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

Percentage Accurate: 51.8% → 98.5%

Time: 9.0s

Alternatives: 6

Speedup: 2.0×

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

real(8) function code(re, im)
    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: 51.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.5%

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

   1. sub-neg 48.5%

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

   2. sqr-neg 48.5%

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

   3. sub-neg 48.5%

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

   4. sqr-neg 48.5%

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

   5. hypot-define 98.7%

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

3. Simplified 98.7%

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

5. Final simplification 98.7%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \]
6. Add Preprocessing

Alternative 2: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.8 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot {\left({\left(2 \cdot im\right)}^{1.5}\right)}^{0.3333333333333333}\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.8e+53)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1.7e-91)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 4.5e+17)
       (* 0.5 (pow (pow (* 2.0 im) 1.5) 0.3333333333333333))
       (if (<= re 1.85e+60)
         (* 0.5 (sqrt (* 2.0 im)))
         (* 0.5 (sqrt (* 2.0 (- re re)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.8e+53) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.7e-91) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 4.5e+17) {
		tmp = 0.5 * pow(pow((2.0 * im), 1.5), 0.3333333333333333);
	} else if (re <= 1.85e+60) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.8d+53)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1.7d-91) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 4.5d+17) then
        tmp = 0.5d0 * (((2.0d0 * im) ** 1.5d0) ** 0.3333333333333333d0)
    else if (re <= 1.85d+60) 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 <= -3.8e+53) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.7e-91) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 4.5e+17) {
		tmp = 0.5 * Math.pow(Math.pow((2.0 * im), 1.5), 0.3333333333333333);
	} else if (re <= 1.85e+60) {
		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 <= -3.8e+53:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1.7e-91:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 4.5e+17:
		tmp = 0.5 * math.pow(math.pow((2.0 * im), 1.5), 0.3333333333333333)
	elif re <= 1.85e+60:
		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 <= -3.8e+53)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1.7e-91)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 4.5e+17)
		tmp = Float64(0.5 * ((Float64(2.0 * im) ^ 1.5) ^ 0.3333333333333333));
	elseif (re <= 1.85e+60)
		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 <= -3.8e+53)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1.7e-91)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 4.5e+17)
		tmp = 0.5 * (((2.0 * im) ^ 1.5) ^ 0.3333333333333333);
	elseif (re <= 1.85e+60)
		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, -3.8e+53], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.7e-91], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.5e+17], N[(0.5 * N[Power[N[Power[N[(2.0 * im), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.85e+60], 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 5 regimes

2. if re < -3.79999999999999997e53

   1. Initial program 28.9%

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

   3. Taylor expanded in re around -inf 89.2%

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

      1. *-commutative 89.2%

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

   5. Simplified 89.2%

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

   if -3.79999999999999997e53 < re < 1.70000000000000013e-91

   1. Initial program 65.8%

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

   3. Taylor expanded in re around 0 79.2%

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

   if 1.70000000000000013e-91 < re < 4.5e17

   1. Initial program 82.0%

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

      1. add-cbrt-cube 81.9%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

      2. pow1/3 80.1%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 80.1%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}^{0.3333333333333333} \]

      4. pow1 80.1%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 80.1%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 80.1%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. hypot-define 90.9%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 90.9%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 90.9%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   5. Taylor expanded in re around 0 71.4%

      \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \color{blue}{im}\right)}^{1.5}\right)}^{0.3333333333333333} \]

   if 4.5e17 < re < 1.84999999999999994e60

   1. Initial program 52.2%

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

   3. Taylor expanded in re around 0 75.5%

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

      1. pow1 75.5%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{im} \cdot \sqrt{2}\right)}^{1}} \]

      2. *-commutative 75.5%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt{2} \cdot \sqrt{im}\right)}}^{1} \]

      3. sqrt-unprod 75.9%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt{2 \cdot im}\right)}}^{1} \]

   5. Applied egg-rr 75.9%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{2 \cdot im}\right)}^{1}} \]
   6. Step-by-step derivation

      1. unpow1 75.9%

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

   7. Simplified 75.9%

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

   if 1.84999999999999994e60 < re

   1. Initial program 20.2%

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

   3. Taylor expanded in re around inf 80.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} - re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.8 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot {\left({\left(2 \cdot im\right)}^{1.5}\right)}^{0.3333333333333333}\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 6.3 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+18} \lor \neg \left(re \leq 1.45 \cdot 10^{+62}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.5e+51)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 6.3e-21)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (or (<= re 1.5e+18) (not (<= re 1.45e+62)))
       (* 0.5 (sqrt (* 2.0 (- re re))))
       (* 0.5 (sqrt (* 2.0 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.5e+51) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 6.3e-21) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if ((re <= 1.5e+18) || !(re <= 1.45e+62)) {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-7.5d+51)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 6.3d-21) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if ((re <= 1.5d+18) .or. (.not. (re <= 1.45d+62))) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.5e+51) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 6.3e-21) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if ((re <= 1.5e+18) || !(re <= 1.45e+62)) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.5e+51:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 6.3e-21:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif (re <= 1.5e+18) or not (re <= 1.45e+62):
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.5e+51)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 6.3e-21)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif ((re <= 1.5e+18) || !(re <= 1.45e+62))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.5e+51)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 6.3e-21)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif ((re <= 1.5e+18) || ~((re <= 1.45e+62)))
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.5e+51], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.3e-21], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.5e+18], N[Not[LessEqual[re, 1.45e+62]], $MachinePrecision]], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.4999999999999999e51

   1. Initial program 28.9%

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

   3. Taylor expanded in re around -inf 89.2%

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

      1. *-commutative 89.2%

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

   5. Simplified 89.2%

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

   if -7.4999999999999999e51 < re < 6.3e-21

   1. Initial program 66.2%

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

   3. Taylor expanded in re around 0 77.7%

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

   if 6.3e-21 < re < 1.5e18 or 1.44999999999999992e62 < re

   1. Initial program 28.9%

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

   3. Taylor expanded in re around inf 80.5%

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

   if 1.5e18 < re < 1.44999999999999992e62

   1. Initial program 52.2%

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

   3. Taylor expanded in re around 0 75.5%

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

      1. pow1 75.5%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{im} \cdot \sqrt{2}\right)}^{1}} \]

      2. *-commutative 75.5%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt{2} \cdot \sqrt{im}\right)}}^{1} \]

      3. sqrt-unprod 75.9%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt{2 \cdot im}\right)}}^{1} \]

   5. Applied egg-rr 75.9%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{2 \cdot im}\right)}^{1}} \]
   6. Step-by-step derivation

      1. unpow1 75.9%

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

   7. Simplified 75.9%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 6.3 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+18} \lor \neg \left(re \leq 1.45 \cdot 10^{+62}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.2e+51)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.2e+51) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.2d+51)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    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 <= -3.2e+51) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.2e+51:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.2e+51)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	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 <= -3.2e+51)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.2e+51], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.2000000000000002e51

   1. Initial program 28.9%

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

   3. Taylor expanded in re around -inf 89.2%

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

      1. *-commutative 89.2%

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

   5. Simplified 89.2%

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

   if -3.2000000000000002e51 < re

   1. Initial program 54.8%

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

   3. Taylor expanded in re around 0 58.9%

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

      1. pow1 58.9%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{im} \cdot \sqrt{2}\right)}^{1}} \]

      2. *-commutative 58.9%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt{2} \cdot \sqrt{im}\right)}}^{1} \]

      3. sqrt-unprod 59.3%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt{2 \cdot im}\right)}}^{1} \]

   5. Applied egg-rr 59.3%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{2 \cdot im}\right)}^{1}} \]
   6. Step-by-step derivation

      1. unpow1 59.3%

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

   7. Simplified 59.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.25 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.25e+52)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (sqrt (* 2.0 (- im re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.25e+52) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.25d+52)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    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 <= -2.25e+52) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.25e+52:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.25e+52)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	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 <= -2.25e+52)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.25e+52], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $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 < -2.25e52

   1. Initial program 28.9%

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

   3. Taylor expanded in re around -inf 89.2%

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

      1. *-commutative 89.2%

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

   5. Simplified 89.2%

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

   if -2.25e52 < re

   1. Initial program 54.8%

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

   3. Taylor expanded in re around 0 60.1%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.25 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.5%

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

3. Taylor expanded in re around 0 48.4%

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

   1. pow1 48.4%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{im} \cdot \sqrt{2}\right)}^{1}} \]

   2. *-commutative 48.4%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt{2} \cdot \sqrt{im}\right)}}^{1} \]

   3. sqrt-unprod 48.8%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt{2 \cdot im}\right)}}^{1} \]

5. Applied egg-rr 48.8%

   \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{2 \cdot im}\right)}^{1}} \]
6. Step-by-step derivation

   1. unpow1 48.8%

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

7. Simplified 48.8%

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

8. Final simplification 48.8%

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

Reproduce

herbie shell --seed 2024052 
(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)))))