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

Percentage Accurate: 40.9% → 89.7%

Time: 9.0s

Alternatives: 9

Speedup: 2.2×

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: 40.9% 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: 89.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (/ (* im 0.5) (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 = (im * 0.5) / 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 = (im * 0.5) / 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 = (im * 0.5) / 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(Float64(im * 0.5) / 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 = (im * 0.5) / 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[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $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 10.0%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 47.8

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

   5. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 97.6

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

   7. Applied rewrites 97.6%

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

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

   1. Initial program 45.7%

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

      1. lower-hypot.f64 89.2

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

   4. Applied rewrites 89.2%

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

Alternative 2: 75.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im} - re\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, im \cdot 2\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+140}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ (* re re) (* im im))) re)))
   (if (<= t_0 0.0)
     (/ (* im 0.5) (sqrt re))
     (if (<= t_0 2e-158)
       (* 0.5 (sqrt (fma re (+ -2.0 (/ re im)) (* im 2.0))))
       (if (<= t_0 1e+140)
         (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
         (* 0.5 (sqrt (* 2.0 (- im re)))))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im))) - re;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (im * 0.5) / sqrt(re);
	} else if (t_0 <= 2e-158) {
		tmp = 0.5 * sqrt(fma(re, (-2.0 + (re / im)), (im * 2.0)));
	} else if (t_0 <= 1e+140) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	elseif (t_0 <= 2e-158)
		tmp = Float64(0.5 * sqrt(fma(re, Float64(-2.0 + Float64(re / im)), Float64(im * 2.0))));
	elseif (t_0 <= 1e+140)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-158], N[(0.5 * N[Sqrt[N[(re * N[(-2.0 + N[(re / im), $MachinePrecision]), $MachinePrecision] + N[(im * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+140], 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], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 10.0%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 47.8

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

   5. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 97.6

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

   7. Applied rewrites 97.6%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 2.00000000000000013e-158

   1. Initial program 18.9%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im + re \cdot \left(\frac{re}{im} - 2\right)}} \]
   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   if 2.00000000000000013e-158 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 1.00000000000000006e140

   1. Initial program 99.6%

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

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 1.00000000000000006e140 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 7.1%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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 59.7

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

   5. Applied rewrites 59.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \leq 2 \cdot 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, im \cdot 2\right)}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \leq 10^{+140}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-159}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \mathsf{fma}\left(-2, \frac{re}{im}, 2\right)}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{2}{re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.2e+14)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 9.5e-159)
     (* 0.5 (sqrt (* im (fma -2.0 (/ re im) 2.0))))
     (if (<= re 2.8e+114)
       (* 0.5 (* im (sqrt (/ 2.0 (+ re (sqrt (fma im im (* re re))))))))
       (/ (* im 0.5) (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.2e+14) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 9.5e-159) {
		tmp = 0.5 * sqrt((im * fma(-2.0, (re / im), 2.0)));
	} else if (re <= 2.8e+114) {
		tmp = 0.5 * (im * sqrt((2.0 / (re + sqrt(fma(im, im, (re * re)))))));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.2e+14)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 9.5e-159)
		tmp = Float64(0.5 * sqrt(Float64(im * fma(-2.0, Float64(re / im), 2.0))));
	elseif (re <= 2.8e+114)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(2.0 / Float64(re + sqrt(fma(im, im, Float64(re * re))))))));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.2e+14], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.5e-159], N[(0.5 * N[Sqrt[N[(im * N[(-2.0 * N[(re / im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.8e+114], N[(0.5 * N[(im * N[Sqrt[N[(2.0 / N[(re + N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.2e14

   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

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -7.2e14 < re < 9.4999999999999997e-159

   1. Initial program 59.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if 9.4999999999999997e-159 < re < 2.8e114

   1. Initial program 29.8%

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

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 26.4%

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

   5. Taylor expanded in re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 53.9

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

   7. Applied rewrites 53.9%

      \[\leadsto 0.5 \cdot \sqrt{\frac{2 \cdot \color{blue}{\left(im \cdot im\right)}}{re + \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 53.9

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

   9. Applied rewrites 84.7%

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

   if 2.8e114 < re

   1. Initial program 6.5%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 42.2

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

   5. Applied rewrites 42.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 78.0

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

   7. Applied rewrites 78.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-159}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \mathsf{fma}\left(-2, \frac{re}{im}, 2\right)}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{2}{re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \mathsf{fma}\left(-2, \frac{re}{im}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.2e+14)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 9.5e-86)
     (* 0.5 (sqrt (* im (fma -2.0 (/ re im) 2.0))))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.2e+14) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 9.5e-86) {
		tmp = 0.5 * sqrt((im * fma(-2.0, (re / im), 2.0)));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.2e+14)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 9.5e-86)
		tmp = Float64(0.5 * sqrt(Float64(im * fma(-2.0, Float64(re / im), 2.0))));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.2e+14], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.5e-86], N[(0.5 * N[Sqrt[N[(im * N[(-2.0 * N[(re / im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.2e14

   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

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -7.2e14 < re < 9.4999999999999996e-86

   1. Initial program 58.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if 9.4999999999999996e-86 < re

   1. Initial program 13.3%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 42.2

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

   5. Applied rewrites 42.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 76.4

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

   7. Applied rewrites 76.4%

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

Alternative 5: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.2e+14)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 9.5e-86)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.2e+14) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 9.5e-86) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) / 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 <= (-7.2d+14)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 9.5d-86) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.2e+14) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 9.5e-86) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.2e+14:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 9.5e-86:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.2e+14)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 9.5e-86)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.2e+14)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 9.5e-86)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.2e+14], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.5e-86], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.2e14

   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

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -7.2e14 < re < 9.4999999999999996e-86

   1. Initial program 58.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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 80.3

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

   5. Applied rewrites 80.3%

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

   if 9.4999999999999996e-86 < re

   1. Initial program 13.3%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 42.2

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

   5. Applied rewrites 42.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 76.4

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

   7. Applied rewrites 76.4%

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

Alternative 6: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.2e+14)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 9.5e-86)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.2e+14) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 9.5e-86) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / 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 <= (-7.2d+14)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 9.5d-86) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.2e+14) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 9.5e-86) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.2e+14:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 9.5e-86:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.2e+14)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 9.5e-86)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.2e+14)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 9.5e-86)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.2e+14], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.5e-86], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.2e14

   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

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -7.2e14 < re < 9.4999999999999996e-86

   1. Initial program 58.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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 80.3

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

   5. Applied rewrites 80.3%

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

   if 9.4999999999999996e-86 < re

   1. Initial program 13.3%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 42.2

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

   5. Applied rewrites 42.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 76.4

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

   7. Applied rewrites 76.4%

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

      1. *-commutative N/A

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

      2. lift-sqrt.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 76.3

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

   9. Applied rewrites 76.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.6 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.6e+14) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.6e+14) {
		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 <= (-5.6d+14)) 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 <= -5.6e+14) {
		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 <= -5.6e+14:
		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 <= -5.6e+14)
		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 <= -5.6e+14)
		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[LessEqual[re, -5.6e+14], 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 < -5.6e14

   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

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -5.6e14 < re

   1. Initial program 39.4%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 57.8

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

   5. Applied rewrites 57.8%

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

Alternative 8: 51.9% accurate, 2.2× 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 39.6%

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

3. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 51.9

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

5. Applied rewrites 51.9%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
6. Add Preprocessing

Alternative 9: 6.1% accurate, 47.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 39.6%

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

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. flip-- N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 24.5%

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

5. Taylor expanded in re around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{0} \cdot im} \]

   3. mul0-lft 6.0

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

7. Applied rewrites 6.0%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]
8. Step-by-step derivation

   1. pow1/2 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{{0}^{\frac{1}{2}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{0} \]

   3. metadata-eval 6.0

      \[\leadsto \color{blue}{0} \]

9. Applied rewrites 6.0%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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)))))