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

Percentage Accurate: 41.1% → 90.3%

Time: 9.4s

Alternatives: 4

Speedup: 1.9×

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: 41.1% 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: 90.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 13.5%

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

   3. Taylor expanded in re around inf 45.1%

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

      1. associate-*r* 45.1%

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

      2. metadata-eval 45.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{{im}^{2}}{re}} \]

      3. *-un-lft-identity 45.1%

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

      4. div-inv 45.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{im}^{2} \cdot \frac{1}{re}}} \]

      5. sqrt-prod 53.8%

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

      6. sqrt-pow1 97.5%

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

      7. metadata-eval 97.5%

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

      8. pow1 97.5%

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

      9. add-log-exp 13.9%

         \[\leadsto 0.5 \cdot \color{blue}{\log \left(e^{im \cdot \sqrt{\frac{1}{re}}}\right)} \]

      10. *-commutative 13.9%

          \[\leadsto 0.5 \cdot \log \left(e^{\color{blue}{\sqrt{\frac{1}{re}} \cdot im}}\right) \]

      11. *-un-lft-identity 13.9%

          \[\leadsto 0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{1}{re}} \cdot im}\right)} \]

      12. log-prod 13.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\sqrt{\frac{1}{re}} \cdot im}\right)\right)} \]

      13. metadata-eval 13.9%

          \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{\frac{1}{re}} \cdot im}\right)\right) \]

      14. *-commutative 13.9%

          \[\leadsto 0.5 \cdot \left(0 + \log \left(e^{\color{blue}{im \cdot \sqrt{\frac{1}{re}}}}\right)\right) \]

      15. add-log-exp 97.5%

          \[\leadsto 0.5 \cdot \left(0 + \color{blue}{im \cdot \sqrt{\frac{1}{re}}}\right) \]

      16. sqrt-div 97.4%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}\right) \]

      17. metadata-eval 97.4%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \frac{\color{blue}{1}}{\sqrt{re}}\right) \]

      18. un-div-inv 97.7%

          \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\frac{im}{\sqrt{re}}}\right) \]

   5. Applied egg-rr 97.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   6. Step-by-step derivation

      1. +-lft-identity 97.7%

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

   7. Simplified 97.7%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 49.9%

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

      1. sub-neg 49.9%

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

      2. sqr-neg 49.9%

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

      3. sub-neg 49.9%

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

      4. sqr-neg 49.9%

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

      5. hypot-define 91.2%

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

   3. Simplified 91.2%

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

4. Final simplification 92.3%

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

Alternative 2: 75.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.7 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.7e-29)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 4.7e+39)
     (* 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 <= -9.7e-29) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 4.7e+39) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-9.7d-29)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 4.7d+39) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9.7e-29) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 4.7e+39) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.7e-29:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 4.7e+39:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.7e-29)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 4.7e+39)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9.7e-29)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 4.7e+39)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.7e-29], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.7e+39], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -9.7000000000000002e-29

   1. Initial program 44.8%

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

   3. Taylor expanded in re around -inf 70.5%

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

      1. *-commutative 70.5%

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

   5. Simplified 70.5%

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

   if -9.7000000000000002e-29 < re < 4.6999999999999999e39

   1. Initial program 56.6%

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

   3. Taylor expanded in re around 0 77.8%

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

   if 4.6999999999999999e39 < re

   1. Initial program 14.0%

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

   3. Taylor expanded in re around inf 57.3%

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

      1. associate-*r* 57.3%

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

      2. metadata-eval 57.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{{im}^{2}}{re}} \]

      3. *-un-lft-identity 57.3%

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

      4. div-inv 57.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{im}^{2} \cdot \frac{1}{re}}} \]

      5. sqrt-prod 66.0%

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

      6. sqrt-pow1 84.8%

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

      7. metadata-eval 84.8%

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

      8. pow1 84.8%

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

      9. *-commutative 84.8%

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

      10. inv-pow 84.8%

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

      11. sqrt-pow1 84.9%

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

      12. metadata-eval 84.9%

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

   5. Applied egg-rr 84.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.7 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-310)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-310) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * (im / sqrt(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 <= (-2d-310)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2e-310) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-310:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-310)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2e-310)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-310], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.999999999999994e-310

   1. Initial program 58.1%

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

   3. Taylor expanded in re around -inf 51.6%

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

      1. *-commutative 51.6%

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

   5. Simplified 51.6%

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

   if -1.999999999999994e-310 < re

   1. Initial program 32.8%

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

   3. Taylor expanded in re around inf 31.8%

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

      1. associate-*r* 31.8%

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

      2. metadata-eval 31.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{{im}^{2}}{re}} \]

      3. *-un-lft-identity 31.8%

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

      4. div-inv 31.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{im}^{2} \cdot \frac{1}{re}}} \]

      5. sqrt-prod 35.5%

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

      6. sqrt-pow1 51.0%

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

      7. metadata-eval 51.0%

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

      8. pow1 51.0%

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

      9. add-log-exp 12.1%

         \[\leadsto 0.5 \cdot \color{blue}{\log \left(e^{im \cdot \sqrt{\frac{1}{re}}}\right)} \]

      10. *-commutative 12.1%

          \[\leadsto 0.5 \cdot \log \left(e^{\color{blue}{\sqrt{\frac{1}{re}} \cdot im}}\right) \]

      11. *-un-lft-identity 12.1%

          \[\leadsto 0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{1}{re}} \cdot im}\right)} \]

      12. log-prod 12.1%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\sqrt{\frac{1}{re}} \cdot im}\right)\right)} \]

      13. metadata-eval 12.1%

          \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{\frac{1}{re}} \cdot im}\right)\right) \]

      14. *-commutative 12.1%

          \[\leadsto 0.5 \cdot \left(0 + \log \left(e^{\color{blue}{im \cdot \sqrt{\frac{1}{re}}}}\right)\right) \]

      15. add-log-exp 51.0%

          \[\leadsto 0.5 \cdot \left(0 + \color{blue}{im \cdot \sqrt{\frac{1}{re}}}\right) \]

      16. sqrt-div 50.9%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}\right) \]

      17. metadata-eval 50.9%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \frac{\color{blue}{1}}{\sqrt{re}}\right) \]

      18. un-div-inv 51.1%

          \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\frac{im}{\sqrt{re}}}\right) \]

   5. Applied egg-rr 51.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   6. Step-by-step derivation

      1. +-lft-identity 51.1%

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

   7. Simplified 51.1%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 inf 18.4%

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

   1. associate-*r* 18.4%

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

   2. metadata-eval 18.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{{im}^{2}}{re}} \]

   3. *-un-lft-identity 18.4%

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

   4. div-inv 18.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{im}^{2} \cdot \frac{1}{re}}} \]

   5. sqrt-prod 20.1%

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

   6. sqrt-pow1 28.9%

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

   7. metadata-eval 28.9%

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

   8. pow1 28.9%

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

   9. add-log-exp 6.9%

      \[\leadsto 0.5 \cdot \color{blue}{\log \left(e^{im \cdot \sqrt{\frac{1}{re}}}\right)} \]

   10. *-commutative 6.9%

       \[\leadsto 0.5 \cdot \log \left(e^{\color{blue}{\sqrt{\frac{1}{re}} \cdot im}}\right) \]

   11. *-un-lft-identity 6.9%

       \[\leadsto 0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{1}{re}} \cdot im}\right)} \]

   12. log-prod 6.9%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\sqrt{\frac{1}{re}} \cdot im}\right)\right)} \]

   13. metadata-eval 6.9%

       \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{\frac{1}{re}} \cdot im}\right)\right) \]

   14. *-commutative 6.9%

       \[\leadsto 0.5 \cdot \left(0 + \log \left(e^{\color{blue}{im \cdot \sqrt{\frac{1}{re}}}}\right)\right) \]

   15. add-log-exp 28.9%

       \[\leadsto 0.5 \cdot \left(0 + \color{blue}{im \cdot \sqrt{\frac{1}{re}}}\right) \]

   16. sqrt-div 28.9%

       \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}\right) \]

   17. metadata-eval 28.9%

       \[\leadsto 0.5 \cdot \left(0 + im \cdot \frac{\color{blue}{1}}{\sqrt{re}}\right) \]

   18. un-div-inv 28.9%

       \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\frac{im}{\sqrt{re}}}\right) \]

5. Applied egg-rr 28.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
6. Step-by-step derivation

   1. +-lft-identity 28.9%

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

7. Simplified 28.9%

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

8. Final simplification 28.9%

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

Reproduce

herbie shell --seed 2024067 
(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)))))