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

Percentage Accurate: 41.3% → 90.5%

Time: 4.3s

Alternatives: 5

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: 41.3% 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: 90.5% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.3%

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

   3. Taylor expanded in re around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(im \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   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. pow1/2 N/A

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

      8. lower-pow.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-pow N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 92.5

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

   7. Applied rewrites 92.5%

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

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

   1. Initial program 43.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 92.5%

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

4. Final simplification 92.5%

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

Alternative 2: 74.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+145}:\\ \;\;\;\;0.5 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-1, \frac{re}{im}, 1\right) \cdot im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))
   (if (<= t_0 0.0)
     (* 0.5 (* im (pow re -0.5)))
     (if (<= t_0 5e+145)
       (* 0.5 (sqrt t_0))
       (* 0.5 (sqrt (* 2.0 (* (fma -1.0 (/ re im) 1.0) im))))))))

C

double code(double re, double im) {
	double t_0 = 2.0 * (sqrt(((re * re) + (im * im))) - re);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else if (t_0 <= 5e+145) {
		tmp = 0.5 * sqrt(t_0);
	} else {
		tmp = 0.5 * sqrt((2.0 * (fma(-1.0, (re / im), 1.0) * im)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	elseif (t_0 <= 5e+145)
		tmp = Float64(0.5 * sqrt(t_0));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(fma(-1.0, Float64(re / im), 1.0) * im))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+145], N[(0.5 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(-1.0 * N[(re / im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.3%

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

   3. Taylor expanded in re around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(im \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   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. pow1/2 N/A

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

      8. lower-pow.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-pow N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 92.5

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

   7. Applied rewrites 92.5%

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

   if 0.0 < (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)) < 4.99999999999999967e145

   1. Initial program 93.2%

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

   if 4.99999999999999967e145 < (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))

   1. Initial program 6.3%

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

   3. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 52.6

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-1, \frac{re}{im}, 1\right) \cdot im\right)} \]

   5. Applied rewrites 52.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{fma}\left(-1, \frac{re}{im}, 1\right) \cdot im\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right) \leq 5 \cdot 10^{+145}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-1, \frac{re}{im}, 1\right) \cdot im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.9% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(-1.0 * N[(re / im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.3%

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

   3. Taylor expanded in re around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(im \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   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. pow1/2 N/A

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

      8. lower-pow.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-pow N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 92.5

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

   7. Applied rewrites 92.5%

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

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

   1. Initial program 43.7%

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

   3. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 58.3

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-1, \frac{re}{im}, 1\right) \cdot im\right)} \]

   5. Applied rewrites 58.3%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{fma}\left(-1, \frac{re}{im}, 1\right) \cdot im\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

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

Alternative 4: 25.6% accurate, N/A× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* im (pow (pow re -1.0) 0.5))))

C

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * ((re ^ -1.0) ^ 0.5)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

3. Taylor expanded in re around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(im \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
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. pow1/2 N/A

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

   8. lower-pow.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 23.4

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

5. Applied rewrites 23.4%

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

6. Final simplification 23.4%

   \[\leadsto 0.5 \cdot \left(im \cdot {\left({re}^{-1}\right)}^{0.5}\right) \]
7. Add Preprocessing

Alternative 5: 25.6% 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 38.5%

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

3. Taylor expanded in re around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(im \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
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. pow1/2 N/A

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

   8. lower-pow.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 23.4

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

5. Applied rewrites 23.4%

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

   1. lift-pow.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-pow N/A

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

   4. metadata-eval N/A

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

   5. lower-pow.f64 23.4

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

7. Applied rewrites 23.4%

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

8. Final simplification 23.4%

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

Reproduce

herbie shell --seed 2025064 
(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)))))