math.sqrt on complex, real part

Percentage Accurate: 41.8% → 88.9%

Time: 3.9s

Alternatives: 8

Speedup: 1.9×

Specification

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: 41.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.9% accurate, 0.1× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* 2.0 (+ (sqrt (+ (* re re) (* im_m im_m))) re)) 0.0)
   (* 0.5 (exp (* (fma 2.0 (log im_m) (log (/ -1.0 re))) 0.5)))
   (* (sqrt (* (+ (hypot im_m re) re) 2.0)) 0.5)))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Exp[N[(N[(2.0 * N[Log[im$95$m], $MachinePrecision] + N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Sqrt[im$95$m ^ 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 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)) < 0.0

   1. Initial program 7.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      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

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

      8. pow1/2 N/A

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

      9. pow-to-exp N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 17.9%

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

   4. Taylor expanded in re around -inf

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

      1. +-commutative N/A

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

      2. log-pow N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 82.1

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

   6. Applied rewrites 82.1%

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

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

   1. Initial program 47.8%

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

      1. lift-*.f64 N/A

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

      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 90.1%

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

Alternative 2: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{-im\_m \cdot \frac{im\_m}{re}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im\_m, re\right) + re\right) \cdot 2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.8e+152)
   (* (sqrt (- (* im_m (/ im_m re)))) 0.5)
   (* (sqrt (* (+ (hypot im_m re) re) 2.0)) 0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.8e+152) {
		tmp = sqrt(-(im_m * (im_m / re))) * 0.5;
	} else {
		tmp = sqrt(((hypot(im_m, re) + re) * 2.0)) * 0.5;
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -1.8e+152) {
		tmp = Math.sqrt(-(im_m * (im_m / re))) * 0.5;
	} else {
		tmp = Math.sqrt(((Math.hypot(im_m, re) + re) * 2.0)) * 0.5;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -1.8e+152:
		tmp = math.sqrt(-(im_m * (im_m / re))) * 0.5
	else:
		tmp = math.sqrt(((math.hypot(im_m, re) + re) * 2.0)) * 0.5
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.8e+152)
		tmp = Float64(sqrt(Float64(-Float64(im_m * Float64(im_m / re)))) * 0.5);
	else
		tmp = Float64(sqrt(Float64(Float64(hypot(im_m, re) + re) * 2.0)) * 0.5);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -1.8e+152)
		tmp = sqrt(-(im_m * (im_m / re))) * 0.5;
	else
		tmp = sqrt(((hypot(im_m, re) + re) * 2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.8e+152], N[(N[Sqrt[(-N[(im$95$m * N[(im$95$m / re), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Sqrt[im$95$m ^ 2 + re ^ 2], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.7999999999999999e152

   1. Initial program 2.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. 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 33.7%

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

   4. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 53.1

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

   6. Applied rewrites 53.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 65.1

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

   8. Applied rewrites 65.1%

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

   if -1.7999999999999999e152 < re

   1. Initial program 47.4%

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

      1. lift-*.f64 N/A

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

      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 86.0%

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

Alternative 3: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.95 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{-im\_m \cdot \frac{im\_m}{re}} \cdot 0.5\\ \mathbf{elif}\;re \leq 4.3 \cdot 10^{-162}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m + im\_m}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{+55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im\_m, im\_m, re \cdot re\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.95e+65)
   (* (sqrt (- (* im_m (/ im_m re)))) 0.5)
   (if (<= re 4.3e-162)
     (* 0.5 (sqrt (+ im_m im_m)))
     (if (<= re 4e+55)
       (* 0.5 (sqrt (* 2.0 (+ (sqrt (fma im_m im_m (* re re))) re))))
       (sqrt re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.95e+65) {
		tmp = sqrt(-(im_m * (im_m / re))) * 0.5;
	} else if (re <= 4.3e-162) {
		tmp = 0.5 * sqrt((im_m + im_m));
	} else if (re <= 4e+55) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(im_m, im_m, (re * re))) + re)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.95e+65)
		tmp = Float64(sqrt(Float64(-Float64(im_m * Float64(im_m / re)))) * 0.5);
	elseif (re <= 4.3e-162)
		tmp = Float64(0.5 * sqrt(Float64(im_m + im_m)));
	elseif (re <= 4e+55)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(im_m, im_m, Float64(re * re))) + re))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.95e+65], N[(N[Sqrt[(-N[(im$95$m * N[(im$95$m / re), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 4.3e-162], N[(0.5 * N[Sqrt[N[(im$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4e+55], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(im$95$m * im$95$m + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.9499999999999999e65

   1. Initial program 8.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 36.7%

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

   4. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 50.4

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

   6. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 58.3

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

   8. Applied rewrites 58.3%

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

   if -1.9499999999999999e65 < re < 4.29999999999999996e-162

   1. Initial program 48.0%

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

   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 76.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 76.7

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

   6. Applied rewrites 76.7%

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

   if 4.29999999999999996e-162 < re < 4.00000000000000004e55

   1. Initial program 73.8%

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

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 73.8

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

   3. Applied rewrites 73.8%

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

   if 4.00000000000000004e55 < re

   1. Initial program 33.5%

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

   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 24.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 24.0

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

   6. Applied rewrites 24.0%

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

   7. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\sqrt{re}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{re}} \]

      2. count-2 N/A

         \[\leadsto \sqrt{re} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{re} \]

      4. lower-sqrt.f64 80.7

         \[\leadsto \sqrt{re} \]

   9. Applied rewrites 80.7%

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

Alternative 4: 71.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.95 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{-im\_m \cdot \frac{im\_m}{re}} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.16 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m + im\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.95e+65)
   (* (sqrt (- (* im_m (/ im_m re)))) 0.5)
   (if (<= re 1.16e+23) (* 0.5 (sqrt (+ im_m im_m))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.95e+65) {
		tmp = sqrt(-(im_m * (im_m / re))) * 0.5;
	} else if (re <= 1.16e+23) {
		tmp = 0.5 * sqrt((im_m + im_m));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-1.95d+65)) then
        tmp = sqrt(-(im_m * (im_m / re))) * 0.5d0
    else if (re <= 1.16d+23) then
        tmp = 0.5d0 * sqrt((im_m + im_m))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -1.95e+65) {
		tmp = Math.sqrt(-(im_m * (im_m / re))) * 0.5;
	} else if (re <= 1.16e+23) {
		tmp = 0.5 * Math.sqrt((im_m + im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -1.95e+65:
		tmp = math.sqrt(-(im_m * (im_m / re))) * 0.5
	elif re <= 1.16e+23:
		tmp = 0.5 * math.sqrt((im_m + im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.95e+65)
		tmp = Float64(sqrt(Float64(-Float64(im_m * Float64(im_m / re)))) * 0.5);
	elseif (re <= 1.16e+23)
		tmp = Float64(0.5 * sqrt(Float64(im_m + im_m)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -1.95e+65)
		tmp = sqrt(-(im_m * (im_m / re))) * 0.5;
	elseif (re <= 1.16e+23)
		tmp = 0.5 * sqrt((im_m + im_m));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.95e+65], N[(N[Sqrt[(-N[(im$95$m * N[(im$95$m / re), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.16e+23], N[(0.5 * N[Sqrt[N[(im$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9499999999999999e65

   1. Initial program 8.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 36.7%

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

   4. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 50.4

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

   6. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 58.3

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

   8. Applied rewrites 58.3%

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

   if -1.9499999999999999e65 < re < 1.16e23

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if 1.16e23 < re

   1. Initial program 38.5%

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

   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 26.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 26.7

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

   6. Applied rewrites 26.7%

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

   7. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\sqrt{re}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{re}} \]

      2. count-2 N/A

         \[\leadsto \sqrt{re} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{re} \]

      4. lower-sqrt.f64 78.0

         \[\leadsto \sqrt{re} \]

   9. Applied rewrites 78.0%

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

Alternative 5: 70.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.95 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{im\_m \cdot im\_m}{re}}\\ \mathbf{elif}\;re \leq 1.16 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m + im\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.95e+65)
   (* 0.5 (sqrt (- (/ (* im_m im_m) re))))
   (if (<= re 1.16e+23) (* 0.5 (sqrt (+ im_m im_m))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.95e+65) {
		tmp = 0.5 * sqrt(-((im_m * im_m) / re));
	} else if (re <= 1.16e+23) {
		tmp = 0.5 * sqrt((im_m + im_m));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-1.95d+65)) then
        tmp = 0.5d0 * sqrt(-((im_m * im_m) / re))
    else if (re <= 1.16d+23) then
        tmp = 0.5d0 * sqrt((im_m + im_m))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -1.95e+65) {
		tmp = 0.5 * Math.sqrt(-((im_m * im_m) / re));
	} else if (re <= 1.16e+23) {
		tmp = 0.5 * Math.sqrt((im_m + im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -1.95e+65:
		tmp = 0.5 * math.sqrt(-((im_m * im_m) / re))
	elif re <= 1.16e+23:
		tmp = 0.5 * math.sqrt((im_m + im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.95e+65)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(Float64(im_m * im_m) / re))));
	elseif (re <= 1.16e+23)
		tmp = Float64(0.5 * sqrt(Float64(im_m + im_m)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -1.95e+65)
		tmp = 0.5 * sqrt(-((im_m * im_m) / re));
	elseif (re <= 1.16e+23)
		tmp = 0.5 * sqrt((im_m + im_m));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.95e+65], N[(0.5 * N[Sqrt[(-N[(N[(im$95$m * im$95$m), $MachinePrecision] / re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.16e+23], N[(0.5 * N[Sqrt[N[(im$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9499999999999999e65

   1. Initial program 8.2%

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

   2. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 50.4

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

   4. Applied rewrites 50.4%

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

   if -1.9499999999999999e65 < re < 1.16e23

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if 1.16e23 < re

   1. Initial program 38.5%

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

   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 26.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 26.7

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

   6. Applied rewrites 26.7%

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

   7. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\sqrt{re}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{re}} \]

      2. count-2 N/A

         \[\leadsto \sqrt{re} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{re} \]

      4. lower-sqrt.f64 78.0

         \[\leadsto \sqrt{re} \]

   9. Applied rewrites 78.0%

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

Alternative 6: 65.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.65 \cdot 10^{+187}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) + re\right)}\\ \mathbf{elif}\;re \leq 1.16 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m + im\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.65e+187)
   (* 0.5 (sqrt (* 2.0 (+ (- re) re))))
   (if (<= re 1.16e+23) (* 0.5 (sqrt (+ im_m im_m))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.65e+187) {
		tmp = 0.5 * sqrt((2.0 * (-re + re)));
	} else if (re <= 1.16e+23) {
		tmp = 0.5 * sqrt((im_m + im_m));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-1.65d+187)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (-re + re)))
    else if (re <= 1.16d+23) then
        tmp = 0.5d0 * sqrt((im_m + im_m))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -1.65e+187) {
		tmp = 0.5 * Math.sqrt((2.0 * (-re + re)));
	} else if (re <= 1.16e+23) {
		tmp = 0.5 * Math.sqrt((im_m + im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -1.65e+187:
		tmp = 0.5 * math.sqrt((2.0 * (-re + re)))
	elif re <= 1.16e+23:
		tmp = 0.5 * math.sqrt((im_m + im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.65e+187)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(-re) + re))));
	elseif (re <= 1.16e+23)
		tmp = Float64(0.5 * sqrt(Float64(im_m + im_m)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -1.65e+187)
		tmp = 0.5 * sqrt((2.0 * (-re + re)));
	elseif (re <= 1.16e+23)
		tmp = 0.5 * sqrt((im_m + im_m));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.65e+187], N[(0.5 * N[Sqrt[N[(2.0 * N[((-re) + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.16e+23], N[(0.5 * N[Sqrt[N[(im$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.6500000000000001e187

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 22.4

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

   4. Applied rewrites 22.4%

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

   if -1.6500000000000001e187 < re < 1.16e23

   1. Initial program 48.7%

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

   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 67.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 67.7

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

   6. Applied rewrites 67.7%

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

   if 1.16e23 < re

   1. Initial program 38.5%

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

   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 26.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 26.7

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

   6. Applied rewrites 26.7%

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

   7. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\sqrt{re}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{re}} \]

      2. count-2 N/A

         \[\leadsto \sqrt{re} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{re} \]

      4. lower-sqrt.f64 78.0

         \[\leadsto \sqrt{re} \]

   9. Applied rewrites 78.0%

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

Alternative 7: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 1.16 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m + im\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 1.16e+23) (* 0.5 (sqrt (+ im_m im_m))) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 1.16e+23) {
		tmp = 0.5 * sqrt((im_m + im_m));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.16d+23) then
        tmp = 0.5d0 * sqrt((im_m + im_m))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 1.16e+23) {
		tmp = 0.5 * Math.sqrt((im_m + im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 1.16e+23:
		tmp = 0.5 * math.sqrt((im_m + im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 1.16e+23)
		tmp = Float64(0.5 * sqrt(Float64(im_m + im_m)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 1.16e+23)
		tmp = 0.5 * sqrt((im_m + im_m));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 1.16e+23], N[(0.5 * N[Sqrt[N[(im$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.16e23

   1. Initial program 42.8%

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

   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 60.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 60.8

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

   6. Applied rewrites 60.8%

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

   if 1.16e23 < re

   1. Initial program 38.5%

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

   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 26.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 26.7

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

   6. Applied rewrites 26.7%

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

   7. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\sqrt{re}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{re}} \]

      2. count-2 N/A

         \[\leadsto \sqrt{re} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{re} \]

      4. lower-sqrt.f64 78.0

         \[\leadsto \sqrt{re} \]

   9. Applied rewrites 78.0%

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

Alternative 8: 26.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sqrt{re} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (sqrt re))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return sqrt(re);
}

Fortran

im_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = sqrt(re)
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return Math.sqrt(re);
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return math.sqrt(re)

Julia

im_m = abs(im)
function code(re, im_m)
	return sqrt(re)
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = sqrt(re);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 41.8%

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

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 53.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 53.0

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

6. Applied rewrites 53.0%

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

7. Taylor expanded in re around inf

   \[\leadsto \color{blue}{\sqrt{re}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{re}} \]

   2. count-2 N/A

      \[\leadsto \sqrt{re} \]

   3. *-commutative N/A

      \[\leadsto \sqrt{re} \]

   4. lower-sqrt.f64 26.0

      \[\leadsto \sqrt{re} \]

9. Applied rewrites 26.0%

   \[\leadsto \color{blue}{\sqrt{re}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64
  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))