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

Percentage Accurate: 41.6% → 89.9%

Time: 3.7s

Alternatives: 9

Speedup: 2.5×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.6% 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: 89.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.4%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 90.4

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

   5. Applied rewrites 90.4%

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

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

   1. Initial program 47.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 89.7%

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

Alternative 2: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im\right) + im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(1 \cdot im\right)\right) \cdot \frac{1}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.1e+84)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -1.9e-125)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 2.8e-37)
       (* 0.5 (sqrt (+ (fma (- (/ re im) 2.0) re im) im)))
       (* (* 0.5 (* 1.0 im)) (/ 1.0 (sqrt re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.1e+84) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -1.9e-125) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 2.8e-37) {
		tmp = 0.5 * sqrt((fma(((re / im) - 2.0), re, im) + im));
	} else {
		tmp = (0.5 * (1.0 * im)) * (1.0 / sqrt(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.1e+84)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -1.9e-125)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 2.8e-37)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(Float64(re / im) - 2.0), re, im) + im)));
	else
		tmp = Float64(Float64(0.5 * Float64(1.0 * im)) * Float64(1.0 / sqrt(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.1e+84], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.9e-125], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.8e-37], N[(0.5 * N[Sqrt[N[(N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(1.0 * im), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.10000000000000019e84

   1. Initial program 25.3%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if -2.10000000000000019e84 < re < -1.9000000000000001e-125

   1. Initial program 77.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 77.3

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

   4. Applied rewrites 77.3%

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

   if -1.9000000000000001e-125 < re < 2.8000000000000001e-37

   1. Initial program 53.5%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 83.4

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

   7. Applied rewrites 83.4%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 83.4

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

   9. Applied rewrites 83.4%

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

   if 2.8000000000000001e-37 < re

   1. Initial program 14.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 40.1%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 74.8

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

   7. Applied rewrites 74.8%

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

   8. Taylor expanded in re around 0

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 74.7

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

   10. Applied rewrites 74.7%

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

Alternative 3: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im}, -2, 2\right) \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(1 \cdot im\right)\right) \cdot \frac{1}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e-10)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 2.8e-37)
     (* 0.5 (sqrt (* (fma (/ re im) -2.0 2.0) im)))
     (* (* 0.5 (* 1.0 im)) (/ 1.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-10) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 2.8e-37) {
		tmp = 0.5 * sqrt((fma((re / im), -2.0, 2.0) * im));
	} else {
		tmp = (0.5 * (1.0 * im)) * (1.0 / sqrt(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e-10)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 2.8e-37)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(re / im), -2.0, 2.0) * im)));
	else
		tmp = Float64(Float64(0.5 * Float64(1.0 * im)) * Float64(1.0 / sqrt(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e-10], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.8e-37], N[(0.5 * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] * -2.0 + 2.0), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(1.0 * im), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.5e-10

   1. Initial program 40.4%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -4.5e-10 < re < 2.8000000000000001e-37

   1. Initial program 58.5%

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

   3. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

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

   if 2.8000000000000001e-37 < re

   1. Initial program 14.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 40.1%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 74.8

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

   7. Applied rewrites 74.8%

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

   8. Taylor expanded in re around 0

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 74.7

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

   10. Applied rewrites 74.7%

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

Alternative 4: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(-2, re, im\right) + im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(1 \cdot im\right)\right) \cdot \frac{1}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e-10)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 2.8e-37)
     (* 0.5 (sqrt (+ (fma -2.0 re im) im)))
     (* (* 0.5 (* 1.0 im)) (/ 1.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-10) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 2.8e-37) {
		tmp = 0.5 * sqrt((fma(-2.0, re, im) + im));
	} else {
		tmp = (0.5 * (1.0 * im)) * (1.0 / sqrt(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e-10)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 2.8e-37)
		tmp = Float64(0.5 * sqrt(Float64(fma(-2.0, re, im) + im)));
	else
		tmp = Float64(Float64(0.5 * Float64(1.0 * im)) * Float64(1.0 / sqrt(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e-10], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.8e-37], N[(0.5 * N[Sqrt[N[(N[(-2.0 * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(1.0 * im), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.5e-10

   1. Initial program 40.4%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -4.5e-10 < re < 2.8000000000000001e-37

   1. Initial program 58.5%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 78.9

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

   7. Applied rewrites 78.9%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 78.9

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

   9. Applied rewrites 78.9%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 79.6%

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

   if 2.8000000000000001e-37 < re

   1. Initial program 14.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 40.1%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 74.8

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

   7. Applied rewrites 74.8%

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

   8. Taylor expanded in re around 0

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 74.7

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

   10. Applied rewrites 74.7%

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

Alternative 5: 77.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(-2, re, im\right) + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{1}{\sqrt{re}} \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e-10)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 2.8e-37)
     (* 0.5 (sqrt (+ (fma -2.0 re im) im)))
     (* 0.5 (* (/ 1.0 (sqrt re)) im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-10) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 2.8e-37) {
		tmp = 0.5 * sqrt((fma(-2.0, re, im) + im));
	} else {
		tmp = 0.5 * ((1.0 / sqrt(re)) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e-10)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 2.8e-37)
		tmp = Float64(0.5 * sqrt(Float64(fma(-2.0, re, im) + im)));
	else
		tmp = Float64(0.5 * Float64(Float64(1.0 / sqrt(re)) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e-10], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.8e-37], N[(0.5 * N[Sqrt[N[(N[(-2.0 * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.5e-10

   1. Initial program 40.4%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -4.5e-10 < re < 2.8000000000000001e-37

   1. Initial program 58.5%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 78.9

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

   7. Applied rewrites 78.9%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 78.9

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

   9. Applied rewrites 78.9%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 79.6%

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

   if 2.8000000000000001e-37 < re

   1. Initial program 14.7%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 26.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 25.9

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

   7. Applied rewrites 25.9%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. lower-*.f64 N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 74.7

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

   10. Applied rewrites 74.7%

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

Alternative 6: 64.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(-2, re, im\right) + im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e-10)
   (* 0.5 (sqrt (* -4.0 re)))
   (* 0.5 (sqrt (+ (fma -2.0 re im) im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-10) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * sqrt((fma(-2.0, re, im) + im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e-10)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(fma(-2.0, re, im) + im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e-10], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(N[(-2.0 * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.5e-10

   1. Initial program 40.4%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -4.5e-10 < re

   1. Initial program 42.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 60.0

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

   7. Applied rewrites 60.0%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 60.0

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

   9. Applied rewrites 60.0%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 59.4%

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

Alternative 7: 64.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e-10)
   (* 0.5 (sqrt (* -4.0 re)))
   (* 0.5 (sqrt (* 2.0 (- im re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-10) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.5d-10)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-10) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5e-10:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e-10)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5e-10)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e-10], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.5e-10

   1. Initial program 40.4%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -4.5e-10 < re

   1. Initial program 42.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 59.4%

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

Alternative 8: 64.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.6e-10) (* 0.5 (sqrt (* -4.0 re))) (* 0.5 (sqrt (+ im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.6e-10) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * sqrt((im + im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.6d-10)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = 0.5d0 * sqrt((im + im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.6e-10) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * Math.sqrt((im + im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.6e-10:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = 0.5 * math.sqrt((im + im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.6e-10)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.6e-10)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = 0.5 * sqrt((im + im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.6e-10], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.5999999999999999e-10

   1. Initial program 40.5%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -1.5999999999999999e-10 < re

   1. Initial program 42.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 60.0

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

   7. Applied rewrites 60.0%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 60.0

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

   9. Applied rewrites 60.0%

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

   10. Taylor expanded in re around 0

       \[\leadsto \frac{1}{2} \cdot \sqrt{im + im} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.0%

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

Alternative 9: 51.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.6%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 51.6

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

5. Applied rewrites 51.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 51.6

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

7. Applied rewrites 51.6%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-+r+ N/A

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

   6. lower-+.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 51.6

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

9. Applied rewrites 51.6%

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

10. Taylor expanded in re around 0

    \[\leadsto \frac{1}{2} \cdot \sqrt{im + im} \]
11. Step-by-step derivation

12. Applied rewrites 51.6%

    \[\leadsto 0.5 \cdot \sqrt{im + im} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))