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

Percentage Accurate: 41.2% → 90.3%

Time: 4.6s

Alternatives: 11

Speedup: 1.7×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))) 0.0)
   (* (pow re -0.5) (* 0.5 im))
   (* (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 = pow(re, -0.5) * (0.5 * im);
	} 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.pow(re, -0.5) * (0.5 * im);
	} 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.pow(re, -0.5) * (0.5 * im)
	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((re ^ -0.5) * Float64(0.5 * im));
	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 = (re ^ -0.5) * (0.5 * im);
	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[Power[re, -0.5], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 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 9.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 9.3%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-lft-identity 99.6

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

   9. Applied rewrites 99.6%

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

   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 45.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 89.0%

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

4. Final simplification 90.3%

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

Alternative 2: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))))
   (if (<= t_0 0.0)
     (* (pow re -0.5) (* 0.5 im))
     (if (<= t_0 5e+68)
       (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
       (* 0.5 (sqrt (* 2.0 (- im re))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = pow(re, -0.5) * (0.5 * im);
	} else if (t_0 <= 5e+68) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64((re ^ -0.5) * Float64(0.5 * im));
	elseif (t_0 <= 5e+68)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Power[re, -0.5], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+68], 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], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 0.0

   1. Initial program 9.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 9.3%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-lft-identity 99.6

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

   9. Applied rewrites 99.6%

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

   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)))) < 5.0000000000000004e68

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 92.6

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

   4. Applied rewrites 92.6%

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

   if 5.0000000000000004e68 < (*.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 9.6%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 56.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 5 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(im \cdot \sqrt{\frac{0.5}{re}}\right) \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))))
   (if (<= t_0 0.0)
     (* (* (* im (sqrt (/ 0.5 re))) (sqrt 2.0)) 0.5)
     (if (<= t_0 5e+68)
       (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
       (* 0.5 (sqrt (* 2.0 (- im re))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((im * sqrt((0.5 / re))) * sqrt(2.0)) * 0.5;
	} else if (t_0 <= 5e+68) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(im * sqrt(Float64(0.5 / re))) * sqrt(2.0)) * 0.5);
	elseif (t_0 <= 5e+68)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * N[Sqrt[N[(0.5 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 5e+68], 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], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 0.0

   1. Initial program 9.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 9.3%

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

   5. Taylor expanded in re around 0

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

      1. pow2 6.8

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

      2. pow2 6.8

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

      3. +-commutative 6.8

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

      4. pow2 6.8

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

      5. pow2 6.8

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

   7. Applied rewrites 6.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 6.8

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

   9. Applied rewrites 6.8%

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

   10. Taylor expanded in re around inf

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. sqrt-unprod N/A

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

       5. *-commutative N/A

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

       6. lower-sqrt.f64 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. lift-/.f64 99.2

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

   12. Applied rewrites 99.2%

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

   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)))) < 5.0000000000000004e68

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 92.6

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

   4. Applied rewrites 92.6%

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

   if 5.0000000000000004e68 < (*.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 9.6%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 56.2%

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

Alternative 4: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \frac{1}{\sqrt{re}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))))
   (if (<= t_0 0.0)
     (* (* 0.5 im) (/ 1.0 (sqrt re)))
     (if (<= t_0 5e+68)
       (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
       (* 0.5 (sqrt (* 2.0 (- im re))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (0.5 * im) * (1.0 / sqrt(re));
	} else if (t_0 <= 5e+68) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(0.5 * im) * Float64(1.0 / sqrt(re)));
	elseif (t_0 <= 5e+68)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(0.5 * im), $MachinePrecision] * N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+68], 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], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 0.0

   1. Initial program 9.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 9.3%

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

   5. Taylor expanded in re around 0

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

      1. pow2 0.0

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

      2. pow2 0.0

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

      3. +-commutative 0.0

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

      4. pow2 0.0

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

      5. pow2 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. sqrt-div N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 99.5

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

   10. Applied rewrites 99.5%

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

   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)))) < 5.0000000000000004e68

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 92.6

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

   4. Applied rewrites 92.6%

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

   if 5.0000000000000004e68 < (*.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 9.6%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 56.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \frac{1}{\sqrt{re}}\\ \mathbf{elif}\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 5 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.000215:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \frac{1}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.000215)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 4.8e-7)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* 0.5 im) (/ 1.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.000215) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 4.8e-7) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) * (1.0 / sqrt(re));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.000215d0)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 4.8d-7) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (0.5d0 * im) * (1.0d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.000215) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 4.8e-7) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) * (1.0 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.000215:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 4.8e-7:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (0.5 * im) * (1.0 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.000215)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 4.8e-7)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(0.5 * im) * Float64(1.0 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.000215)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 4.8e-7)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (0.5 * im) * (1.0 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.000215], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.8e-7], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.14999999999999995e-4

   1. Initial program 40.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 75.3

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

   5. Applied rewrites 75.3%

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

   if -2.14999999999999995e-4 < re < 4.79999999999999957e-7

   1. Initial program 56.6%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 77.8%

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

   if 4.79999999999999957e-7 < re

   1. Initial program 12.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 40.2%

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

   5. Taylor expanded in re around 0

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

      1. pow2 25.6

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

      2. pow2 25.6

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

      3. +-commutative 25.6

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

      4. pow2 25.6

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

      5. pow2 25.6

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

   7. Applied rewrites 25.6%

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

   8. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. sqrt-div N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 75.0

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

   10. Applied rewrites 75.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.000215:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \frac{1}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.000215:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{re}} \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.000215)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 4.8e-7)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* (/ 1.0 (sqrt re)) im) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.000215) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 4.8e-7) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = ((1.0 / sqrt(re)) * im) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.000215d0)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 4.8d-7) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = ((1.0d0 / sqrt(re)) * im) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.000215) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 4.8e-7) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = ((1.0 / Math.sqrt(re)) * im) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.000215:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 4.8e-7:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = ((1.0 / math.sqrt(re)) * im) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.000215)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 4.8e-7)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(Float64(1.0 / sqrt(re)) * im) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.000215)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 4.8e-7)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = ((1.0 / sqrt(re)) * im) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.000215], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.8e-7], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.14999999999999995e-4

   1. Initial program 40.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 75.3

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

   5. Applied rewrites 75.3%

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

   if -2.14999999999999995e-4 < re < 4.79999999999999957e-7

   1. Initial program 56.6%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 77.8%

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

   if 4.79999999999999957e-7 < re

   1. Initial program 12.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 40.2%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(im, re\right) - re\right) \cdot 2} \cdot 0.5} \]
   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. lift--.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. sqrt-prod N/A

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

      12. *-commutative N/A

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

      13. pow1/2 N/A

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

      14. pow-to-exp N/A

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

      15. lower-exp.f64 N/A

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

   6. Applied rewrites 38.0%

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

      1. lift-hypot.f64 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 12.0

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

   8. Applied rewrites 12.0%

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

   9. Taylor expanded in re around inf

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

       1. exp-to-pow N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. pow2 N/A

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

       6. pow2 N/A

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

       7. pow1/2 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. sqrt-div N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. lower-sqrt.f64 75.0

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

   11. Applied rewrites 75.0%

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

Alternative 7: 69.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.000215:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.000215)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 1.9e+84)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (sqrt (/ (* im im) re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.000215) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 1.9e+84) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * sqrt(((im * im) / re));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.000215d0)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 1.9d+84) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * sqrt(((im * im) / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.000215) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 1.9e+84) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * Math.sqrt(((im * im) / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.000215:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 1.9e+84:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * math.sqrt(((im * im) / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.000215)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 1.9e+84)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	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 <= -0.000215)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 1.9e+84)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * sqrt(((im * im) / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.000215], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+84], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 < -2.14999999999999995e-4

   1. Initial program 40.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 75.3

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

   5. Applied rewrites 75.3%

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

   if -2.14999999999999995e-4 < re < 1.9e84

   1. Initial program 52.6%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 73.3%

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

   if 1.9e84 < re

   1. Initial program 6.9%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 50.1

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

   5. Applied rewrites 50.1%

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

Alternative 8: 65.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.000215:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 1.92 \cdot 10^{+183}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.000215)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 1.92e+183)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (sqrt (* 2.0 (- re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.000215) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 1.92e+183) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re - 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 <= (-0.000215d0)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 1.92d+183) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.000215) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 1.92e+183) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.000215:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 1.92e+183:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.000215)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 1.92e+183)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.000215)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 1.92e+183)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.000215], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.92e+183], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.14999999999999995e-4

   1. Initial program 40.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 75.3

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

   5. Applied rewrites 75.3%

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

   if -2.14999999999999995e-4 < re < 1.92000000000000008e183

   1. Initial program 47.5%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 67.6%

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

   if 1.92000000000000008e183 < re

   1. Initial program 2.6%

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

   3. Taylor expanded in re around inf

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

   5. Applied rewrites 22.8%

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

Alternative 9: 60.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-151}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5e-151)
   (* 0.5 (sqrt (* -4.0 re)))
   (* 0.5 (sqrt (* 2.0 (- im re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-151) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (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 (im <= 2.5d-151) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-151) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5e-151:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5e-151)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5e-151)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5e-151], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.50000000000000002e-151

   1. Initial program 31.6%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 43.2

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

   5. Applied rewrites 43.2%

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

   if 2.50000000000000002e-151 < im

   1. Initial program 44.3%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 66.4%

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

Alternative 10: 62.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7e-125) (* 0.5 (sqrt (* -4.0 re))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7e-125) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-7d-125)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7e-125) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7e-125:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7e-125)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7e-125)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -6.99999999999999995e-125

   1. Initial program 50.3%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if -6.99999999999999995e-125 < re

   1. Initial program 36.3%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 60.3%

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

Alternative 11: 25.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt(((-4.0d0) * re))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.2%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
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 25.7

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

5. Applied rewrites 25.7%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025086 
(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)))))