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

Percentage Accurate: 41.6% → 90.6%

Time: 8.4s

Alternatives: 5

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: 41.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 11.2%

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

   3. Taylor expanded in re around inf 50.4%

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

   4. Taylor expanded in im around 0 95.7%

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

      1. unpow-1 95.7%

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

      2. metadata-eval 95.7%

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

      3. pow-sqr 95.7%

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

      4. rem-sqrt-square 95.7%

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

      5. rem-square-sqrt 95.2%

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

      6. fabs-sqr 95.2%

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

      7. rem-square-sqrt 95.7%

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

      8. exp-to-pow 90.7%

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

      9. metadata-eval 90.7%

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

      10. distribute-rgt-neg-in 90.7%

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

      11. exp-neg 90.7%

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

      12. exp-to-pow 95.7%

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

      13. unpow1/2 95.7%

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

      14. associate-/l* 96.0%

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

      15. *-rgt-identity 96.0%

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

   6. Simplified 96.0%

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

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

   1. Initial program 41.6%

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

      1. sub-neg 41.6%

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

      2. sqr-neg 41.6%

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

      3. sub-neg 41.6%

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

      4. sqr-neg 41.6%

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

      5. hypot-define 91.0%

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

   3. Simplified 91.0%

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

4. Final simplification 91.9%

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

Alternative 2: 74.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq -1.5 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -9.2 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -1.1 \cdot 10^{-74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{-65}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re -1.5e+45)
     t_0
     (if (<= re -9.2e-33)
       t_1
       (if (<= re -1.1e-74)
         t_0
         (if (<= re 6.8e-65) t_1 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((im * 2.0));
	double tmp;
	if (re <= -1.5e+45) {
		tmp = t_0;
	} else if (re <= -9.2e-33) {
		tmp = t_1;
	} else if (re <= -1.1e-74) {
		tmp = t_0;
	} else if (re <= 6.8e-65) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((im * 2.0d0))
    if (re <= (-1.5d+45)) then
        tmp = t_0
    else if (re <= (-9.2d-33)) then
        tmp = t_1
    else if (re <= (-1.1d-74)) then
        tmp = t_0
    else if (re <= 6.8d-65) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((im * 2.0));
	double tmp;
	if (re <= -1.5e+45) {
		tmp = t_0;
	} else if (re <= -9.2e-33) {
		tmp = t_1;
	} else if (re <= -1.1e-74) {
		tmp = t_0;
	} else if (re <= 6.8e-65) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((im * 2.0))
	tmp = 0
	if re <= -1.5e+45:
		tmp = t_0
	elif re <= -9.2e-33:
		tmp = t_1
	elif re <= -1.1e-74:
		tmp = t_0
	elif re <= 6.8e-65:
		tmp = t_1
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	tmp = 0.0
	if (re <= -1.5e+45)
		tmp = t_0;
	elseif (re <= -9.2e-33)
		tmp = t_1;
	elseif (re <= -1.1e-74)
		tmp = t_0;
	elseif (re <= 6.8e-65)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * sqrt((im * 2.0));
	tmp = 0.0;
	if (re <= -1.5e+45)
		tmp = t_0;
	elseif (re <= -9.2e-33)
		tmp = t_1;
	elseif (re <= -1.1e-74)
		tmp = t_0;
	elseif (re <= 6.8e-65)
		tmp = t_1;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.5e+45], t$95$0, If[LessEqual[re, -9.2e-33], t$95$1, If[LessEqual[re, -1.1e-74], t$95$0, If[LessEqual[re, 6.8e-65], t$95$1, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.50000000000000005e45 or -9.19999999999999942e-33 < re < -1.10000000000000005e-74

   1. Initial program 40.2%

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

   3. Taylor expanded in re around -inf 80.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 80.4%

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

   5. Simplified 80.4%

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

   if -1.50000000000000005e45 < re < -9.19999999999999942e-33 or -1.10000000000000005e-74 < re < 6.79999999999999973e-65

   1. Initial program 50.9%

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

   3. Taylor expanded in re around 0 82.1%

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

      1. *-commutative 82.1%

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

   5. Simplified 82.1%

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

   if 6.79999999999999973e-65 < re

   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 44.1%

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

   4. Taylor expanded in im around 0 71.2%

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

      1. unpow-1 71.2%

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

      2. metadata-eval 71.2%

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

      3. pow-sqr 71.3%

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

      4. rem-sqrt-square 71.3%

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

      5. rem-square-sqrt 71.0%

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

      6. fabs-sqr 71.0%

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

      7. rem-square-sqrt 71.3%

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

      8. exp-to-pow 67.4%

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

      9. metadata-eval 67.4%

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

      10. distribute-rgt-neg-in 67.4%

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

      11. exp-neg 67.4%

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

      12. exp-to-pow 71.3%

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

      13. unpow1/2 71.3%

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

      14. associate-/l* 71.4%

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

      15. *-rgt-identity 71.4%

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

   6. Simplified 71.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -9.2 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq -1.1 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{-65}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{if}\;re \leq -2.8 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -3 \cdot 10^{-75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-65}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))))
   (if (<= re -2.8e+52)
     t_0
     (if (<= re -1.9e-32)
       (* 0.5 (sqrt (* 2.0 (- im re))))
       (if (<= re -3e-75)
         t_0
         (if (<= re 3.4e-65)
           (* 0.5 (sqrt (* im 2.0)))
           (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double tmp;
	if (re <= -2.8e+52) {
		tmp = t_0;
	} else if (re <= -1.9e-32) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= -3e-75) {
		tmp = t_0;
	} else if (re <= 3.4e-65) {
		tmp = 0.5 * sqrt((im * 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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    if (re <= (-2.8d+52)) then
        tmp = t_0
    else if (re <= (-1.9d-32)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= (-3d-75)) then
        tmp = t_0
    else if (re <= 3.4d-65) then
        tmp = 0.5d0 * sqrt((im * 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 t_0 = 0.5 * Math.sqrt((re * -4.0));
	double tmp;
	if (re <= -2.8e+52) {
		tmp = t_0;
	} else if (re <= -1.9e-32) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= -3e-75) {
		tmp = t_0;
	} else if (re <= 3.4e-65) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	tmp = 0
	if re <= -2.8e+52:
		tmp = t_0
	elif re <= -1.9e-32:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= -3e-75:
		tmp = t_0
	elif re <= 3.4e-65:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	tmp = 0.0
	if (re <= -2.8e+52)
		tmp = t_0;
	elseif (re <= -1.9e-32)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= -3e-75)
		tmp = t_0;
	elseif (re <= 3.4e-65)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	tmp = 0.0;
	if (re <= -2.8e+52)
		tmp = t_0;
	elseif (re <= -1.9e-32)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= -3e-75)
		tmp = t_0;
	elseif (re <= 3.4e-65)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.8e+52], t$95$0, If[LessEqual[re, -1.9e-32], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -3e-75], t$95$0, If[LessEqual[re, 3.4e-65], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.8e52 or -1.90000000000000004e-32 < re < -2.9999999999999999e-75

   1. Initial program 40.2%

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

   3. Taylor expanded in re around -inf 80.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 80.4%

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

   5. Simplified 80.4%

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

   if -2.8e52 < re < -1.90000000000000004e-32

   1. Initial program 63.2%

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

   3. Taylor expanded in re around 0 75.5%

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

   if -2.9999999999999999e-75 < re < 3.39999999999999987e-65

   1. Initial program 49.2%

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

   3. Taylor expanded in re around 0 83.5%

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

      1. *-commutative 83.5%

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

   5. Simplified 83.5%

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

   if 3.39999999999999987e-65 < re

   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 44.1%

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

   4. Taylor expanded in im around 0 71.2%

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

      1. unpow-1 71.2%

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

      2. metadata-eval 71.2%

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

      3. pow-sqr 71.3%

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

      4. rem-sqrt-square 71.3%

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

      5. rem-square-sqrt 71.0%

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

      6. fabs-sqr 71.0%

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

      7. rem-square-sqrt 71.3%

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

      8. exp-to-pow 67.4%

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

      9. metadata-eval 67.4%

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

      10. distribute-rgt-neg-in 67.4%

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

      11. exp-neg 67.4%

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

      12. exp-to-pow 71.3%

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

      13. unpow1/2 71.3%

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

      14. associate-/l* 71.4%

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

      15. *-rgt-identity 71.4%

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

   6. Simplified 71.4%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -3 \cdot 10^{-75}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-65}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+47} \lor \neg \left(re \leq -2.5 \cdot 10^{-32}\right) \land re \leq -3.5 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -1.5e+47) (and (not (<= re -2.5e-32)) (<= re -3.5e-70)))
   (* 0.5 (sqrt (* re -4.0)))
   (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -1.5e+47) || (!(re <= -2.5e-32) && (re <= -3.5e-70))) {
		tmp = 0.5 * sqrt((re * -4.0));
	} 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 <= (-1.5d+47)) .or. (.not. (re <= (-2.5d-32))) .and. (re <= (-3.5d-70))) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    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 <= -1.5e+47) || (!(re <= -2.5e-32) && (re <= -3.5e-70))) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -1.5e+47) or (not (re <= -2.5e-32) and (re <= -3.5e-70)):
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -1.5e+47) || (!(re <= -2.5e-32) && (re <= -3.5e-70)))
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	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 <= -1.5e+47) || (~((re <= -2.5e-32)) && (re <= -3.5e-70)))
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -1.5e+47], And[N[Not[LessEqual[re, -2.5e-32]], $MachinePrecision], LessEqual[re, -3.5e-70]]], N[(0.5 * N[Sqrt[N[(re * -4.0), $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 < -1.5000000000000001e47 or -2.5e-32 < re < -3.49999999999999974e-70

   1. Initial program 40.2%

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

   3. Taylor expanded in re around -inf 80.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 80.4%

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

   5. Simplified 80.4%

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

   if -1.5000000000000001e47 < re < -2.5e-32 or -3.49999999999999974e-70 < re

   1. Initial program 34.7%

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

   3. Taylor expanded in re around 0 61.1%

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

      1. *-commutative 61.1%

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

   5. Simplified 61.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+47} \lor \neg \left(re \leq -2.5 \cdot 10^{-32}\right) \land re \leq -3.5 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.2% 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 36.1%

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

3. Taylor expanded in re around 0 52.1%

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

   1. *-commutative 52.1%

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

5. Simplified 52.1%

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

6. Final simplification 52.1%

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

Reproduce

herbie shell --seed 2024044 
(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)))))