math.sqrt on complex, real part

Percentage Accurate: 41.8% → 81.9%

Time: 4.9s

Alternatives: 7

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.8% 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: 81.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{im \cdot im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+152)
   (* 0.5 (sqrt (- (/ (* im im) re))))
   (* 0.5 (sqrt (* 2.0 (+ (hypot re im) re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+152) {
		tmp = 0.5 * sqrt(-((im * im) / re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) + re)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+152) {
		tmp = 0.5 * Math.sqrt(-((im * im) / re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) + re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+152:
		tmp = 0.5 * math.sqrt(-((im * im) / re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) + re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+152)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(Float64(im * im) / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e+152)
		tmp = 0.5 * sqrt(-((im * im) / re));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+152], N[(0.5 * N[Sqrt[(-N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.7999999999999999e152

   1. Initial program 2.7%

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

   2. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 53.1

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

   4. Applied rewrites 53.1%

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

   if -1.7999999999999999e152 < re

   1. Initial program 47.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 86.0

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

   3. Applied rewrites 86.0%

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

Alternative 2: 57.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\\ \mathbf{if}\;re \leq -1.4 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{im \cdot im}{re}}\\ \mathbf{elif}\;re \leq -92000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{t\_0 - re}}\\ \mathbf{elif}\;re \leq 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\left(t\_0 + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{re} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (fma im im (* re re)))))
   (if (<= re -1.4e+152)
     (* 0.5 (sqrt (- (/ (* im im) re))))
     (if (<= re -92000000000000.0)
       (* 0.5 (sqrt (* 2.0 (/ (* im im) (- t_0 re)))))
       (if (<= re 1e-158)
         (* 0.5 (sqrt (+ im im)))
         (if (<= re 4e+55)
           (* (sqrt (* (+ t_0 re) 2.0)) 0.5)
           (* 0.5 (* (sqrt re) 2.0))))))))

C

double code(double re, double im) {
	double t_0 = sqrt(fma(im, im, (re * re)));
	double tmp;
	if (re <= -1.4e+152) {
		tmp = 0.5 * sqrt(-((im * im) / re));
	} else if (re <= -92000000000000.0) {
		tmp = 0.5 * sqrt((2.0 * ((im * im) / (t_0 - re))));
	} else if (re <= 1e-158) {
		tmp = 0.5 * sqrt((im + im));
	} else if (re <= 4e+55) {
		tmp = sqrt(((t_0 + re) * 2.0)) * 0.5;
	} else {
		tmp = 0.5 * (sqrt(re) * 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = sqrt(fma(im, im, Float64(re * re)))
	tmp = 0.0
	if (re <= -1.4e+152)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(Float64(im * im) / re))));
	elseif (re <= -92000000000000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im * im) / Float64(t_0 - re)))));
	elseif (re <= 1e-158)
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	elseif (re <= 4e+55)
		tmp = Float64(sqrt(Float64(Float64(t_0 + re) * 2.0)) * 0.5);
	else
		tmp = Float64(0.5 * Float64(sqrt(re) * 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[re, -1.4e+152], N[(0.5 * N[Sqrt[(-N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -92000000000000.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e-158], N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4e+55], N[(N[Sqrt[N[(N[(t$95$0 + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Sqrt[re], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -1.4000000000000001e152

   1. Initial program 2.7%

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

   2. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 53.1

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

   4. Applied rewrites 53.1%

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

   if -1.4000000000000001e152 < re < -9.2e13

   1. Initial program 22.6%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip-+ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 21.6%

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

   4. Taylor expanded in re around 0

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

      1. pow2 N/A

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

      2. lower-*.f64 53.4

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

   6. Applied rewrites 53.4%

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

   if -9.2e13 < re < 1.00000000000000006e-158

   1. Initial program 49.9%

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

   2. Taylor expanded in re around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 40.3

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

   4. Applied rewrites 40.3%

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

   if 1.00000000000000006e-158 < re < 4.00000000000000004e55

   1. Initial program 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      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

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

   3. Applied rewrites 74.1%

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

   if 4.00000000000000004e55 < re

   1. Initial program 33.5%

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

   2. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. rem-square-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 80.7

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

   4. Applied rewrites 80.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{re} \cdot 2\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 55.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{im \cdot im}{re}}\\ \mathbf{elif}\;re \leq 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{re} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e+70)
   (* 0.5 (sqrt (- (/ (* im im) re))))
   (if (<= re 1e-158)
     (* 0.5 (sqrt (+ im im)))
     (if (<= re 4e+55)
       (* (sqrt (* (+ (sqrt (fma im im (* re re))) re) 2.0)) 0.5)
       (* 0.5 (* (sqrt re) 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+70) {
		tmp = 0.5 * sqrt(-((im * im) / re));
	} else if (re <= 1e-158) {
		tmp = 0.5 * sqrt((im + im));
	} else if (re <= 4e+55) {
		tmp = sqrt(((sqrt(fma(im, im, (re * re))) + re) * 2.0)) * 0.5;
	} else {
		tmp = 0.5 * (sqrt(re) * 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e+70)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(Float64(im * im) / re))));
	elseif (re <= 1e-158)
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	elseif (re <= 4e+55)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(im, im, Float64(re * re))) + re) * 2.0)) * 0.5);
	else
		tmp = Float64(0.5 * Float64(sqrt(re) * 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e+70], N[(0.5 * N[Sqrt[(-N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e-158], N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4e+55], N[(N[Sqrt[N[(N[(N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Sqrt[re], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.8999999999999999e70

   1. Initial program 7.9%

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

   2. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 50.4

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

   4. Applied rewrites 50.4%

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

   if -1.8999999999999999e70 < re < 1.00000000000000006e-158

   1. Initial program 47.9%

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

   2. Taylor expanded in re around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 38.7

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

   4. Applied rewrites 38.7%

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

   if 1.00000000000000006e-158 < re < 4.00000000000000004e55

   1. Initial program 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      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

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

   3. Applied rewrites 74.1%

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

   if 4.00000000000000004e55 < re

   1. Initial program 33.5%

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

   2. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. rem-square-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 80.7

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

   4. Applied rewrites 80.7%

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

Alternative 4: 49.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{im \cdot im}{re}}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{re} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e+70)
   (* 0.5 (sqrt (- (/ (* im im) re))))
   (if (<= re 9e+22) (* 0.5 (sqrt (+ im im))) (* 0.5 (* (sqrt re) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+70) {
		tmp = 0.5 * sqrt(-((im * im) / re));
	} else if (re <= 9e+22) {
		tmp = 0.5 * sqrt((im + im));
	} else {
		tmp = 0.5 * (sqrt(re) * 2.0);
	}
	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 <= (-1.9d+70)) then
        tmp = 0.5d0 * sqrt(-((im * im) / re))
    else if (re <= 9d+22) then
        tmp = 0.5d0 * sqrt((im + im))
    else
        tmp = 0.5d0 * (sqrt(re) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+70) {
		tmp = 0.5 * Math.sqrt(-((im * im) / re));
	} else if (re <= 9e+22) {
		tmp = 0.5 * Math.sqrt((im + im));
	} else {
		tmp = 0.5 * (Math.sqrt(re) * 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e+70:
		tmp = 0.5 * math.sqrt(-((im * im) / re))
	elif re <= 9e+22:
		tmp = 0.5 * math.sqrt((im + im))
	else:
		tmp = 0.5 * (math.sqrt(re) * 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e+70)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(Float64(im * im) / re))));
	elseif (re <= 9e+22)
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	else
		tmp = Float64(0.5 * Float64(sqrt(re) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.9e+70)
		tmp = 0.5 * sqrt(-((im * im) / re));
	elseif (re <= 9e+22)
		tmp = 0.5 * sqrt((im + im));
	else
		tmp = 0.5 * (sqrt(re) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e+70], N[(0.5 * N[Sqrt[(-N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9e+22], N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[re], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.8999999999999999e70

   1. Initial program 7.9%

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

   2. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 50.4

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

   4. Applied rewrites 50.4%

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

   if -1.8999999999999999e70 < re < 8.9999999999999996e22

   1. Initial program 54.3%

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

   2. Taylor expanded in re around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 37.2

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

   4. Applied rewrites 37.2%

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

   if 8.9999999999999996e22 < re

   1. Initial program 38.5%

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

   2. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. rem-square-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 78.0

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

   4. Applied rewrites 78.0%

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

Alternative 5: 43.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.06 \cdot 10^{+187}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) + re\right)}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{re} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.06e+187)
   (* 0.5 (sqrt (* 2.0 (+ (- re) re))))
   (if (<= re 9e+22) (* 0.5 (sqrt (+ im im))) (* 0.5 (* (sqrt re) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.06e+187) {
		tmp = 0.5 * sqrt((2.0 * (-re + re)));
	} else if (re <= 9e+22) {
		tmp = 0.5 * sqrt((im + im));
	} else {
		tmp = 0.5 * (sqrt(re) * 2.0);
	}
	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 <= (-1.06d+187)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (-re + re)))
    else if (re <= 9d+22) then
        tmp = 0.5d0 * sqrt((im + im))
    else
        tmp = 0.5d0 * (sqrt(re) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.06e+187) {
		tmp = 0.5 * Math.sqrt((2.0 * (-re + re)));
	} else if (re <= 9e+22) {
		tmp = 0.5 * Math.sqrt((im + im));
	} else {
		tmp = 0.5 * (Math.sqrt(re) * 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.06e+187:
		tmp = 0.5 * math.sqrt((2.0 * (-re + re)))
	elif re <= 9e+22:
		tmp = 0.5 * math.sqrt((im + im))
	else:
		tmp = 0.5 * (math.sqrt(re) * 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.06e+187)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(-re) + re))));
	elseif (re <= 9e+22)
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	else
		tmp = Float64(0.5 * Float64(sqrt(re) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.06e+187)
		tmp = 0.5 * sqrt((2.0 * (-re + re)));
	elseif (re <= 9e+22)
		tmp = 0.5 * sqrt((im + im));
	else
		tmp = 0.5 * (sqrt(re) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.06e+187], N[(0.5 * N[Sqrt[N[(2.0 * N[((-re) + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9e+22], N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[re], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.06e187

   1. Initial program 2.6%

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

   2. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 22.3

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

   4. Applied rewrites 22.3%

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

   if -1.06e187 < re < 8.9999999999999996e22

   1. Initial program 48.7%

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

   2. Taylor expanded in re around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 34.2

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

   4. Applied rewrites 34.2%

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

   if 8.9999999999999996e22 < re

   1. Initial program 38.5%

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

   2. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. rem-square-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 78.0

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

   4. Applied rewrites 78.0%

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

Alternative 6: 41.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{re} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9e+22) (* 0.5 (sqrt (+ im im))) (* 0.5 (* (sqrt re) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9e+22) {
		tmp = 0.5 * sqrt((im + im));
	} else {
		tmp = 0.5 * (sqrt(re) * 2.0);
	}
	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 <= 9d+22) then
        tmp = 0.5d0 * sqrt((im + im))
    else
        tmp = 0.5d0 * (sqrt(re) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9e+22) {
		tmp = 0.5 * Math.sqrt((im + im));
	} else {
		tmp = 0.5 * (Math.sqrt(re) * 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9e+22:
		tmp = 0.5 * math.sqrt((im + im))
	else:
		tmp = 0.5 * (math.sqrt(re) * 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9e+22)
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	else
		tmp = Float64(0.5 * Float64(sqrt(re) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9e+22)
		tmp = 0.5 * sqrt((im + im));
	else
		tmp = 0.5 * (sqrt(re) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9e+22], N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[re], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.9999999999999996e22

   1. Initial program 42.8%

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

   2. Taylor expanded in re around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 30.7

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

   4. Applied rewrites 30.7%

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

   if 8.9999999999999996e22 < re

   1. Initial program 38.5%

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

   2. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. rem-square-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 78.0

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

   4. Applied rewrites 78.0%

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

Alternative 7: 26.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

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((im + im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

2. Taylor expanded in re around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f64 26.5

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

4. Applied rewrites 26.5%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64
  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))