math.sqrt on complex, real part

Percentage Accurate: 41.3% → 88.7%

Time: 9.5s

Alternatives: 13

Speedup: 1.7×

Specification

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: 41.3% 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: 88.7% accurate, 0.1× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im_m im_m)))))) 0.0)
   (* 0.5 (exp (* (+ (log (/ im_m (- re))) (log im_m)) 0.5)))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Exp[N[(N[(N[Log[N[(im$95$m / (-re)), $MachinePrecision]], $MachinePrecision] + N[Log[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 8.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 8.3

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 8.3

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 8.3

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

   4. Applied rewrites 8.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-hypot.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. pow-to-exp N/A

          \[\leadsto \color{blue}{e^{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)\right) \cdot \frac{1}{2}}} \cdot \frac{1}{2} \]

      12. lower-exp.f64 N/A

          \[\leadsto \color{blue}{e^{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)\right) \cdot \frac{1}{2}}} \cdot \frac{1}{2} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\color{blue}{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)\right) \cdot \frac{1}{2}}} \cdot \frac{1}{2} \]

   6. Applied rewrites 8.3%

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

   7. Taylor expanded in re around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 43.6

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

   9. Applied rewrites 43.6%

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{-1}{re}\right) + \log \left(im \cdot im\right)\right)} \cdot 0.5} \cdot 0.5 \]
   10. Step-by-step derivation

   11. Applied rewrites 90.7%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 43.9%

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

      1. lift-sqrt.f64 N/A

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

      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. 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-*.f64 N/A

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

      5. lower-hypot.f64 88.7

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

   4. Applied rewrites 88.7%

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

4. Final simplification 88.9%

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

Alternative 2: 89.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im\_m \cdot im\_m}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot e^{0.5 \cdot \mathsf{fma}\left(\log im\_m, 2, -\log \left(-re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im_m im_m)))))) 0.0)
   (* 0.5 (exp (* 0.5 (fma (log im_m) 2.0 (- (log (- re)))))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Exp[N[(0.5 * N[(N[Log[im$95$m], $MachinePrecision] * 2.0 + (-N[Log[(-re)], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 9.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 9.6

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 9.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 9.6

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

   4. Applied rewrites 9.6%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-hypot.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. pow-to-exp N/A

          \[\leadsto \color{blue}{e^{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)\right) \cdot \frac{1}{2}}} \cdot \frac{1}{2} \]

      12. lower-exp.f64 N/A

          \[\leadsto \color{blue}{e^{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)\right) \cdot \frac{1}{2}}} \cdot \frac{1}{2} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\color{blue}{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)\right) \cdot \frac{1}{2}}} \cdot \frac{1}{2} \]

   6. Applied rewrites 9.6%

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

   7. Taylor expanded in re around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 52.3

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

   9. Applied rewrites 52.3%

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{-1}{re}\right) + \log \left(im \cdot im\right)\right)} \cdot 0.5} \cdot 0.5 \]
   10. Step-by-step derivation

   11. Applied rewrites 90.7%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.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. lift-sqrt.f64 N/A

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

      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. 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-*.f64 N/A

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

      5. lower-hypot.f64 89.0

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

   4. Applied rewrites 89.0%

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

4. Final simplification 89.2%

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

Developer Target 1: 48.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t\_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t\_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Reproduce

herbie shell --seed 2024228 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< re 0) (* 1/2 (* (sqrt 2) (sqrt (/ (* im im) (- (modulus re im) re))))) (* 1/2 (sqrt (* 2 (+ (modulus re im) re))))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))