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

Percentage Accurate: 42.6% → 88.2%

Time: 3.4s

Alternatives: 8

Speedup: 1.7×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.6% 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: 88.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9.2 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9.2e-12)
   (* (sqrt (* (- (hypot im re) re) 2.0)) 0.5)
   (* (* 0.5 im) (pow re -0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9.2e-12) {
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = (0.5 * im) * pow(re, -0.5);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9.2e-12) {
		tmp = Math.sqrt(((Math.hypot(im, re) - re) * 2.0)) * 0.5;
	} else {
		tmp = (0.5 * im) * Math.pow(re, -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9.2e-12:
		tmp = math.sqrt(((math.hypot(im, re) - re) * 2.0)) * 0.5
	else:
		tmp = (0.5 * im) * math.pow(re, -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9.2e-12)
		tmp = Float64(sqrt(Float64(Float64(hypot(im, re) - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(0.5 * im) * (re ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9.2e-12)
		tmp = sqrt(((hypot(im, re) - re) * 2.0)) * 0.5;
	else
		tmp = (0.5 * im) * (re ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9.2e-12], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.19999999999999957e-12

   1. Initial program 53.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 94.7%

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

   if 9.19999999999999957e-12 < re

   1. Initial program 8.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 78.5

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

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\left(0.5 \cdot \left(1 \cdot im\right)\right) \cdot {re}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9.2 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.2e+115)
   (* (sqrt (* (- re) (fma (/ im re) (/ im re) 4.0))) 0.5)
   (if (<= re 2.1e-18)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* 0.5 im) (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.2e+115) {
		tmp = sqrt((-re * fma((im / re), (im / re), 4.0))) * 0.5;
	} else if (re <= 2.1e-18) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) * pow(re, -0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.2e+115)
		tmp = Float64(sqrt(Float64(Float64(-re) * fma(Float64(im / re), Float64(im / re), 4.0))) * 0.5);
	elseif (re <= 2.1e-18)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(0.5 * im) * (re ^ -0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.2e+115], N[(N[Sqrt[N[((-re) * N[(N[(im / re), $MachinePrecision] * N[(im / re), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 2.1e-18], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.2e115

   1. Initial program 15.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 100.0%

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

   5. Taylor expanded in re around inf

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

      1. pow2 2.5

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

      2. pow2 2.5

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

      3. +-commutative 2.5

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

      4. pow2 2.5

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

      5. pow2 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in re around -inf

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

      1. associate-*r* N/A

         \[\leadsto \sqrt{\left(-1 \cdot re\right) \cdot \color{blue}{\left(4 + \frac{{im}^{2}}{{re}^{2}}\right)}} \cdot \frac{1}{2} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{neg}\left(re\right)\right) \cdot \left(\color{blue}{4} + \frac{{im}^{2}}{{re}^{2}}\right)} \cdot \frac{1}{2} \]

      3. lift-neg.f64 N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \left(\color{blue}{4} + \frac{{im}^{2}}{{re}^{2}}\right)} \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \color{blue}{\left(4 + \frac{{im}^{2}}{{re}^{2}}\right)}} \cdot \frac{1}{2} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \left(\frac{{im}^{2}}{{re}^{2}} + \color{blue}{4}\right)} \cdot \frac{1}{2} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \left(\frac{im \cdot im}{{re}^{2}} + 4\right)} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \left(\frac{im \cdot im}{re \cdot re} + 4\right)} \cdot \frac{1}{2} \]

      8. times-frac N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \left(\frac{im}{re} \cdot \frac{im}{re} + 4\right)} \cdot \frac{1}{2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \color{blue}{\frac{im}{re}}, 4\right)} \cdot \frac{1}{2} \]

      10. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{\color{blue}{im}}{re}, 4\right)} \cdot \frac{1}{2} \]

      11. lower-/.f64 89.1

          \[\leadsto \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{im}{\color{blue}{re}}, 4\right)} \cdot 0.5 \]

   10. Applied rewrites 89.1%

       \[\leadsto \sqrt{\color{blue}{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)}} \cdot 0.5 \]

   if -2.2e115 < re < 2.1e-18

   1. Initial program 62.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 82.2%

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

   if 2.1e-18 < re

   1. Initial program 10.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{0.5}{re}} \cdot im\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.2e+115)
   (* (sqrt (* (- re) (fma (/ im re) (/ im re) 4.0))) 0.5)
   (if (<= re 2.1e-18)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* (sqrt (/ 0.5 re)) im) (* (sqrt 2.0) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.2e+115) {
		tmp = sqrt((-re * fma((im / re), (im / re), 4.0))) * 0.5;
	} else if (re <= 2.1e-18) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (sqrt((0.5 / re)) * im) * (sqrt(2.0) * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.2e+115)
		tmp = Float64(sqrt(Float64(Float64(-re) * fma(Float64(im / re), Float64(im / re), 4.0))) * 0.5);
	elseif (re <= 2.1e-18)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(sqrt(Float64(0.5 / re)) * im) * Float64(sqrt(2.0) * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.2e+115], N[(N[Sqrt[N[((-re) * N[(N[(im / re), $MachinePrecision] * N[(im / re), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 2.1e-18], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(0.5 / re), $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.2e115

   1. Initial program 15.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 100.0%

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

   5. Taylor expanded in re around inf

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

      1. pow2 2.5

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

      2. pow2 2.5

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

      3. +-commutative 2.5

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

      4. pow2 2.5

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

      5. pow2 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in re around -inf

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

      1. associate-*r* N/A

         \[\leadsto \sqrt{\left(-1 \cdot re\right) \cdot \color{blue}{\left(4 + \frac{{im}^{2}}{{re}^{2}}\right)}} \cdot \frac{1}{2} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{neg}\left(re\right)\right) \cdot \left(\color{blue}{4} + \frac{{im}^{2}}{{re}^{2}}\right)} \cdot \frac{1}{2} \]

      3. lift-neg.f64 N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \left(\color{blue}{4} + \frac{{im}^{2}}{{re}^{2}}\right)} \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \color{blue}{\left(4 + \frac{{im}^{2}}{{re}^{2}}\right)}} \cdot \frac{1}{2} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \left(\frac{{im}^{2}}{{re}^{2}} + \color{blue}{4}\right)} \cdot \frac{1}{2} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \left(\frac{im \cdot im}{{re}^{2}} + 4\right)} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \left(\frac{im \cdot im}{re \cdot re} + 4\right)} \cdot \frac{1}{2} \]

      8. times-frac N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \left(\frac{im}{re} \cdot \frac{im}{re} + 4\right)} \cdot \frac{1}{2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \color{blue}{\frac{im}{re}}, 4\right)} \cdot \frac{1}{2} \]

      10. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{\color{blue}{im}}{re}, 4\right)} \cdot \frac{1}{2} \]

      11. lower-/.f64 89.1

          \[\leadsto \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{im}{\color{blue}{re}}, 4\right)} \cdot 0.5 \]

   10. Applied rewrites 89.1%

       \[\leadsto \sqrt{\color{blue}{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)}} \cdot 0.5 \]

   if -2.2e115 < re < 2.1e-18

   1. Initial program 62.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 82.2%

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

   if 2.1e-18 < re

   1. Initial program 10.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 38.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-hypot.f64 N/A

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

      15. lower-sqrt.f64 38.1

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

   6. Applied rewrites 38.1%

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

   7. Taylor expanded in re around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. *-commutative N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 77.6

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

   9. Applied rewrites 77.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 77.6%

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

Alternative 4: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{0.5}{re}} \cdot im\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e+165)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 2.1e-18)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* (sqrt (/ 0.5 re)) im) (* (sqrt 2.0) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+165) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 2.1e-18) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (sqrt((0.5 / re)) * im) * (sqrt(2.0) * 0.5);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.9d+165)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 2.1d-18) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (sqrt((0.5d0 / re)) * im) * (sqrt(2.0d0) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+165) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 2.1e-18) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (Math.sqrt((0.5 / re)) * im) * (Math.sqrt(2.0) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e+165:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 2.1e-18:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (math.sqrt((0.5 / re)) * im) * (math.sqrt(2.0) * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e+165)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 2.1e-18)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(sqrt(Float64(0.5 / re)) * im) * Float64(sqrt(2.0) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.9e+165)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 2.1e-18)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (sqrt((0.5 / re)) * im) * (sqrt(2.0) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e+165], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e-18], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(0.5 / re), $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.89999999999999995e165

   1. Initial program 4.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.89999999999999995e165 < re < 2.1e-18

   1. Initial program 61.8%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 80.7%

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

   if 2.1e-18 < re

   1. Initial program 10.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 38.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-hypot.f64 N/A

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

      15. lower-sqrt.f64 38.1

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

   6. Applied rewrites 38.1%

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

   7. Taylor expanded in re around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. *-commutative N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 77.6

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

   9. Applied rewrites 77.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 77.6%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{0.5}{re}} \cdot im\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e+165)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 3.6e-17)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (sqrt (* im (/ im re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+165) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 3.6e-17) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * sqrt((im * (im / re)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.9d+165)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 3.6d-17) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * sqrt((im * (im / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+165) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 3.6e-17) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * Math.sqrt((im * (im / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e+165:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 3.6e-17:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * math.sqrt((im * (im / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e+165)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 3.6e-17)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(im / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.9e+165)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 3.6e-17)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * sqrt((im * (im / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e+165], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.6e-17], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.89999999999999995e165

   1. Initial program 4.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.89999999999999995e165 < re < 3.59999999999999995e-17

   1. Initial program 61.8%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 80.7%

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

   if 3.59999999999999995e-17 < re

   1. Initial program 10.4%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 51.3

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

   5. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 59.1

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

   7. Applied rewrites 59.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{+245}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.4e-5)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 5.4e+245)
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (sqrt (* 2.0 (- re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.4e-5) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 5.4e+245) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.4d-5)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 5.4d+245) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.4e-5) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 5.4e+245) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.4e-5:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 5.4e+245:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.4e-5)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 5.4e+245)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.4e-5)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 5.4e+245)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.4e-5], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.4e+245], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.3999999999999999e-5

   1. Initial program 28.6%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   if -4.3999999999999999e-5 < re < 5.39999999999999983e245

   1. Initial program 48.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 69.3%

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

   if 5.39999999999999983e245 < re

   1. Initial program 2.4%

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

   3. Taylor expanded in re around inf

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

   5. Applied rewrites 38.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{+245}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.4e-5) (* 0.5 (sqrt (* -4.0 re))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.4e-5) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.4d-5)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.4e-5) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.4e-5:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.4e-5)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.4e-5)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.4e-5], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.3999999999999999e-5

   1. Initial program 28.6%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   if -4.3999999999999999e-5 < re

   1. Initial program 43.7%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 63.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
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 20.4

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

5. Applied rewrites 20.4%

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

6. Final simplification 20.4%

   \[\leadsto 0.5 \cdot \sqrt{-4 \cdot re} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025080 
(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)))))