im) im)) 0.5)
       (* (* (sqrt 2.0) (/ (* (sqrt 0.5) im) (sqrt re))) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7e+146) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= -9e-125) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(im, im, (re * re))) - re)));
	} else if (re <= 2.1e-35) {
		tmp = sqrt((fma(-2.0, re, im) + im)) * 0.5;
	} else {
		tmp = (sqrt(2.0) * ((sqrt(0.5) * im) / sqrt(re))) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7e+146)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= -9e-125)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(im, im, Float64(re * re))) - re))));
	elseif (re <= 2.1e-35)
		tmp = Float64(sqrt(Float64(fma(-2.0, re, im) + im)) * 0.5);
	else
		tmp = Float64(Float64(sqrt(2.0) * Float64(Float64(sqrt(0.5) * im) / sqrt(re))) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7e+146], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, -9e-125], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e-35], N[(N[Sqrt[N[(N[(-2.0 * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.0000000000000002e146

   1. Initial program 4.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 4.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 4.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 84.4

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

   7. Applied rewrites 84.4%

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

   if -7.0000000000000002e146 < re < -9.00000000000000024e-125

   1. Initial program 79.6%

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

      1. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 79.6

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

   4. Applied rewrites 79.6%

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

   if -9.00000000000000024e-125 < re < 2.1e-35

   1. Initial program 50.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.4

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 50.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 90.7

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 84.1%

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

   if 2.1e-35 < re

   1. Initial program 10.5%

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

      1. 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. *-commutative N/A

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

      3. lower-*.f64 10.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 10.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 38.4

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

   4. Applied rewrites 38.4%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 75.6

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

   7. Applied rewrites 75.6%

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

   9. Applied rewrites 75.6%

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

   11. Applied rewrites 75.6%

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

Alternative 9: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq -6.5 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im\right) + im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{\left(\frac{im}{re} \cdot im\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7e+146)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re -6.5e-89)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma im im (* re re))) re))))
     (if (<= re 3.9e+135)
       (* (sqrt (+ (fma (- (/ re im) 2.0) re im) im)) 0.5)
       (* (* 0.5 (sqrt 2.0)) (sqrt (* (* (/ im re) im) 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7e+146) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= -6.5e-89) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(im, im, (re * re))) - re)));
	} else if (re <= 3.9e+135) {
		tmp = sqrt((fma(((re / im) - 2.0), re, im) + im)) * 0.5;
	} else {
		tmp = (0.5 * sqrt(2.0)) * sqrt((((im / re) * im) * 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7e+146)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= -6.5e-89)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(im, im, Float64(re * re))) - re))));
	elseif (re <= 3.9e+135)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(re / im) - 2.0), re, im) + im)) * 0.5);
	else
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * sqrt(Float64(Float64(Float64(im / re) * im) * 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7e+146], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, -6.5e-89], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.9e+135], N[(N[Sqrt[N[(N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(im / re), $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.0000000000000002e146

   1. Initial program 4.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 4.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 4.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 84.4

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

   7. Applied rewrites 84.4%

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

   if -7.0000000000000002e146 < re < -6.50000000000000034e-89

   1. Initial program 78.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 78.1

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

   4. Applied rewrites 78.1%

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

   if -6.50000000000000034e-89 < re < 3.90000000000000032e135

   1. Initial program 46.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 46.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 46.1

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 75.4

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

   7. Applied rewrites 75.4%

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

   9. Applied rewrites 75.4%

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

   if 3.90000000000000032e135 < re

   1. Initial program 5.3%

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

   3. Taylor expanded in re around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 15.2

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

   5. Applied rewrites 15.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 15.1

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

   7. Applied rewrites 15.1%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 58.9

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

   10. Applied rewrites 58.9%

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

   12. Applied rewrites 78.0%

       \[\leadsto \left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{\left(\frac{im}{re} \cdot im\right) \cdot 0.5} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00024:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im\right) + im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{im \cdot im}{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00024)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 3.9e+135)
     (* (sqrt (+ (fma (- (/ re im) 2.0) re im) im)) 0.5)
     (* (sqrt (/ (* im im) re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00024) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 3.9e+135) {
		tmp = sqrt((fma(((re / im) - 2.0), re, im) + im)) * 0.5;
	} else {
		tmp = sqrt(((im * im) / re)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00024)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 3.9e+135)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(re / im) - 2.0), re, im) + im)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(Float64(im * im) / re)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00024], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 3.9e+135], N[(N[Sqrt[N[(N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.40000000000000006e-4

   1. Initial program 50.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.4

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 50.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 77.7

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

   7. Applied rewrites 77.7%

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

   if -2.40000000000000006e-4 < re < 3.90000000000000032e135

   1. Initial program 50.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 50.1

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 84.7

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 73.8

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

   7. Applied rewrites 73.8%

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

   9. Applied rewrites 73.8%

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

   if 3.90000000000000032e135 < re

   1. Initial program 5.3%

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

      1. 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. *-commutative N/A

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

      3. lower-*.f64 5.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 5.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 31.9

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

   4. Applied rewrites 31.9%

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

   5. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 59.2

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

   7. Applied rewrites 59.2%

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

Alternative 11: 68.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00029:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{im \cdot im}{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00029)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 1.5e-30)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (sqrt (/ (* im im) re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00029) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 1.5e-30) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = sqrt(((im * im) / re)) * 0.5;
	}
	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) :: tmp
    if (re <= (-0.00029d0)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else if (re <= 1.5d-30) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = sqrt(((im * im) / re)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00029) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 1.5e-30) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = Math.sqrt(((im * im) / re)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00029:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	elif re <= 1.5e-30:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = math.sqrt(((im * im) / re)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00029)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 1.5e-30)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(sqrt(Float64(Float64(im * im) / re)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00029)
		tmp = sqrt((-4.0 * re)) * 0.5;
	elseif (re <= 1.5e-30)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = sqrt(((im * im) / re)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00029], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.5e-30], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.9e-4

   1. Initial program 50.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.4

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 50.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 77.7

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

   7. Applied rewrites 77.7%

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

   if -2.9e-4 < re < 1.49999999999999995e-30

   1. Initial program 57.4%

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

   3. Taylor expanded in re around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 81.1

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

   5. Applied rewrites 81.1%

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

   if 1.49999999999999995e-30 < re

   1. Initial program 10.5%

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

      1. 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. *-commutative N/A

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

      3. lower-*.f64 10.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 10.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 38.4

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

   4. Applied rewrites 38.4%

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

   5. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 52.7

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

   7. Applied rewrites 52.7%

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

Alternative 12: 64.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00024:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00024) (* (sqrt (* -4.0 re)) 0.5) (* (sqrt (+ im im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00024) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = sqrt((im + im)) * 0.5;
	}
	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) :: tmp
    if (re <= (-0.00024d0)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else
        tmp = sqrt((im + im)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00024) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = Math.sqrt((im + im)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00024:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	else:
		tmp = math.sqrt((im + im)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00024)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00024)
		tmp = sqrt((-4.0 * re)) * 0.5;
	else
		tmp = sqrt((im + im)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00024], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.40000000000000006e-4

   1. Initial program 50.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.4

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 50.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 77.7

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

   7. Applied rewrites 77.7%

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

   if -2.40000000000000006e-4 < re

   1. Initial program 39.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 39.8

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 39.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 72.6

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 60.5

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

   7. Applied rewrites 60.5%

      \[\leadsto \sqrt{\color{blue}{2 \cdot im}} \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 60.5%

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

Alternative 13: 51.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

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 = sqrt((im + im)) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 42.1

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 42.1

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

   7. lift-sqrt.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-hypot.f64 78.6

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

4. Applied rewrites 78.6%

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

5. Taylor expanded in re around 0

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

   1. lower-*.f64 52.9

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

7. Applied rewrites 52.9%

   \[\leadsto \sqrt{\color{blue}{2 \cdot im}} \cdot 0.5 \]
8. Step-by-step derivation

9. Applied rewrites 52.9%

   \[\leadsto \sqrt{im + \color{blue}{im}} \cdot 0.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024360 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))