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

Percentage Accurate: 41.4% → 86.5%

Time: 3.2s

Alternatives: 7

Speedup: 1.7×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.4% 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: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.2 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right) \cdot \sqrt{2}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.2e+125)
   (* (sqrt (* (- (hypot im re) re) 2.0)) 0.5)
   (* (* (* (* im (sqrt 0.5)) (sqrt (/ 1.0 re))) (sqrt 2.0)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.2e+125) {
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = (((im * sqrt(0.5)) * sqrt((1.0 / re))) * sqrt(2.0)) * 0.5;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.2e+125) {
		tmp = Math.sqrt(((Math.hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = (((im * Math.sqrt(0.5)) * Math.sqrt((1.0 / re))) * Math.sqrt(2.0)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.2e+125:
		tmp = math.sqrt(((math.hypot(im, re) - re) * 2.0)) * 0.5
	else:
		tmp = (((im * math.sqrt(0.5)) * math.sqrt((1.0 / re))) * math.sqrt(2.0)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.2e+125)
		tmp = Float64(sqrt(Float64(Float64(hypot(im, re) - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(im * sqrt(0.5)) * sqrt(Float64(1.0 / re))) * sqrt(2.0)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.2e+125)
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	else
		tmp = (((im * sqrt(0.5)) * sqrt((1.0 / re))) * sqrt(2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.2e+125], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(im * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.19999999999999983e125

   1. Initial program 47.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 47.7

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

      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 47.7

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 87.0

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

   3. Applied rewrites 87.0%

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

   if 3.19999999999999983e125 < re

   1. Initial program 5.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 5.0

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

      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 5.0

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 35.2

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

   3. Applied rewrites 35.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-hypot.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. exp-to-pow N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 35.2%

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

      1. lift-hypot.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow1/2 N/A

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

      5. unpow1 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-hypot.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-hypot.f64 N/A

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

      17. pow-prod-down N/A

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

      18. lift-pow.f64 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. lower-pow.f64 27.7

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 83.4

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

   10. Applied rewrites 83.4%

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

Alternative 2: 77.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -7.5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-104}:\\ \;\;\;\;\left(\sqrt{im - re} \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right) \cdot \sqrt{2}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.05e+126)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -7.5e-79)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 1.3e-104)
       (* (* (sqrt (- im re)) (sqrt 2.0)) 0.5)
       (* (* (* (* im (sqrt 0.5)) (sqrt (/ 1.0 re))) (sqrt 2.0)) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.05e+126) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -7.5e-79) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 1.3e-104) {
		tmp = (sqrt((im - re)) * sqrt(2.0)) * 0.5;
	} else {
		tmp = (((im * sqrt(0.5)) * sqrt((1.0 / re))) * sqrt(2.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.05e+126)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -7.5e-79)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 1.3e-104)
		tmp = Float64(Float64(sqrt(Float64(im - re)) * sqrt(2.0)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(im * sqrt(0.5)) * sqrt(Float64(1.0 / re))) * sqrt(2.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.05e+126], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -7.5e-79], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.3e-104], N[(N[(N[Sqrt[N[(im - re), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(im * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.05e126

   1. Initial program 14.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 14.8

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

   3. Applied rewrites 14.8%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. exp-to-pow N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. flip3-+ N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

   5. Applied rewrites 0.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(re \cdot re\right)}^{3} + {\left(im \cdot im\right)}^{3}}{\mathsf{fma}\left(re \cdot re, re \cdot re, \left(im \cdot im\right) \cdot \left(im \cdot im\right) - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-log.f64 N/A

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

      4. rem-exp-log N/A

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

      5. lift-log.f64 N/A

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

      6. exp-lft-sqr-rev N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in re around -inf

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

      1. lower-*.f64 86.3

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

   10. Applied rewrites 86.3%

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

   if -1.05e126 < re < -7.49999999999999969e-79

   1. Initial program 76.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 76.8

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

   3. Applied rewrites 76.8%

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

   if -7.49999999999999969e-79 < re < 1.30000000000000001e-104

   1. Initial program 57.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 57.2

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

      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 57.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 93.3

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

   3. Applied rewrites 93.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-hypot.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. exp-to-pow N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 92.7%

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

      1. lift-hypot.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow1/2 N/A

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

      5. unpow1 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-hypot.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-hypot.f64 N/A

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

      17. pow-prod-down N/A

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

      18. lift-pow.f64 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. lower-pow.f64 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 83.1%

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

   if 1.30000000000000001e-104 < re

   1. Initial program 18.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 18.9

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

      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 18.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 45.0

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

   3. Applied rewrites 45.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-hypot.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. exp-to-pow N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 44.8%

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

      1. lift-hypot.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow1/2 N/A

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

      5. unpow1 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-hypot.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-hypot.f64 N/A

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

      17. pow-prod-down N/A

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

      18. lift-pow.f64 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. lower-pow.f64 40.7

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 67.5

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

   10. Applied rewrites 67.5%

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

Alternative 3: 77.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -7.5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-104}:\\ \;\;\;\;\left(\sqrt{im - re} \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.05e+126)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -7.5e-79)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 1.3e-104)
       (* (* (sqrt (- im re)) (sqrt 2.0)) 0.5)
       (* (* (* im (* (sqrt 0.5) (sqrt 2.0))) (sqrt (/ 1.0 re))) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.05e+126) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -7.5e-79) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 1.3e-104) {
		tmp = (sqrt((im - re)) * sqrt(2.0)) * 0.5;
	} else {
		tmp = ((im * (sqrt(0.5) * sqrt(2.0))) * sqrt((1.0 / re))) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.05e+126)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -7.5e-79)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 1.3e-104)
		tmp = Float64(Float64(sqrt(Float64(im - re)) * sqrt(2.0)) * 0.5);
	else
		tmp = Float64(Float64(Float64(im * Float64(sqrt(0.5) * sqrt(2.0))) * sqrt(Float64(1.0 / re))) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.05e+126], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -7.5e-79], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.3e-104], N[(N[(N[Sqrt[N[(im - re), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(im * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.05e126

   1. Initial program 14.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 14.8

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

   3. Applied rewrites 14.8%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. exp-to-pow N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. flip3-+ N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

   5. Applied rewrites 0.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(re \cdot re\right)}^{3} + {\left(im \cdot im\right)}^{3}}{\mathsf{fma}\left(re \cdot re, re \cdot re, \left(im \cdot im\right) \cdot \left(im \cdot im\right) - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-log.f64 N/A

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

      4. rem-exp-log N/A

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

      5. lift-log.f64 N/A

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

      6. exp-lft-sqr-rev N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in re around -inf

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

      1. lower-*.f64 86.3

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

   10. Applied rewrites 86.3%

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

   if -1.05e126 < re < -7.49999999999999969e-79

   1. Initial program 76.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 76.8

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

   3. Applied rewrites 76.8%

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

   if -7.49999999999999969e-79 < re < 1.30000000000000001e-104

   1. Initial program 57.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 57.2

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

      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 57.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 93.3

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

   3. Applied rewrites 93.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-hypot.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. exp-to-pow N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 92.7%

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

      1. lift-hypot.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow1/2 N/A

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

      5. unpow1 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-hypot.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-hypot.f64 N/A

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

      17. pow-prod-down N/A

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

      18. lift-pow.f64 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. lower-pow.f64 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 83.1%

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

   if 1.30000000000000001e-104 < re

   1. Initial program 18.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 18.9

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

      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 18.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 45.0

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

   3. Applied rewrites 45.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-hypot.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. exp-to-pow N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 44.8%

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

      1. lift-hypot.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow1/2 N/A

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

      5. unpow1 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-hypot.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-hypot.f64 N/A

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

      17. pow-prod-down N/A

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

      18. lift-pow.f64 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. lower-pow.f64 40.7

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 67.1

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

   10. Applied rewrites 67.1%

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

Alternative 4: 66.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -7.5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{im} \cdot \sqrt{2}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.05e+126)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -7.5e-79)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (* (* (sqrt im) (sqrt 2.0)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.05e+126) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -7.5e-79) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else {
		tmp = (sqrt(im) * sqrt(2.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.05e+126)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -7.5e-79)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	else
		tmp = Float64(Float64(sqrt(im) * sqrt(2.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.05e+126], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -7.5e-79], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[im], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.05e126

   1. Initial program 14.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 14.8

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

   3. Applied rewrites 14.8%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. exp-to-pow N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. flip3-+ N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

   5. Applied rewrites 0.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(re \cdot re\right)}^{3} + {\left(im \cdot im\right)}^{3}}{\mathsf{fma}\left(re \cdot re, re \cdot re, \left(im \cdot im\right) \cdot \left(im \cdot im\right) - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-log.f64 N/A

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

      4. rem-exp-log N/A

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

      5. lift-log.f64 N/A

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

      6. exp-lft-sqr-rev N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in re around -inf

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

      1. lower-*.f64 86.3

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

   10. Applied rewrites 86.3%

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

   if -1.05e126 < re < -7.49999999999999969e-79

   1. Initial program 76.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 76.8

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

   3. Applied rewrites 76.8%

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

   if -7.49999999999999969e-79 < re

   1. Initial program 38.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 38.5

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

      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 38.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 69.7

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

   3. Applied rewrites 69.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-hypot.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. exp-to-pow N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 69.3%

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

      1. lift-hypot.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow1/2 N/A

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

      5. unpow1 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-hypot.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-hypot.f64 N/A

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

      17. pow-prod-down N/A

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

      18. lift-pow.f64 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. lower-pow.f64 67.3

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

   7. Applied rewrites 67.3%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 59.9%

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

Alternative 5: 64.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{im} \cdot \sqrt{2}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e-46)
   (* 0.5 (sqrt (* -4.0 re)))
   (* (* (sqrt im) (sqrt 2.0)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-46) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = (sqrt(im) * sqrt(2.0)) * 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 <= (-2.4d-46)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = (sqrt(im) * sqrt(2.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-46) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = (Math.sqrt(im) * Math.sqrt(2.0)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.4e-46:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = (math.sqrt(im) * math.sqrt(2.0)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.4e-46)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(Float64(sqrt(im) * sqrt(2.0)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.4e-46)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = (sqrt(im) * sqrt(2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.4e-46], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[im], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.40000000000000013e-46

   1. Initial program 45.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 45.1

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

   3. Applied rewrites 45.1%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. exp-to-pow N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. flip3-+ N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

   5. Applied rewrites 16.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(re \cdot re\right)}^{3} + {\left(im \cdot im\right)}^{3}}{\mathsf{fma}\left(re \cdot re, re \cdot re, \left(im \cdot im\right) \cdot \left(im \cdot im\right) - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-log.f64 N/A

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

      4. rem-exp-log N/A

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

      5. lift-log.f64 N/A

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

      6. exp-lft-sqr-rev N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in re around -inf

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

      1. lower-*.f64 74.8

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

   10. Applied rewrites 74.8%

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

   if -2.40000000000000013e-46 < re

   1. Initial program 39.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 39.9

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

      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 39.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 70.9

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

   3. Applied rewrites 70.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-hypot.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. exp-to-pow N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 70.5%

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

      1. lift-hypot.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow1/2 N/A

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

      5. unpow1 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-hypot.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-hypot.f64 N/A

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

      17. pow-prod-down N/A

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

      18. lift-pow.f64 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. lower-pow.f64 68.5

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

   7. Applied rewrites 68.5%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 59.9%

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

Alternative 6: 64.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e-46) (* 0.5 (sqrt (* -4.0 re))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-46) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.4d-46)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-46) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.4e-46:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.4e-46)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.4e-46)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.4e-46], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.40000000000000013e-46

   1. Initial program 45.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 45.1

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

   3. Applied rewrites 45.1%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. exp-to-pow N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. flip3-+ N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

   5. Applied rewrites 16.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(re \cdot re\right)}^{3} + {\left(im \cdot im\right)}^{3}}{\mathsf{fma}\left(re \cdot re, re \cdot re, \left(im \cdot im\right) \cdot \left(im \cdot im\right) - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-log.f64 N/A

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

      4. rem-exp-log N/A

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

      5. lift-log.f64 N/A

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

      6. exp-lft-sqr-rev N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in re around -inf

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

      1. lower-*.f64 74.8

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

   10. Applied rewrites 74.8%

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

   if -2.40000000000000013e-46 < re

   1. Initial program 39.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 39.9

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

   3. Applied rewrites 39.9%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. exp-to-pow N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. flip3-+ N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

   5. Applied rewrites 12.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(re \cdot re\right)}^{3} + {\left(im \cdot im\right)}^{3}}{\mathsf{fma}\left(re \cdot re, re \cdot re, \left(im \cdot im\right) \cdot \left(im \cdot im\right) - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-log.f64 N/A

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

      4. rem-exp-log N/A

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

      5. lift-log.f64 N/A

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

      6. exp-lft-sqr-rev N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 6.1

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

   7. Applied rewrites 6.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 60.2%

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

Alternative 7: 27.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 41.4

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

3. Applied rewrites 41.4%

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

   1. lift-fma.f64 N/A

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

   2. pow2 N/A

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

   3. exp-to-pow N/A

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

   4. lift-log.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. flip3-+ N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(e^{\log re \cdot 2}\right)}^{3} + {\left(im \cdot im\right)}^{3}}{e^{\log re \cdot 2} \cdot e^{\log re \cdot 2} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - e^{\log re \cdot 2} \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]

5. Applied rewrites 13.5%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\frac{{\left(re \cdot re\right)}^{3} + {\left(im \cdot im\right)}^{3}}{\mathsf{fma}\left(re \cdot re, re \cdot re, \left(im \cdot im\right) \cdot \left(im \cdot im\right) - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)}}} - re\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. rem-exp-log N/A

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

   3. lift-log.f64 N/A

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

   4. rem-exp-log N/A

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

   5. lift-log.f64 N/A

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

   6. exp-lft-sqr-rev N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-*.f64 4.4

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

7. Applied rewrites 4.4%

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

8. Taylor expanded in re around -inf

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

   1. lower-*.f64 27.4

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

10. Applied rewrites 27.4%

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

Reproduce

herbie shell --seed 2025108 
(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)))))