math.sqrt on complex, real part

Percentage Accurate: 41.9% → 85.5%

Time: 4.8s

Alternatives: 6

Speedup: N/A×

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: 41.9% 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: 85.5% accurate, N/A× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))) 0.0)
   (* 0.5 (sqrt (* 2.0 (* -0.5 (* im (/ im re))))))
   (* (pow (* (+ (hypot im re) re) 2.0) 0.5) 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 * sqrt((2.0 * (-0.5 * (im * (im / re)))));
	} else {
		tmp = pow(((hypot(im, re) + re) * 2.0), 0.5) * 0.5;
	}
	return tmp;
}

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)))) <= 0.0)
		tmp = 0.5 * sqrt((2.0 * (-0.5 * (im * (im / re)))));
	else
		tmp = (((hypot(im, re) + re) * 2.0) ^ 0.5) * 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[Sqrt[N[(2.0 * N[(-0.5 * N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 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 6.5%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 55.6

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

   7. Applied rewrites 55.6%

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

   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 45.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 89.6%

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

4. Final simplification 85.5%

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

Alternative 2: 58.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \left(im \cdot \frac{im}{re}\right)\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))))))
   (if (<= t_0 0.0)
     (* 0.5 (sqrt (* 2.0 (* -0.5 (* im (/ im re))))))
     (if (<= t_0 1e+77) t_0 (* 0.5 (sqrt (* 2.0 im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * sqrt((2.0 * (-0.5 * (im * (im / re)))));
	} else if (t_0 <= 1e+77) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
    if (t_0 <= 0.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((-0.5d0) * (im * (im / re)))))
    else if (t_0 <= 1d+77) then
        tmp = t_0
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (-0.5 * (im * (im / re)))));
	} else if (t_0 <= 1e+77) {
		tmp = t_0;
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = 0.5 * math.sqrt((2.0 * (-0.5 * (im * (im / re)))))
	elif t_0 <= 1e+77:
		tmp = t_0
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(-0.5 * Float64(im * Float64(im / re))))));
	elseif (t_0 <= 1e+77)
		tmp = t_0;
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 0.5 * sqrt((2.0 * (-0.5 * (im * (im / re)))));
	elseif (t_0 <= 1e+77)
		tmp = t_0;
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 0.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(-0.5 * N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+77], t$95$0, N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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 6.5%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 55.6

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

   7. Applied rewrites 55.6%

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

   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)))) < 9.99999999999999983e76

   1. Initial program 93.4%

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

   if 9.99999999999999983e76 < (*.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 3.5%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 28.6%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \left(im \cdot \frac{im}{re}\right)\right)}\\ \mathbf{elif}\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \leq 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 32.7% accurate, N/A× speedup

Math

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

FPCore

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

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 * sqrt((2.0 * (-0.5 * (im * (im / 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 ((0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))) <= 0.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((-0.5d0) * (im * (im / 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 ((0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)))) <= 0.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (-0.5 * (im * (im / re)))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	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 * math.sqrt((2.0 * (-0.5 * (im * (im / re)))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	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 * sqrt(Float64(2.0 * Float64(-0.5 * Float64(im * Float64(im / 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 ((0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)))) <= 0.0)
		tmp = 0.5 * sqrt((2.0 * (-0.5 * (im * (im / re)))));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	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[Sqrt[N[(2.0 * N[(-0.5 * N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $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 6.5%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 55.6

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

   7. Applied rewrites 55.6%

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

   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 45.5%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 32.4%

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

4. Final simplification 35.2%

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

Alternative 4: 26.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot im} \end{array} \]

FPCore

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

C

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

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 * im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

3. Taylor expanded in re around 0

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

5. Applied rewrites 28.9%

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

6. Final simplification 28.9%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
7. Add Preprocessing

Alternative 5: 24.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot e^{\log \left(im \cdot 2\right) \cdot 0.5} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (exp (* (log (* im 2.0)) 0.5))))

C

double code(double re, double im) {
	return 0.5 * exp((log((im * 2.0)) * 0.5));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return 0.5 * math.exp((math.log((im * 2.0)) * 0.5))

Julia

function code(re, im)
	return Float64(0.5 * exp(Float64(log(Float64(im * 2.0)) * 0.5)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * exp((log((im * 2.0)) * 0.5));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Exp[N[(N[Log[N[(im * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.8%

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

3. Taylor expanded in re around 0

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

5. Applied rewrites 28.9%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 26.9%

   \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(im \cdot 2\right) \cdot 0.5}} \]
8. Add Preprocessing

Alternative 6: 24.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {\left(e^{0.5}\right)}^{\left(\log 2 - -1 \cdot \log im\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (pow (exp 0.5) (- (log 2.0) (* -1.0 (log im))))))

C

double code(double re, double im) {
	return 0.5 * pow(exp(0.5), (log(2.0) - (-1.0 * log(im))));
}

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 * (exp(0.5d0) ** (log(2.0d0) - ((-1.0d0) * log(im))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.pow(Math.exp(0.5), (Math.log(2.0) - (-1.0 * Math.log(im))));
}

Python

def code(re, im):
	return 0.5 * math.pow(math.exp(0.5), (math.log(2.0) - (-1.0 * math.log(im))))

Julia

function code(re, im)
	return Float64(0.5 * (exp(0.5) ^ Float64(log(2.0) - Float64(-1.0 * log(im)))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (exp(0.5) ^ (log(2.0) - (-1.0 * log(im))));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Power[N[Exp[0.5], $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(-1.0 * N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.8%

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

3. Taylor expanded in re around 0

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

5. Applied rewrites 28.9%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 26.9%

   \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(im \cdot 2\right) \cdot 0.5}} \]
8. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. log-prod N/A

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

   4. lower-+.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. +-commutative N/A

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

   9. pow2 N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot e^{\left(\log im + \mathsf{Rewrite=>}\left(lower-log.f64, \log 2\right)\right) \cdot \frac{1}{2}} \]

9. Applied rewrites 26.7%

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

10. Taylor expanded in im around inf

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

    1. exp-prod N/A

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

    2. lower-pow.f64 N/A

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

    3. lower-exp.f64 N/A

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

    4. fp-cancel-sign-sub-inv N/A

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

    5. lower--.f64 N/A

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

    6. lift-log.f64 N/A

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

    7. metadata-eval N/A

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

    8. lower-*.f64 N/A

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

    9. log-div N/A

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

    10. metadata-eval N/A

        \[\leadsto \frac{1}{2} \cdot {\left(e^{\frac{1}{2}}\right)}^{\left(\log 2 - 1 \cdot \left(0 - \log \color{blue}{im}\right)\right)} \]

    11. lower--.f64 N/A

        \[\leadsto \frac{1}{2} \cdot {\left(e^{\frac{1}{2}}\right)}^{\left(\log 2 - 1 \cdot \left(0 - \color{blue}{\log im}\right)\right)} \]

    12. lower-log.f64 26.6

        \[\leadsto 0.5 \cdot {\left(e^{0.5}\right)}^{\left(\log 2 - 1 \cdot \left(0 - \log im\right)\right)} \]

12. Applied rewrites 26.6%

    \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.5}\right)}^{\left(\log 2 - 1 \cdot \left(0 - \log im\right)\right)}} \]

13. Final simplification 26.6%

    \[\leadsto 0.5 \cdot {\left(e^{0.5}\right)}^{\left(\log 2 - -1 \cdot \log im\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform c (if (< re 0) (* 1/2 (* (sqrt 2) (sqrt (/ (* im im) (- (modulus re im) re))))) (* 1/2 (sqrt (* 2 (+ (modulus re im) re))))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))