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

Percentage Accurate: 41.7% → 90.0%

Time: 6.1s

Alternatives: 7

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: 41.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 90.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)))) <= 0.0) {
		tmp = (sqrt((1.0 / re)) * 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 ((0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)))) <= 0.0) {
		tmp = (Math.sqrt((1.0 / re)) * 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 (0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))) <= 0.0:
		tmp = (math.sqrt((1.0 / re)) * 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(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re)))) <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 / re)) * 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 ((0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)))) <= 0.0)
		tmp = (sqrt((1.0 / re)) * 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[(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], 0.0], N[(N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision] * 0.5), $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 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 0.0

   1. Initial program 8.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 8.2

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

      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 8.2

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

      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 8.2

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

   4. Applied rewrites 8.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

   6. Applied rewrites 8.2%

      \[\leadsto \color{blue}{\left({\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{0.25} \cdot {\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{0.25}\right)} \cdot 0.5 \]

   7. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 99.5

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

   9. Applied rewrites 99.5%

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

   if 0.0 < (*.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 46.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 46.3

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

      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 46.3

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

      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 91.8

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

   4. Applied rewrites 91.8%

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

Alternative 2: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.8 \cdot 10^{+121}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -3.4 \cdot 10^{-137}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 230000000:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, 2 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{re}} \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.8e+121)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -3.4e-137)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 230000000.0)
       (* 0.5 (sqrt (fma (- (/ re im) 2.0) re (* 2.0 im))))
       (* (* (sqrt (/ 1.0 re)) im) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.8e+121) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -3.4e-137) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 230000000.0) {
		tmp = 0.5 * sqrt(fma(((re / im) - 2.0), re, (2.0 * im)));
	} else {
		tmp = (sqrt((1.0 / re)) * im) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.8e+121)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -3.4e-137)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 230000000.0)
		tmp = Float64(0.5 * sqrt(fma(Float64(Float64(re / im) - 2.0), re, Float64(2.0 * im))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / re)) * im) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.8e+121], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -3.4e-137], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 230000000.0], N[(0.5 * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + N[(2.0 * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.79999999999999967e121

   1. Initial program 13.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

   if -7.79999999999999967e121 < re < -3.40000000000000014e-137

   1. Initial program 82.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 82.3

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

   4. Applied rewrites 82.3%

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

   if -3.40000000000000014e-137 < re < 2.3e8

   1. Initial program 49.4%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if 2.3e8 < re

   1. Initial program 8.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 8.3

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

      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 8.3

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

      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 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

   6. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\left({\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{0.25} \cdot {\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{0.25}\right)} \cdot 0.5 \]

   7. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 71.8

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

   9. Applied rewrites 71.8%

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

Alternative 3: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 230000000:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, 2 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{re}} \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.4e-50)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 230000000.0)
     (* 0.5 (sqrt (fma (- (/ re im) 2.0) re (* 2.0 im))))
     (* (* (sqrt (/ 1.0 re)) im) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.4e-50) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 230000000.0) {
		tmp = 0.5 * sqrt(fma(((re / im) - 2.0), re, (2.0 * im)));
	} else {
		tmp = (sqrt((1.0 / re)) * im) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.4e-50)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 230000000.0)
		tmp = Float64(0.5 * sqrt(fma(Float64(Float64(re / im) - 2.0), re, Float64(2.0 * im))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / re)) * im) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.4e-50], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 230000000.0], N[(0.5 * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + N[(2.0 * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.40000000000000014e-50

   1. Initial program 47.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -3.40000000000000014e-50 < re < 2.3e8

   1. Initial program 55.1%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if 2.3e8 < re

   1. Initial program 8.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 8.3

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

      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 8.3

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

      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 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

   6. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\left({\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{0.25} \cdot {\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{0.25}\right)} \cdot 0.5 \]

   7. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 71.8

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

   9. Applied rewrites 71.8%

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

Alternative 4: 75.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{re}} \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.4e-50)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 2.15e-32)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* (sqrt (/ 1.0 re)) im) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.4e-50) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 2.15e-32) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (sqrt((1.0 / 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 <= (-3.4d-50)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 2.15d-32) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (sqrt((1.0d0 / re)) * im) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.4e-50) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 2.15e-32) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (Math.sqrt((1.0 / re)) * im) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.4e-50:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 2.15e-32:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (math.sqrt((1.0 / re)) * im) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.4e-50)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 2.15e-32)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / re)) * im) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.4e-50)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 2.15e-32)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (sqrt((1.0 / re)) * im) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.4e-50], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.15e-32], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.40000000000000014e-50

   1. Initial program 47.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -3.40000000000000014e-50 < re < 2.14999999999999995e-32

   1. Initial program 57.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 82.8

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

   5. Applied rewrites 82.8%

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

   if 2.14999999999999995e-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. 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 51.2

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

   4. Applied rewrites 51.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

   6. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\left({\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{0.25} \cdot {\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{0.25}\right)} \cdot 0.5 \]

   7. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 68.9

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

   9. Applied rewrites 68.9%

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

Alternative 5: 69.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.4e-50)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 5.4e+160)
     (* (sqrt (+ im im)) 0.5)
     (* 0.5 (sqrt (/ (* im im) re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.4e-50) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 5.4e+160) {
		tmp = sqrt((im + im)) * 0.5;
	} 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 <= (-3.4d-50)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 5.4d+160) then
        tmp = sqrt((im + im)) * 0.5d0
    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 <= -3.4e-50) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 5.4e+160) {
		tmp = Math.sqrt((im + im)) * 0.5;
	} else {
		tmp = 0.5 * Math.sqrt(((im * im) / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.4e-50:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 5.4e+160:
		tmp = math.sqrt((im + im)) * 0.5
	else:
		tmp = 0.5 * math.sqrt(((im * im) / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.4e-50)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 5.4e+160)
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	else
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * im) / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.4e-50)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 5.4e+160)
		tmp = sqrt((im + im)) * 0.5;
	else
		tmp = 0.5 * sqrt(((im * im) / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.4e-50], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.4e+160], N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[Sqrt[N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.40000000000000014e-50

   1. Initial program 47.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -3.40000000000000014e-50 < re < 5.4e160

   1. Initial program 46.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 46.3

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

      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 46.3

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

      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 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 73.6

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

   7. Applied rewrites 73.6%

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

   9. Applied rewrites 73.6%

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

   if 5.4e160 < re

   1. Initial program 2.4%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 54.5

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

   5. Applied rewrites 54.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.4e-50) (* 0.5 (sqrt (* -4.0 re))) (* (sqrt (+ im im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.4e-50) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} 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 <= (-3.4d-50)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    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 <= -3.4e-50) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = Math.sqrt((im + im)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.4e-50:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = math.sqrt((im + im)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.4e-50)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	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 <= -3.4e-50)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = sqrt((im + im)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.4e-50], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.40000000000000014e-50

   1. Initial program 47.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -3.40000000000000014e-50 < 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 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 64.2

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

   7. Applied rewrites 64.2%

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

   9. Applied rewrites 64.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.4% 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 41.9%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
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 41.9

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

   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 41.9

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

   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 82.0

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

4. Applied rewrites 82.0%

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

5. Taylor expanded in re around 0

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

   1. lower-*.f64 53.6

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

7. Applied rewrites 53.6%

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

9. Applied rewrites 53.6%

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

10. Final simplification 53.6%

    \[\leadsto \sqrt{im + im} \cdot 0.5 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025016 
(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)))))