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

Percentage Accurate: 40.6% → 90.1%

Time: 4.2s

Alternatives: 9

Speedup: 1.7×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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: 40.6% 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: 90.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)) 0.0)
   (* (pow re -0.5) (* im 0.5))
   (* (sqrt (* (- (hypot im re) re) 2.0)) 0.5)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.6%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 92.8

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

   5. Applied rewrites 92.8%

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

      1. lift-pow.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 92.8

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 92.8

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

   7. Applied rewrites 92.8%

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

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

   1. Initial program 52.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 89.0%

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

Alternative 2: 79.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -4.9 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 950000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im + im\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4e+149)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -4.9e-82)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma im im (* re re))) re))))
     (if (<= re 950000000.0)
       (* (sqrt (fma (- (/ re im) 2.0) re (+ im im))) 0.5)
       (* (pow re -0.5) (* im 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4e+149) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -4.9e-82) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(im, im, (re * re))) - re)));
	} else if (re <= 950000000.0) {
		tmp = sqrt(fma(((re / im) - 2.0), re, (im + im))) * 0.5;
	} else {
		tmp = pow(re, -0.5) * (im * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4e+149)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -4.9e-82)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(im, im, Float64(re * re))) - re))));
	elseif (re <= 950000000.0)
		tmp = Float64(sqrt(fma(Float64(Float64(re / im) - 2.0), re, Float64(im + im))) * 0.5);
	else
		tmp = Float64((re ^ -0.5) * Float64(im * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4e+149], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -4.9e-82], 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, 950000000.0], N[(N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + N[(im + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.0000000000000002e149

   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. Taylor expanded in re around -inf

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

      1. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if -4.0000000000000002e149 < re < -4.9000000000000003e-82

   1. Initial program 85.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 85.8

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

   4. Applied rewrites 85.8%

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

   if -4.9000000000000003e-82 < re < 9.5e8

   1. Initial program 51.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 84.5%

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

   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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 81.0

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

   7. Applied rewrites 81.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 81.0

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

   9. Applied rewrites 81.0%

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

   if 9.5e8 < re

   1. Initial program 10.7%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 82.1

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

   5. Applied rewrites 82.1%

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

      1. lift-pow.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 82.1

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -4.9 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 950000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im + im\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -4.9 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 950000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im + im\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \frac{1}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4e+149)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -4.9e-82)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma im im (* re re))) re))))
     (if (<= re 950000000.0)
       (* (sqrt (fma (- (/ re im) 2.0) re (+ im im))) 0.5)
       (* (* 0.5 im) (/ 1.0 (sqrt re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4e+149) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -4.9e-82) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(im, im, (re * re))) - re)));
	} else if (re <= 950000000.0) {
		tmp = sqrt(fma(((re / im) - 2.0), re, (im + im))) * 0.5;
	} else {
		tmp = (0.5 * im) * (1.0 / sqrt(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4e+149)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -4.9e-82)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(im, im, Float64(re * re))) - re))));
	elseif (re <= 950000000.0)
		tmp = Float64(sqrt(fma(Float64(Float64(re / im) - 2.0), re, Float64(im + im))) * 0.5);
	else
		tmp = Float64(Float64(0.5 * im) * Float64(1.0 / sqrt(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4e+149], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -4.9e-82], 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, 950000000.0], N[(N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + N[(im + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.0000000000000002e149

   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. Taylor expanded in re around -inf

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

      1. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if -4.0000000000000002e149 < re < -4.9000000000000003e-82

   1. Initial program 85.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 85.8

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

   4. Applied rewrites 85.8%

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

   if -4.9000000000000003e-82 < re < 9.5e8

   1. Initial program 51.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 84.5%

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

   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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 81.0

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

   7. Applied rewrites 81.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 81.0

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

   9. Applied rewrites 81.0%

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

   if 9.5e8 < re

   1. Initial program 10.7%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in re around 0

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 82.0

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

   8. Applied rewrites 82.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -4.9 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 950000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im + im\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \frac{1}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 950000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im + im\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \frac{1}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.2e-15)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 950000000.0)
     (* (sqrt (fma (- (/ re im) 2.0) re (+ im im))) 0.5)
     (* (* 0.5 im) (/ 1.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-15) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 950000000.0) {
		tmp = sqrt(fma(((re / im) - 2.0), re, (im + im))) * 0.5;
	} else {
		tmp = (0.5 * im) * (1.0 / sqrt(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.2e-15)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 950000000.0)
		tmp = Float64(sqrt(fma(Float64(Float64(re / im) - 2.0), re, Float64(im + im))) * 0.5);
	else
		tmp = Float64(Float64(0.5 * im) * Float64(1.0 / sqrt(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.2e-15], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 950000000.0], N[(N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + N[(im + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.1999999999999998e-15

   1. Initial program 54.4%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -6.1999999999999998e-15 < re < 9.5e8

   1. Initial program 55.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 86.4%

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

   5. Taylor expanded in re around 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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 78.6

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

   9. Applied rewrites 78.6%

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

   if 9.5e8 < re

   1. Initial program 10.7%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in re around 0

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 82.0

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

   8. Applied rewrites 82.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 950000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im + im\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \frac{1}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2800:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \frac{1}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.2e-15)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 2800.0)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* 0.5 im) (/ 1.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-15) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 2800.0) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) * (1.0 / sqrt(re));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6.2d-15)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 2800.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (0.5d0 * im) * (1.0d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-15) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 2800.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) * (1.0 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.2e-15:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 2800.0:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (0.5 * im) * (1.0 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.2e-15)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 2800.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(0.5 * im) * Float64(1.0 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.2e-15)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 2800.0)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (0.5 * im) * (1.0 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.2e-15], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2800.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.1999999999999998e-15

   1. Initial program 54.4%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -6.1999999999999998e-15 < re < 2800

   1. Initial program 56.1%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 78.9%

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

   if 2800 < 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

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in re around 0

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 81.0

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

   8. Applied rewrites 81.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2800:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \frac{1}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2800:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{re}} \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.2e-15)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 2800.0)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* (/ 1.0 (sqrt re)) im) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-15) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 2800.0) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = ((1.0 / sqrt(re)) * 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 <= (-6.2d-15)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 2800.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = ((1.0d0 / sqrt(re)) * im) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-15) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 2800.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = ((1.0 / Math.sqrt(re)) * im) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.2e-15:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 2800.0:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = ((1.0 / math.sqrt(re)) * im) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.2e-15)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 2800.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(Float64(1.0 / sqrt(re)) * im) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.2e-15)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 2800.0)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = ((1.0 / sqrt(re)) * im) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.2e-15], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2800.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.1999999999999998e-15

   1. Initial program 54.4%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -6.1999999999999998e-15 < re < 2800

   1. Initial program 56.1%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 78.9%

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

   if 2800 < 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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 33.4%

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

   5. Taylor expanded in re around 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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 21.9

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

   7. Applied rewrites 21.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 21.9

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

   9. Applied rewrites 21.9%

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

   10. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. sqrt-unprod N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. *-commutative N/A

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

       6. *-lft-identity N/A

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

       7. lower-*.f64 N/A

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

       8. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. lower-/.f64 N/A

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

       11. lower-sqrt.f64 80.9

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

   12. Applied rewrites 80.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2800:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{re}} \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 67000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.2e-15)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 67000000.0)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (sqrt (* im (/ im re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-15) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 67000000.0) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * sqrt((im * (im / re)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6.2d-15)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 67000000.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * sqrt((im * (im / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-15) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 67000000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * Math.sqrt((im * (im / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.2e-15:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 67000000.0:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * math.sqrt((im * (im / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.2e-15)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 67000000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(im / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.2e-15)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 67000000.0)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * sqrt((im * (im / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.2e-15], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 67000000.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.1999999999999998e-15

   1. Initial program 54.4%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -6.1999999999999998e-15 < re < 6.7e7

   1. Initial program 56.1%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 78.9%

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

   if 6.7e7 < 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

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.2

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

   5. Applied rewrites 49.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 60.5

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

   7. Applied rewrites 60.5%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 67000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.2e-15) (* 0.5 (sqrt (* -4.0 re))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-15) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	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 <= (-6.2d-15)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-15) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.2e-15:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.2e-15)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.2e-15)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.2e-15], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.1999999999999998e-15

   1. Initial program 54.4%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -6.1999999999999998e-15 < re

   1. Initial program 42.8%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 62.4%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 25.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{-4 \cdot re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (sqrt (* -4.0 re))))

C

double code(double re, double im) {
	return 0.5 * sqrt((-4.0 * 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(((-4.0d0) * re))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((-4.0 * re));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((-4.0 * re))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(-4.0 * re)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((-4.0 * re));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.5%

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

3. Taylor expanded in re around -inf

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

   1. lower-*.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Final simplification 25.3%

   \[\leadsto 0.5 \cdot \sqrt{-4 \cdot re} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025084 
(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)))))