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

Percentage Accurate: 42.2% → 83.5%

Time: 2.8s

Alternatives: 8

Speedup: 2.6×

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: 42.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{\left(\log \left(\frac{1}{re}\right) + \log \left({im}^{2}\right)\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4e+36)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (exp (* (+ (log (/ 1.0 re)) (log (pow im 2.0))) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4e+36) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * exp(((log((1.0 / re)) + log(pow(im, 2.0))) * 0.5));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4e+36) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * Math.exp(((Math.log((1.0 / re)) + Math.log(Math.pow(im, 2.0))) * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4e+36:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * math.exp(((math.log((1.0 / re)) + math.log(math.pow(im, 2.0))) * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4e+36)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * exp(Float64(Float64(log(Float64(1.0 / re)) + log((im ^ 2.0))) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4e+36)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * exp(((log((1.0 / re)) + log((im ^ 2.0))) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4e+36], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Exp[N[(N[(N[Log[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] + N[Log[N[Power[im, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.00000000000000017e36

   1. Initial program 42.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. distribute-lft-neg-out N/A

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

      9. sqr-neg-rev N/A

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

      10. lower-hypot.f64 79.6

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

   3. Applied rewrites 79.6%

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

   if 4.00000000000000017e36 < re

   1. Initial program 42.2%

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

   2. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot \left(\sqrt{{re}^{2}} - re\right) + {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \color{blue}{\sqrt{{re}^{2}} - re}, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - \color{blue}{re}, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

   4. Applied rewrites 18.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(-0.25, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)}} \]

   6. Applied rewrites 15.5%

      \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \frac{1}{re}\right) \cdot im, im, \left(re - re\right) \cdot 2\right)\right) \cdot 0.5}} \]

   7. Taylor expanded in re around inf

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-pow.f64 16.6

         \[\leadsto 0.5 \cdot e^{\left(\log \left(\frac{1}{re}\right) + \log \left({im}^{2}\right)\right) \cdot 0.5} \]

   9. Applied rewrites 16.6%

      \[\leadsto 0.5 \cdot e^{\color{blue}{\left(\log \left(\frac{1}{re}\right) + \log \left({im}^{2}\right)\right)} \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.6 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{im}{re}, im, \left(re - re\right) \cdot 2\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.6e+36)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* (sqrt (fma (/ im re) im (* (- re re) 2.0))) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.6e+36) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = sqrt(fma((im / re), im, ((re - re) * 2.0))) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.6e+36)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(sqrt(fma(Float64(im / re), im, Float64(Float64(re - re) * 2.0))) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.6e+36], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(im / re), $MachinePrecision] * im + N[(N[(re - re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.6000000000000001e36

   1. Initial program 42.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. distribute-lft-neg-out N/A

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

      9. sqr-neg-rev N/A

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

      10. lower-hypot.f64 79.6

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

   3. Applied rewrites 79.6%

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

   if 2.6000000000000001e36 < re

   1. Initial program 42.2%

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

   2. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot \left(\sqrt{{re}^{2}} - re\right) + {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \color{blue}{\sqrt{{re}^{2}} - re}, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - \color{blue}{re}, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

   4. Applied rewrites 18.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(-0.25, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)}} \]

   6. Applied rewrites 16.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \frac{1}{re}\right) \cdot im, im, \left(re - re\right) \cdot 2\right)} \cdot 0.5} \]

   7. Taylor expanded in re around inf

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

      1. lower-/.f64 18.1

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

   9. Applied rewrites 18.1%

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

Alternative 3: 74.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.95 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -8 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{im}{re}, im, \left(re - re\right) \cdot 2\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.95e+152)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -8e-112)
     (* (sqrt (* (- (sqrt (fma im im (* re re))) re) 2.0)) 0.5)
     (if (<= re 2.6e+36)
       (* (sqrt (+ im im)) 0.5)
       (* (sqrt (fma (/ im re) im (* (- re re) 2.0))) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.95e+152) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -8e-112) {
		tmp = sqrt(((sqrt(fma(im, im, (re * re))) - re) * 2.0)) * 0.5;
	} else if (re <= 2.6e+36) {
		tmp = sqrt((im + im)) * 0.5;
	} else {
		tmp = sqrt(fma((im / re), im, ((re - re) * 2.0))) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.95e+152)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -8e-112)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(im, im, Float64(re * re))) - re) * 2.0)) * 0.5);
	elseif (re <= 2.6e+36)
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	else
		tmp = Float64(sqrt(fma(Float64(im / re), im, Float64(Float64(re - re) * 2.0))) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.95e+152], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -8e-112], N[(N[Sqrt[N[(N[(N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 2.6e+36], N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(N[(im / re), $MachinePrecision] * im + N[(N[(re - re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.9500000000000001e152

   1. Initial program 42.2%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 25.7

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

   4. Applied rewrites 25.7%

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

   if -2.9500000000000001e152 < re < -7.9999999999999996e-112

   1. Initial program 42.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 42.2

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

   3. Applied rewrites 42.2%

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

   if -7.9999999999999996e-112 < re < 2.6000000000000001e36

   1. Initial program 42.2%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 52.8

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

   4. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.8

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 52.8

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

   6. Applied rewrites 52.8%

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

   if 2.6000000000000001e36 < re

   1. Initial program 42.2%

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

   2. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot \left(\sqrt{{re}^{2}} - re\right) + {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \color{blue}{\sqrt{{re}^{2}} - re}, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - \color{blue}{re}, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

   4. Applied rewrites 18.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(-0.25, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)}} \]

   6. Applied rewrites 16.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \frac{1}{re}\right) \cdot im, im, \left(re - re\right) \cdot 2\right)} \cdot 0.5} \]

   7. Taylor expanded in re around inf

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

      1. lower-/.f64 18.1

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

   9. Applied rewrites 18.1%

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

Alternative 4: 70.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{im}{re}, im, \left(re - re\right) \cdot 2\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e-98)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 2.6e+36)
     (* (sqrt (+ im im)) 0.5)
     (* (sqrt (fma (/ im re) im (* (- re re) 2.0))) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-98) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 2.6e+36) {
		tmp = sqrt((im + im)) * 0.5;
	} else {
		tmp = sqrt(fma((im / re), im, ((re - re) * 2.0))) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e-98)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 2.6e+36)
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	else
		tmp = Float64(sqrt(fma(Float64(im / re), im, Float64(Float64(re - re) * 2.0))) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e-98], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.6e+36], N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(N[(im / re), $MachinePrecision] * im + N[(N[(re - re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.1e-98

   1. Initial program 42.2%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 25.7

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

   4. Applied rewrites 25.7%

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

   if -3.1e-98 < re < 2.6000000000000001e36

   1. Initial program 42.2%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 52.8

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

   4. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.8

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 52.8

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

   6. Applied rewrites 52.8%

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

   if 2.6000000000000001e36 < re

   1. Initial program 42.2%

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

   2. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot \left(\sqrt{{re}^{2}} - re\right) + {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \color{blue}{\sqrt{{re}^{2}} - re}, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - \color{blue}{re}, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}} + \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)} \]

   4. Applied rewrites 18.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2, \sqrt{{re}^{2}} - re, {im}^{2} \cdot \mathsf{fma}\left(-0.25, \frac{{im}^{2}}{{\left(\sqrt{{re}^{2}}\right)}^{3}}, \frac{1}{\sqrt{{re}^{2}}}\right)\right)}} \]

   6. Applied rewrites 16.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \frac{1}{re}\right) \cdot im, im, \left(re - re\right) \cdot 2\right)} \cdot 0.5} \]

   7. Taylor expanded in re around inf

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

      1. lower-/.f64 18.1

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

   9. Applied rewrites 18.1%

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

Alternative 5: 66.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, re, im + im\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot \left(2 \cdot \frac{im}{re}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e-98)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 7.5e-31)
     (* (sqrt (fma -2.0 re (+ im im))) 0.5)
     (* 0.5 (sqrt (* re (* 2.0 (/ im re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-98) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 7.5e-31) {
		tmp = sqrt(fma(-2.0, re, (im + im))) * 0.5;
	} else {
		tmp = 0.5 * sqrt((re * (2.0 * (im / re))));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e-98)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 7.5e-31)
		tmp = Float64(sqrt(fma(-2.0, re, Float64(im + im))) * 0.5);
	else
		tmp = Float64(0.5 * sqrt(Float64(re * Float64(2.0 * Float64(im / re)))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e-98], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.5e-31], N[(N[Sqrt[N[(-2.0 * re + N[(im + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[Sqrt[N[(re * N[(2.0 * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.1e-98

   1. Initial program 42.2%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 25.7

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

   4. Applied rewrites 25.7%

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

   if -3.1e-98 < re < 7.49999999999999975e-31

   1. Initial program 42.2%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 54.2

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

   4. Applied rewrites 54.2%

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

   5. Taylor expanded in re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 55.2

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

   7. Applied rewrites 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.2

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 55.2

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

   9. Applied rewrites 55.2%

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

   if 7.49999999999999975e-31 < re

   1. Initial program 42.2%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 54.2

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

   4. Applied rewrites 54.2%

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

   5. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 42.6

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

   7. Applied rewrites 42.6%

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

   8. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 43.1

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

   10. Applied rewrites 43.1%

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

Alternative 6: 66.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(2 + -2 \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot \left(2 \cdot \frac{im}{re}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e-98)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 7.5e-31)
     (* 0.5 (sqrt (* im (+ 2.0 (* -2.0 (/ re im))))))
     (* 0.5 (sqrt (* re (* 2.0 (/ im re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-98) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 7.5e-31) {
		tmp = 0.5 * sqrt((im * (2.0 + (-2.0 * (re / im)))));
	} else {
		tmp = 0.5 * sqrt((re * (2.0 * (im / re))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.1d-98)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 7.5d-31) then
        tmp = 0.5d0 * sqrt((im * (2.0d0 + ((-2.0d0) * (re / im)))))
    else
        tmp = 0.5d0 * sqrt((re * (2.0d0 * (im / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-98) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 7.5e-31) {
		tmp = 0.5 * Math.sqrt((im * (2.0 + (-2.0 * (re / im)))));
	} else {
		tmp = 0.5 * Math.sqrt((re * (2.0 * (im / re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e-98:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 7.5e-31:
		tmp = 0.5 * math.sqrt((im * (2.0 + (-2.0 * (re / im)))))
	else:
		tmp = 0.5 * math.sqrt((re * (2.0 * (im / re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e-98)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 7.5e-31)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(2.0 + Float64(-2.0 * Float64(re / im))))));
	else
		tmp = Float64(0.5 * sqrt(Float64(re * Float64(2.0 * Float64(im / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1e-98)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 7.5e-31)
		tmp = 0.5 * sqrt((im * (2.0 + (-2.0 * (re / im)))));
	else
		tmp = 0.5 * sqrt((re * (2.0 * (im / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e-98], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.5e-31], N[(0.5 * N[Sqrt[N[(im * N[(2.0 + N[(-2.0 * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(re * N[(2.0 * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.1e-98

   1. Initial program 42.2%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 25.7

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

   4. Applied rewrites 25.7%

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

   if -3.1e-98 < re < 7.49999999999999975e-31

   1. Initial program 42.2%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 54.2

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

   4. Applied rewrites 54.2%

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

   if 7.49999999999999975e-31 < re

   1. Initial program 42.2%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 54.2

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

   4. Applied rewrites 54.2%

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

   5. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 42.6

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

   7. Applied rewrites 42.6%

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

   8. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 43.1

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

   10. Applied rewrites 43.1%

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

Alternative 7: 63.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e-98) (* 0.5 (sqrt (* -4.0 re))) (* (sqrt (+ im im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-98) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = sqrt((im + im)) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.1d-98)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = sqrt((im + im)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-98) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = Math.sqrt((im + im)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e-98:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = math.sqrt((im + im)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e-98)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1e-98)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = sqrt((im + im)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e-98], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.1e-98

   1. Initial program 42.2%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 25.7

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

   4. Applied rewrites 25.7%

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

   if -3.1e-98 < re

   1. Initial program 42.2%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 52.8

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

   4. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.8

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 52.8

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

   6. Applied rewrites 52.8%

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

Alternative 8: 52.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.2%

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

2. Taylor expanded in im around inf

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

   1. lower-*.f64 52.8

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

4. Applied rewrites 52.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 52.8

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f64 52.8

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

6. Applied rewrites 52.8%

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

Reproduce

herbie shell --seed 2025149 
(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)))))