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

Percentage Accurate: 40.6% → 90.6%

Time: 9.1s

Alternatives: 4

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: 40.6% 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

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: 90.6% 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 7.6%

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

   3. Taylor expanded in re around inf 93.2%

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

      1. *-commutative 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-*r* 93.3%

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

      4. *-commutative 93.3%

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

      5. associate-*l* 93.3%

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

      6. *-commutative 93.3%

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

   5. Simplified 93.3%

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

      1. sqrt-unprod 94.4%

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

      2. metadata-eval 94.4%

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

      3. metadata-eval 94.4%

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

   7. Applied egg-rr 94.4%

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

   8. Taylor expanded in re around 0 94.4%

      \[\leadsto 0.5 \cdot \left(1 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow1/2 94.4%

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

      2. rem-exp-log 88.9%

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

      3. exp-neg 88.9%

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

      4. exp-prod 88.9%

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

      5. distribute-lft-neg-out 88.9%

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

      6. exp-neg 88.9%

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

      7. exp-to-pow 94.3%

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

      8. unpow1/2 94.3%

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

      9. associate-*r/ 94.4%

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

      10. *-rgt-identity 94.4%

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

   10. Simplified 94.4%

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

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

   1. Initial program 51.6%

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

      1. sub-neg 51.6%

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

      2. sqr-neg 51.6%

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

      3. sub-neg 51.6%

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

      4. sqr-neg 51.6%

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

      5. hypot-define 90.6%

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

   3. Simplified 90.6%

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

4. Final simplification 91.1%

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

Alternative 2: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 5.1 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.3e+100)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 5.1e-54)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.3e+100) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 5.1e-54) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} 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 <= (-3.3d+100)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 5.1d-54) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    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 <= -3.3e+100) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 5.1e-54) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.3e+100:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 5.1e-54:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.3e+100)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 5.1e-54)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	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 <= -3.3e+100)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 5.1e-54)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.3e+100], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.1e-54], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.3000000000000001e100

   1. Initial program 33.0%

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

   3. Taylor expanded in re around -inf 84.0%

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

      1. *-commutative 84.0%

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

   5. Simplified 84.0%

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

   if -3.3000000000000001e100 < re < 5.1000000000000001e-54

   1. Initial program 64.4%

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

   3. Taylor expanded in re around 0 79.6%

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

   if 5.1000000000000001e-54 < re

   1. Initial program 11.3%

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

   3. Taylor expanded in re around inf 67.5%

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

      1. *-commutative 67.5%

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

      2. *-commutative 67.5%

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

      3. associate-*r* 67.6%

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

      4. *-commutative 67.6%

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

      5. associate-*l* 67.6%

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

      6. *-commutative 67.6%

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

   5. Simplified 67.6%

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

      1. sqrt-unprod 68.4%

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

      2. metadata-eval 68.4%

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

      3. metadata-eval 68.4%

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

   7. Applied egg-rr 68.4%

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

   8. Taylor expanded in re around 0 68.4%

      \[\leadsto 0.5 \cdot \left(1 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow1/2 68.4%

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

      2. rem-exp-log 64.6%

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

      3. exp-neg 64.6%

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

      4. exp-prod 64.6%

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

      5. distribute-lft-neg-out 64.6%

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

      6. exp-neg 64.6%

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

      7. exp-to-pow 68.3%

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

      8. unpow1/2 68.3%

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

      9. associate-*r/ 68.4%

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

      10. *-rgt-identity 68.4%

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

   10. Simplified 68.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 5.1 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.7e+100)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.7e+100) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.7d+100)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.7e+100) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.7e+100:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.7e+100)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.7e+100)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.7e+100], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.69999999999999997e100

   1. Initial program 33.0%

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

   3. Taylor expanded in re around -inf 84.0%

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

      1. *-commutative 84.0%

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

   5. Simplified 84.0%

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

   if -1.69999999999999997e100 < re

   1. Initial program 47.7%

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

   3. Taylor expanded in re around 0 64.8%

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

      1. expm1-log1p-u 61.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{im} \cdot \sqrt{2}\right)\right)} \]

      2. expm1-undefine 51.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im} \cdot \sqrt{2}\right)} - 1\right)} \]

      3. sqrt-unprod 51.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{im \cdot 2}}\right)} - 1\right) \]

   5. Applied egg-rr 51.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im \cdot 2}\right)} - 1\right)} \]
   6. Step-by-step derivation

      1. log1p-undefine 51.4%

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

      2. rem-exp-log 55.1%

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

      3. +-commutative 55.1%

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

      4. associate--l+ 65.2%

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

      5. metadata-eval 65.2%

         \[\leadsto 0.5 \cdot \left(\sqrt{im \cdot 2} + \color{blue}{0}\right) \]

      6. +-rgt-identity 65.2%

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

   7. Simplified 65.2%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.4%

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

3. Taylor expanded in re around 0 57.6%

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

   1. expm1-log1p-u 54.6%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{im} \cdot \sqrt{2}\right)\right)} \]

   2. expm1-undefine 46.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im} \cdot \sqrt{2}\right)} - 1\right)} \]

   3. sqrt-unprod 46.0%

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{im \cdot 2}}\right)} - 1\right) \]

5. Applied egg-rr 46.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im \cdot 2}\right)} - 1\right)} \]
6. Step-by-step derivation

   1. log1p-undefine 46.0%

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

   2. rem-exp-log 49.3%

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

   3. +-commutative 49.3%

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

   4. associate--l+ 58.0%

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

   5. metadata-eval 58.0%

      \[\leadsto 0.5 \cdot \left(\sqrt{im \cdot 2} + \color{blue}{0}\right) \]

   6. +-rgt-identity 58.0%

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

7. Simplified 58.0%

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

8. Final simplification 58.0%

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

Reproduce

herbie shell --seed 2024078 
(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)))))