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

Percentage Accurate: 41.9% → 78.3%

Time: 6.6s

Alternatives: 6

Speedup: 1.7×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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: 41.9% 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: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+148}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)}\\ \mathbf{elif}\;re \leq -1.16 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, 2 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1e+148)
   (* 0.5 (sqrt (* (- re) (fma (/ im re) (/ im re) 4.0))))
   (if (<= re -1.16e-54)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 1.2e+69)
       (* 0.5 (sqrt (fma (- (/ re im) 2.0) re (* 2.0 im))))
       (* (/ im (sqrt re)) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e+148) {
		tmp = 0.5 * sqrt((-re * fma((im / re), (im / re), 4.0)));
	} else if (re <= -1.16e-54) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 1.2e+69) {
		tmp = 0.5 * sqrt(fma(((re / im) - 2.0), re, (2.0 * im)));
	} else {
		tmp = (im / sqrt(re)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e+148)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-re) * fma(Float64(im / re), Float64(im / re), 4.0))));
	elseif (re <= -1.16e-54)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 1.2e+69)
		tmp = Float64(0.5 * sqrt(fma(Float64(Float64(re / im) - 2.0), re, Float64(2.0 * im))));
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e+148], N[(0.5 * N[Sqrt[N[((-re) * N[(N[(im / re), $MachinePrecision] * N[(im / re), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.16e-54], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.2e+69], N[(0.5 * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + N[(2.0 * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1e148

   1. Initial program 7.2%

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

   3. Taylor expanded in re around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -1e148 < re < -1.16e-54

   1. Initial program 89.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 89.6

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

   4. Applied rewrites 89.6%

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

   if -1.16e-54 < re < 1.2000000000000001e69

   1. Initial program 50.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if 1.2000000000000001e69 < re

   1. Initial program 7.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 83.0

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

   5. Applied rewrites 83.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 83.0

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

   7. Applied rewrites 83.7%

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

Alternative 2: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{+147}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -1.16 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, 2 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.5e+147)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -1.16e-54)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 1.2e+69)
       (* 0.5 (sqrt (fma (- (/ re im) 2.0) re (* 2.0 im))))
       (* (/ im (sqrt re)) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.5e+147) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -1.16e-54) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 1.2e+69) {
		tmp = 0.5 * sqrt(fma(((re / im) - 2.0), re, (2.0 * im)));
	} else {
		tmp = (im / sqrt(re)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.5e+147)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -1.16e-54)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 1.2e+69)
		tmp = Float64(0.5 * sqrt(fma(Float64(Float64(re / im) - 2.0), re, Float64(2.0 * im))));
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.5e+147], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.16e-54], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.2e+69], N[(0.5 * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + N[(2.0 * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.49999999999999975e147

   1. Initial program 7.2%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if -3.49999999999999975e147 < re < -1.16e-54

   1. Initial program 89.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 89.6

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

   4. Applied rewrites 89.6%

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

   if -1.16e-54 < re < 1.2000000000000001e69

   1. Initial program 50.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if 1.2000000000000001e69 < re

   1. Initial program 7.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 83.0

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

   5. Applied rewrites 83.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 83.0

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

   7. Applied rewrites 83.7%

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

Alternative 3: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im}, -2, 2\right) \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.7e-6)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 1.2e+69)
     (* 0.5 (sqrt (* (fma (/ re im) -2.0 2.0) im)))
     (* (/ im (sqrt re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.7e-6) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 1.2e+69) {
		tmp = 0.5 * sqrt((fma((re / im), -2.0, 2.0) * im));
	} else {
		tmp = (im / sqrt(re)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.7e-6)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 1.2e+69)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(re / im), -2.0, 2.0) * im)));
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.7e-6], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.2e+69], N[(0.5 * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] * -2.0 + 2.0), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.70000000000000003e-6

   1. Initial program 50.7%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   if -1.70000000000000003e-6 < re < 1.2000000000000001e69

   1. Initial program 52.8%

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

   3. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if 1.2000000000000001e69 < re

   1. Initial program 7.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 83.0

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

   5. Applied rewrites 83.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 83.0

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

   7. Applied rewrites 83.7%

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

Alternative 4: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.7e-6)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 1.2e+69)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (/ im (sqrt re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.7e-6) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 1.2e+69) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im / sqrt(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 <= (-1.7d-6)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 1.2d+69) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (im / sqrt(re)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.7e-6) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 1.2e+69) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im / Math.sqrt(re)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.7e-6:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 1.2e+69:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im / math.sqrt(re)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.7e-6)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 1.2e+69)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.7e-6)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 1.2e+69)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im / sqrt(re)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.7e-6], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.2e+69], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.70000000000000003e-6

   1. Initial program 50.7%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   if -1.70000000000000003e-6 < re < 1.2000000000000001e69

   1. Initial program 52.8%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if 1.2000000000000001e69 < re

   1. Initial program 7.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 83.0

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

   5. Applied rewrites 83.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 83.0

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

   7. Applied rewrites 83.7%

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

Alternative 5: 64.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.05e-53) (* 0.5 (sqrt (* -4.0 re))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.05e-53) {
		tmp = 0.5 * sqrt((-4.0 * 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 <= (-1.05d-53)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.05e-53) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.05e-53:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.05e-53)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.05e-53)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.05e-53], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.04999999999999989e-53

   1. Initial program 55.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if -1.04999999999999989e-53 < re

   1. Initial program 39.2%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

Alternative 6: 26.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.3%

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

3. Taylor expanded in re around -inf

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

   1. lower-*.f64 27.6

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

5. Applied rewrites 27.6%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(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)))))