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

Percentage Accurate: 40.8% → 89.7%

Time: 4.5s

Alternatives: 7

Speedup: N/A×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.7% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $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 13.1%

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

   3. Taylor expanded in re around inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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 48.8%

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

      1. lift-*.f64 N/A

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

      2. 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 91.9%

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

4. Final simplification 92.9%

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

Alternative 2: 63.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+76}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+76)))
     (* 0.5 (* im (pow re -0.5)))
     t_0)))

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) || !(t_0 <= 2e+76)) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+76))) then
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+76)) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+76):
		tmp = 0.5 * (im * math.pow(re, -0.5))
	else:
		tmp = t_0
	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) || !(t_0 <= 2e+76))
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+76)))
		tmp = 0.5 * (im * (re ^ -0.5));
	else
		tmp = t_0;
	end
	tmp_2 = 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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+76]], $MachinePrecision]], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

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 or 2.0000000000000001e76 < (*.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 5.7%

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

   3. Taylor expanded in re around inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 42.2

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

   5. Applied rewrites 42.2%

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

   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)))) < 2.0000000000000001e76

   1. Initial program 96.4%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0 \lor \neg \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 2 \cdot 10^{+76}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 26.7% accurate, N/A× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* im (pow re -0.5))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.9%

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

3. Taylor expanded in re around inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. inv-pow N/A

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

   8. sqrt-pow1 N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 25.6

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

5. Applied rewrites 25.6%

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

6. Final simplification 25.6%

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

Alternative 4: 26.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot \left({re}^{-0.25} \cdot {re}^{-0.25}\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (* im (* (pow re -0.25) (pow re -0.25)))))

C

double code(double re, double im) {
	return 0.5 * (im * (pow(re, -0.25) * pow(re, -0.25)));
}

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 * (im * ((re ** (-0.25d0)) * (re ** (-0.25d0))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (im * (Math.pow(re, -0.25) * Math.pow(re, -0.25)));
}

Python

def code(re, im):
	return 0.5 * (im * (math.pow(re, -0.25) * math.pow(re, -0.25)))

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * Float64((re ^ -0.25) * (re ^ -0.25))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * ((re ^ -0.25) * (re ^ -0.25)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * N[(N[Power[re, -0.25], $MachinePrecision] * N[Power[re, -0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.9%

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

3. Taylor expanded in re around inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. inv-pow N/A

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

   8. sqrt-pow1 N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 25.6

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

5. Applied rewrites 25.6%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {re}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\color{blue}{re}}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {re}^{\frac{-1}{4}}\right)\right) \]

   7. lower-pow.f64 25.5

      \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot {re}^{\color{blue}{-0.25}}\right)\right) \]

7. Applied rewrites 25.5%

   \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot \color{blue}{{re}^{-0.25}}\right)\right) \]

8. Final simplification 25.5%

   \[\leadsto 0.5 \cdot \left(im \cdot \left({re}^{-0.25} \cdot {re}^{-0.25}\right)\right) \]
9. Add Preprocessing

Alternative 5: 26.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot \left({re}^{-0.25} \cdot \left({re}^{-0.125} \cdot {re}^{-0.125}\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (* im (* (pow re -0.25) (* (pow re -0.125) (pow re -0.125))))))

C

double code(double re, double im) {
	return 0.5 * (im * (pow(re, -0.25) * (pow(re, -0.125) * pow(re, -0.125))));
}

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 * (im * ((re ** (-0.25d0)) * ((re ** (-0.125d0)) * (re ** (-0.125d0)))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (im * (Math.pow(re, -0.25) * (Math.pow(re, -0.125) * Math.pow(re, -0.125))));
}

Python

def code(re, im):
	return 0.5 * (im * (math.pow(re, -0.25) * (math.pow(re, -0.125) * math.pow(re, -0.125))))

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * Float64((re ^ -0.25) * Float64((re ^ -0.125) * (re ^ -0.125)))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * ((re ^ -0.25) * ((re ^ -0.125) * (re ^ -0.125))));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * N[(N[Power[re, -0.25], $MachinePrecision] * N[(N[Power[re, -0.125], $MachinePrecision] * N[Power[re, -0.125], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.9%

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

3. Taylor expanded in re around inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. inv-pow N/A

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

   8. sqrt-pow1 N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 25.6

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

5. Applied rewrites 25.6%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {re}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\color{blue}{re}}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {re}^{\frac{-1}{4}}\right)\right) \]

   7. lower-pow.f64 25.5

      \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot {re}^{\color{blue}{-0.25}}\right)\right) \]

7. Applied rewrites 25.5%

   \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot \color{blue}{{re}^{-0.25}}\right)\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {re}^{\color{blue}{\frac{-1}{4}}}\right)\right) \]

   2. sqr-pow N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot \left({re}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \color{blue}{{re}^{\left(\frac{\frac{-1}{4}}{2}\right)}}\right)\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot \left({re}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \color{blue}{{re}^{\left(\frac{\frac{-1}{4}}{2}\right)}}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot \left({re}^{\frac{-1}{8}} \cdot {re}^{\left(\frac{\frac{-1}{4}}{2}\right)}\right)\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot \left({re}^{\frac{-1}{8}} \cdot {\color{blue}{re}}^{\left(\frac{\frac{-1}{4}}{2}\right)}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot \left({re}^{\frac{-1}{8}} \cdot {re}^{\frac{-1}{8}}\right)\right)\right) \]

   7. lower-pow.f64 25.5

      \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot \left({re}^{-0.125} \cdot {re}^{\color{blue}{-0.125}}\right)\right)\right) \]

9. Applied rewrites 25.5%

   \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot \left({re}^{-0.125} \cdot \color{blue}{{re}^{-0.125}}\right)\right)\right) \]

10. Final simplification 25.5%

    \[\leadsto 0.5 \cdot \left(im \cdot \left({re}^{-0.25} \cdot \left({re}^{-0.125} \cdot {re}^{-0.125}\right)\right)\right) \]
11. Add Preprocessing

Alternative 6: 24.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot \left({re}^{-0.25} \cdot {\left(e^{0.25}\right)}^{\log \left(\frac{--1}{re}\right)}\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (* im (* (pow re -0.25) (pow (exp 0.25) (log (/ (- -1.0) re)))))))

C

double code(double re, double im) {
	return 0.5 * (im * (pow(re, -0.25) * pow(exp(0.25), log((-(-1.0) / 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 * (im * ((re ** (-0.25d0)) * (exp(0.25d0) ** log((-(-1.0d0) / re)))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (im * (Math.pow(re, -0.25) * Math.pow(Math.exp(0.25), Math.log((-(-1.0) / re)))));
}

Python

def code(re, im):
	return 0.5 * (im * (math.pow(re, -0.25) * math.pow(math.exp(0.25), math.log((-(-1.0) / re)))))

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * Float64((re ^ -0.25) * (exp(0.25) ^ log(Float64(Float64(-(-1.0)) / re))))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * ((re ^ -0.25) * (exp(0.25) ^ log((-(-1.0) / re)))));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * N[(N[Power[re, -0.25], $MachinePrecision] * N[Power[N[Exp[0.25], $MachinePrecision], N[Log[N[((--1.0) / re), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.9%

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

3. Taylor expanded in re around inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. inv-pow N/A

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

   8. sqrt-pow1 N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 25.6

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

5. Applied rewrites 25.6%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {re}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\color{blue}{re}}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {re}^{\frac{-1}{4}}\right)\right) \]

   7. lower-pow.f64 25.5

      \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot {re}^{\color{blue}{-0.25}}\right)\right) \]

7. Applied rewrites 25.5%

   \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot \color{blue}{{re}^{-0.25}}\right)\right) \]

8. Taylor expanded in re around -inf

   \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot e^{\frac{1}{4} \cdot \left(\log -1 + \log \left(\frac{-1}{re}\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. exp-prod N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\left(\log -1 + \color{blue}{\log \left(\frac{-1}{re}\right)}\right)}\right)\right) \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\left(\log -1 + \color{blue}{\log \left(\frac{-1}{re}\right)}\right)}\right)\right) \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\left(\log -1 + \log \color{blue}{\left(\frac{-1}{re}\right)}\right)}\right)\right) \]

   4. sum-log N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

   5. lower-log.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

   7. lower-/.f64 23.7

      \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot {\left(e^{0.25}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

10. Applied rewrites 23.7%

    \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot {\left(e^{0.25}\right)}^{\color{blue}{\log \left(-1 \cdot \frac{-1}{re}\right)}}\right)\right) \]

11. Final simplification 23.7%

    \[\leadsto 0.5 \cdot \left(im \cdot \left({re}^{-0.25} \cdot {\left(e^{0.25}\right)}^{\log \left(\frac{--1}{re}\right)}\right)\right) \]
12. Add Preprocessing

Alternative 7: 24.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot \left({re}^{-0.25} \cdot e^{\log \left(e^{0.25}\right) \cdot \left(-\log re\right)}\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (* im (* (pow re -0.25) (exp (* (log (exp 0.25)) (- (log re))))))))

C

double code(double re, double im) {
	return 0.5 * (im * (pow(re, -0.25) * exp((log(exp(0.25)) * -log(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 * (im * ((re ** (-0.25d0)) * exp((log(exp(0.25d0)) * -log(re)))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (im * (Math.pow(re, -0.25) * Math.exp((Math.log(Math.exp(0.25)) * -Math.log(re)))));
}

Python

def code(re, im):
	return 0.5 * (im * (math.pow(re, -0.25) * math.exp((math.log(math.exp(0.25)) * -math.log(re)))))

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * Float64((re ^ -0.25) * exp(Float64(log(exp(0.25)) * Float64(-log(re)))))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * ((re ^ -0.25) * exp((log(exp(0.25)) * -log(re)))));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * N[(N[Power[re, -0.25], $MachinePrecision] * N[Exp[N[(N[Log[N[Exp[0.25], $MachinePrecision]], $MachinePrecision] * (-N[Log[re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.9%

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

3. Taylor expanded in re around inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. inv-pow N/A

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

   8. sqrt-pow1 N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 25.6

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

5. Applied rewrites 25.6%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {re}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\color{blue}{re}}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {re}^{\frac{-1}{4}}\right)\right) \]

   7. lower-pow.f64 25.5

      \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot {re}^{\color{blue}{-0.25}}\right)\right) \]

7. Applied rewrites 25.5%

   \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot \color{blue}{{re}^{-0.25}}\right)\right) \]

8. Taylor expanded in re around -inf

   \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot e^{\frac{1}{4} \cdot \left(\log -1 + \log \left(\frac{-1}{re}\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. exp-prod N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\left(\log -1 + \color{blue}{\log \left(\frac{-1}{re}\right)}\right)}\right)\right) \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\left(\log -1 + \color{blue}{\log \left(\frac{-1}{re}\right)}\right)}\right)\right) \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\left(\log -1 + \log \color{blue}{\left(\frac{-1}{re}\right)}\right)}\right)\right) \]

   4. sum-log N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

   5. lower-log.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

   7. lower-/.f64 23.7

      \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot {\left(e^{0.25}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

10. Applied rewrites 23.7%

    \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot {\left(e^{0.25}\right)}^{\color{blue}{\log \left(-1 \cdot \frac{-1}{re}\right)}}\right)\right) \]
11. Step-by-step derivation

    1. lift-pow.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

    2. lift-log.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

    4. lift-/.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot {\left(e^{\frac{1}{4}}\right)}^{\log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

    5. pow-to-exp N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot e^{\log \left(e^{\frac{1}{4}}\right) \cdot \log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

    6. lower-exp.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot e^{\log \left(e^{\frac{1}{4}}\right) \cdot \log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot e^{\log \left(e^{\frac{1}{4}}\right) \cdot \log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

    8. lower-log.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot e^{\log \left(e^{\frac{1}{4}}\right) \cdot \log \left(-1 \cdot \frac{-1}{re}\right)}\right)\right) \]

    9. associate-*r/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot e^{\log \left(e^{\frac{1}{4}}\right) \cdot \log \left(\frac{-1 \cdot -1}{re}\right)}\right)\right) \]

    10. metadata-eval N/A

        \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot e^{\log \left(e^{\frac{1}{4}}\right) \cdot \log \left(\frac{1}{re}\right)}\right)\right) \]

    11. log-rec N/A

        \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot e^{\log \left(e^{\frac{1}{4}}\right) \cdot \left(\mathsf{neg}\left(\log re\right)\right)}\right)\right) \]

    12. lower-neg.f64 N/A

        \[\leadsto \frac{1}{2} \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{\frac{-1}{4}} \cdot e^{\log \left(e^{\frac{1}{4}}\right) \cdot \left(-\log re\right)}\right)\right) \]

    13. lower-log.f64 23.6

        \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot e^{\log \left(e^{0.25}\right) \cdot \left(-\log re\right)}\right)\right) \]

12. Applied rewrites 23.6%

    \[\leadsto 0.5 \cdot \left(\left(1 \cdot im\right) \cdot \left({re}^{-0.25} \cdot e^{\log \left(e^{0.25}\right) \cdot \left(-\log re\right)}\right)\right) \]

13. Final simplification 23.6%

    \[\leadsto 0.5 \cdot \left(im \cdot \left({re}^{-0.25} \cdot e^{\log \left(e^{0.25}\right) \cdot \left(-\log re\right)}\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))))