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

Percentage Accurate: 41.7% → 90.3%

Time: 4.0s

Alternatives: 10

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: 41.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)))) <= 0.0) {
		tmp = 0.5 * ((1.0 * im) * Math.pow(re, -0.5));
	} else {
		tmp = Math.sqrt(((Math.hypot(im, re) - re) * 2.0)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))) <= 0.0:
		tmp = 0.5 * ((1.0 * im) * math.pow(re, -0.5))
	else:
		tmp = math.sqrt(((math.hypot(im, re) - re) * 2.0)) * 0.5
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(1.0 * im), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 10.1%

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

   2. Taylor expanded in re around inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 89.0%

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

Alternative 2: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re} \cdot \frac{im}{re}, 0.5, 1\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 1300000:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im}, -2, 2\right) \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(1 \cdot im\right) \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1e+75)
   (*
    (sqrt (* (- (* (- re) (fma (* (/ im re) (/ im re)) 0.5 1.0)) re) 2.0))
    0.5)
   (if (<= re -5.5e-6)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 1300000.0)
       (* 0.5 (sqrt (* (fma (/ re im) -2.0 2.0) im)))
       (* 0.5 (* (* 1.0 im) (pow re -0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e+75) {
		tmp = sqrt((((-re * fma(((im / re) * (im / re)), 0.5, 1.0)) - re) * 2.0)) * 0.5;
	} else if (re <= -5.5e-6) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 1300000.0) {
		tmp = 0.5 * sqrt((fma((re / im), -2.0, 2.0) * im));
	} else {
		tmp = 0.5 * ((1.0 * im) * pow(re, -0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e+75)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(-re) * fma(Float64(Float64(im / re) * Float64(im / re)), 0.5, 1.0)) - re) * 2.0)) * 0.5);
	elseif (re <= -5.5e-6)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 1300000.0)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(re / im), -2.0, 2.0) * im)));
	else
		tmp = Float64(0.5 * Float64(Float64(1.0 * im) * (re ^ -0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e+75], N[(N[Sqrt[N[(N[(N[((-re) * N[(N[(N[(im / re), $MachinePrecision] * N[(im / re), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, -5.5e-6], 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, 1300000.0], N[(0.5 * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] * -2.0 + 2.0), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(1.0 * im), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -9.99999999999999927e74

   1. Initial program 28.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in re around inf

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

      1. pow2 2.5

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

      2. pow2 2.5

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

      3. +-commutative 2.5

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

      4. pow2 2.5

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

      5. pow2 2.5

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

   6. Applied rewrites 2.5%

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

   7. Taylor expanded in re around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 81.5

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

   9. Applied rewrites 81.5%

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

   if -9.99999999999999927e74 < re < -5.4999999999999999e-6

   1. Initial program 76.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 76.7

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

   3. Applied rewrites 76.7%

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

   if -5.4999999999999999e-6 < re < 1.3e6

   1. Initial program 55.7%

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

   2. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 76.8

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

   4. Applied rewrites 76.8%

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

   if 1.3e6 < re

   1. Initial program 13.9%

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

   2. Taylor expanded in re around inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 75.9

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

   4. Applied rewrites 75.9%

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

Alternative 3: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re} \cdot \frac{im}{re}, 0.5, 1\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 1300000:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im}, -2, 2\right) \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \sqrt{\frac{0.5}{re}}\right) \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1e+75)
   (*
    (sqrt (* (- (* (- re) (fma (* (/ im re) (/ im re)) 0.5 1.0)) re) 2.0))
    0.5)
   (if (<= re -5.5e-6)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 1300000.0)
       (* 0.5 (sqrt (* (fma (/ re im) -2.0 2.0) im)))
       (* 0.5 (* (* im (sqrt (/ 0.5 re))) (sqrt 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e+75) {
		tmp = sqrt((((-re * fma(((im / re) * (im / re)), 0.5, 1.0)) - re) * 2.0)) * 0.5;
	} else if (re <= -5.5e-6) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 1300000.0) {
		tmp = 0.5 * sqrt((fma((re / im), -2.0, 2.0) * im));
	} else {
		tmp = 0.5 * ((im * sqrt((0.5 / re))) * sqrt(2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e+75)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(-re) * fma(Float64(Float64(im / re) * Float64(im / re)), 0.5, 1.0)) - re) * 2.0)) * 0.5);
	elseif (re <= -5.5e-6)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 1300000.0)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(re / im), -2.0, 2.0) * im)));
	else
		tmp = Float64(0.5 * Float64(Float64(im * sqrt(Float64(0.5 / re))) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e+75], N[(N[Sqrt[N[(N[(N[((-re) * N[(N[(N[(im / re), $MachinePrecision] * N[(im / re), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, -5.5e-6], 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, 1300000.0], N[(0.5 * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] * -2.0 + 2.0), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(im * N[Sqrt[N[(0.5 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -9.99999999999999927e74

   1. Initial program 28.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in re around inf

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

      1. pow2 2.5

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

      2. pow2 2.5

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

      3. +-commutative 2.5

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

      4. pow2 2.5

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

      5. pow2 2.5

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

   6. Applied rewrites 2.5%

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

   7. Taylor expanded in re around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 81.5

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

   9. Applied rewrites 81.5%

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

   if -9.99999999999999927e74 < re < -5.4999999999999999e-6

   1. Initial program 76.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 76.7

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

   3. Applied rewrites 76.7%

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

   if -5.4999999999999999e-6 < re < 1.3e6

   1. Initial program 55.7%

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

   2. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 76.8

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

   4. Applied rewrites 76.8%

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

   if 1.3e6 < re

   1. Initial program 13.9%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 25.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 24.9

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

   6. Applied rewrites 24.9%

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

   7. Taylor expanded in re around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. *-commutative N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 75.6

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

   9. Applied rewrites 75.6%

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

Alternative 4: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 1300000:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im}, -2, 2\right) \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \sqrt{\frac{0.5}{re}}\right) \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e+65)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -5.5e-6)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 1300000.0)
       (* 0.5 (sqrt (* (fma (/ re im) -2.0 2.0) im)))
       (* 0.5 (* (* im (sqrt (/ 0.5 re))) (sqrt 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e+65) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -5.5e-6) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 1300000.0) {
		tmp = 0.5 * sqrt((fma((re / im), -2.0, 2.0) * im));
	} else {
		tmp = 0.5 * ((im * sqrt((0.5 / re))) * sqrt(2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.6e+65)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -5.5e-6)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 1300000.0)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(re / im), -2.0, 2.0) * im)));
	else
		tmp = Float64(0.5 * Float64(Float64(im * sqrt(Float64(0.5 / re))) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e+65], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -5.5e-6], 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, 1300000.0], N[(0.5 * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] * -2.0 + 2.0), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(im * N[Sqrt[N[(0.5 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.59999999999999978e65

   1. Initial program 30.5%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 80.7

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

   4. Applied rewrites 80.7%

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

   if -3.59999999999999978e65 < re < -5.4999999999999999e-6

   1. Initial program 75.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 75.9

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

   3. Applied rewrites 75.9%

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

   if -5.4999999999999999e-6 < re < 1.3e6

   1. Initial program 55.7%

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

   2. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 76.8

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

   4. Applied rewrites 76.8%

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

   if 1.3e6 < re

   1. Initial program 13.9%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 25.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 24.9

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

   6. Applied rewrites 24.9%

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

   7. Taylor expanded in re around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. *-commutative N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 75.6

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

   9. Applied rewrites 75.6%

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

Alternative 5: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im}, -2, 2\right) \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e+65)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -5.5e-6)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 2.05e+70)
       (* 0.5 (sqrt (* (fma (/ re im) -2.0 2.0) im)))
       (* 0.5 (sqrt (* im (/ im re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e+65) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -5.5e-6) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 2.05e+70) {
		tmp = 0.5 * sqrt((fma((re / im), -2.0, 2.0) * im));
	} else {
		tmp = 0.5 * sqrt((im * (im / re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.6e+65)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -5.5e-6)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 2.05e+70)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(re / im), -2.0, 2.0) * im)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(im / re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e+65], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -5.5e-6], 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, 2.05e+70], N[(0.5 * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] * -2.0 + 2.0), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.59999999999999978e65

   1. Initial program 30.5%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 80.7

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

   4. Applied rewrites 80.7%

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

   if -3.59999999999999978e65 < re < -5.4999999999999999e-6

   1. Initial program 75.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 75.9

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

   3. Applied rewrites 75.9%

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

   if -5.4999999999999999e-6 < re < 2.0500000000000001e70

   1. Initial program 53.3%

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

   2. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 73.7

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

   4. Applied rewrites 73.7%

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

   if 2.0500000000000001e70 < re

   1. Initial program 9.3%

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

   2. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.9

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

   4. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 57.5

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

   6. Applied rewrites 57.5%

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

Alternative 6: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im}, -2, 2\right) \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.3e-5)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 2.05e+70)
     (* 0.5 (sqrt (* (fma (/ re im) -2.0 2.0) im)))
     (* 0.5 (sqrt (* im (/ im re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.3e-5) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 2.05e+70) {
		tmp = 0.5 * sqrt((fma((re / im), -2.0, 2.0) * im));
	} else {
		tmp = 0.5 * sqrt((im * (im / re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.3e-5)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 2.05e+70)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(re / im), -2.0, 2.0) * im)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(im / re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.3e-5], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.05e+70], N[(0.5 * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] * -2.0 + 2.0), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.3e-5

   1. Initial program 40.7%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

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

   if -2.3e-5 < re < 2.0500000000000001e70

   1. Initial program 53.4%

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

   2. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 73.7

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

   4. Applied rewrites 73.7%

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

   if 2.0500000000000001e70 < re

   1. Initial program 9.3%

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

   2. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.9

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

   4. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 57.5

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

   6. Applied rewrites 57.5%

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

Alternative 7: 70.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.3e-5)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 2.05e+70)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (sqrt (* im (/ im re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.3e-5) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 2.05e+70) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * sqrt((im * (im / re)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.3d-5)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 2.05d+70) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * sqrt((im * (im / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.3e-5) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 2.05e+70) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * Math.sqrt((im * (im / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.3e-5:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 2.05e+70:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * math.sqrt((im * (im / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.3e-5)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 2.05e+70)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(im / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.3e-5)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 2.05e+70)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * sqrt((im * (im / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.3e-5], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.05e+70], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.3e-5

   1. Initial program 40.7%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

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

   if -2.3e-5 < re < 2.0500000000000001e70

   1. Initial program 53.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 73.7%

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

   if 2.0500000000000001e70 < re

   1. Initial program 9.3%

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

   2. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.9

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

   4. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 57.5

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

   6. Applied rewrites 57.5%

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

Alternative 8: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{+198}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.5e-6)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 4.6e+198)
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (sqrt (* 2.0 (- re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.5e-6) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 4.6e+198) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-7.5d-6)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 4.6d+198) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.5e-6) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 4.6e+198) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.5e-6:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 4.6e+198:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.5e-6)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 4.6e+198)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.5e-6)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 4.6e+198)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.5e-6], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.6e+198], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.50000000000000019e-6

   1. Initial program 40.7%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 75.1

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

   4. Applied rewrites 75.1%

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

   if -7.50000000000000019e-6 < re < 4.6000000000000001e198

   1. Initial program 47.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 66.4%

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

   if 4.6000000000000001e198 < re

   1. Initial program 2.6%

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

   2. Taylor expanded in re around inf

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

   4. Applied rewrites 22.8%

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

Alternative 9: 64.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.5e-6) (* 0.5 (sqrt (* -4.0 re))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.5e-6) {
		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 <= (-7.5d-6)) 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 <= -7.5e-6) {
		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 <= -7.5e-6:
		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 <= -7.5e-6)
		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 <= -7.5e-6)
		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, -7.5e-6], 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 < -7.50000000000000019e-6

   1. Initial program 40.7%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 75.1

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

   4. Applied rewrites 75.1%

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

   if -7.50000000000000019e-6 < re

   1. Initial program 42.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 60.3%

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

Alternative 10: 26.0% 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 41.7%

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

2. Taylor expanded in re around -inf

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

   1. lower-*.f64 26.0

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

4. Applied rewrites 26.0%

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

Reproduce

herbie shell --seed 2025103 
(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)))))