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

Percentage Accurate: 41.7% → 87.8%

Time: 7.2s

Alternatives: 7

Speedup: 1.9×

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: 41.7% 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: 87.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.12 \cdot 10^{+35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.12e+35)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* (/ im (sqrt re)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.12e+35) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = (im / sqrt(re)) * 0.5;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.12e+35) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = (im / Math.sqrt(re)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.12e+35:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = (im / math.sqrt(re)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.12e+35)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, 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 <= 1.12e+35)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = (im / sqrt(re)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.12e+35], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.12000000000000003e35

   1. Initial program 48.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 91.7

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

   4. Applied rewrites 91.7%

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

   if 1.12000000000000003e35 < re

   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-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 32.8

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

   4. Applied rewrites 32.8%

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

   5. Taylor expanded in re around inf

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

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 82.0

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

   7. Applied rewrites 82.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.0

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

   9. Applied rewrites 82.8%

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

Alternative 2: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{+89}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot re}\\ \mathbf{elif}\;re \leq -3.2 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 8.9 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.8e+89)
   (* (* 0.5 (sqrt 2.0)) (sqrt (* -2.0 re)))
   (if (<= re -3.2e-58)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 8.9e+33)
       (* 0.5 (sqrt (* 2.0 (- im re))))
       (* (/ im (sqrt re)) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.8e+89) {
		tmp = (0.5 * sqrt(2.0)) * sqrt((-2.0 * re));
	} else if (re <= -3.2e-58) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 8.9e+33) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im / sqrt(re)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.8e+89)
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * sqrt(Float64(-2.0 * re)));
	elseif (re <= -3.2e-58)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 8.9e+33)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.8e+89], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-2.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -3.2e-58], 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, 8.9e+33], 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 4 regimes

2. if re < -6.8000000000000004e89

   1. Initial program 24.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-hypot.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

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

      6. lift-fma.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 24.2

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in re around -inf

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

      1. lower-*.f64 80.5

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

   9. Applied rewrites 80.5%

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

   if -6.8000000000000004e89 < re < -3.2000000000000001e-58

   1. Initial program 86.3%

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

      1. lift-+.f64 N/A

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

      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 86.3

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

   4. Applied rewrites 86.3%

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

   if -3.2000000000000001e-58 < re < 8.9000000000000005e33

   1. Initial program 51.2%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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 78.5

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

   5. Applied rewrites 78.5%

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

   if 8.9000000000000005e33 < re

   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-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 32.8

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

   4. Applied rewrites 32.8%

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

   5. Taylor expanded in re around inf

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

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 82.0

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

   7. Applied rewrites 82.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.0

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

   9. Applied rewrites 82.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{+89}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot re}\\ \mathbf{elif}\;re \leq -3.2 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 8.9 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+32}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot re}\\ \mathbf{elif}\;re \leq 8.9 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.5e+32)
   (* (* 0.5 (sqrt 2.0)) (sqrt (* -2.0 re)))
   (if (<= re 8.9e+33)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (/ im (sqrt re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.5e+32) {
		tmp = (0.5 * sqrt(2.0)) * sqrt((-2.0 * re));
	} else if (re <= 8.9e+33) {
		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 <= (-1.5d+32)) then
        tmp = (0.5d0 * sqrt(2.0d0)) * sqrt(((-2.0d0) * re))
    else if (re <= 8.9d+33) 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 <= -1.5e+32) {
		tmp = (0.5 * Math.sqrt(2.0)) * Math.sqrt((-2.0 * re));
	} else if (re <= 8.9e+33) {
		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 <= -1.5e+32:
		tmp = (0.5 * math.sqrt(2.0)) * math.sqrt((-2.0 * re))
	elif re <= 8.9e+33:
		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 <= -1.5e+32)
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * sqrt(Float64(-2.0 * re)));
	elseif (re <= 8.9e+33)
		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 <= -1.5e+32)
		tmp = (0.5 * sqrt(2.0)) * sqrt((-2.0 * re));
	elseif (re <= 8.9e+33)
		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, -1.5e+32], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-2.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.9e+33], 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 < -1.5e32

   1. Initial program 35.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-hypot.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

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

      6. lift-fma.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 35.5

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in re around -inf

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

      1. lower-*.f64 77.1

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

   9. Applied rewrites 77.1%

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

   if -1.5e32 < re < 8.9000000000000005e33

   1. Initial program 55.6%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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 76.6

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

   5. Applied rewrites 76.6%

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

   if 8.9000000000000005e33 < re

   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-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 32.8

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

   4. Applied rewrites 32.8%

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

   5. Taylor expanded in re around inf

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

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 82.0

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

   7. Applied rewrites 82.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.0

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

   9. Applied rewrites 82.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+32}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot re}\\ \mathbf{elif}\;re \leq 8.9 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 8.9 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.5e+32)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 8.9e+33)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (/ im (sqrt re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.5e+32) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 8.9e+33) {
		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 <= (-1.5d+32)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 8.9d+33) 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 <= -1.5e+32) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 8.9e+33) {
		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 <= -1.5e+32:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 8.9e+33:
		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 <= -1.5e+32)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 8.9e+33)
		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 <= -1.5e+32)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 8.9e+33)
		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, -1.5e+32], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.9e+33], 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 < -1.5e32

   1. Initial program 35.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if -1.5e32 < re < 8.9000000000000005e33

   1. Initial program 55.6%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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 76.6

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

   5. Applied rewrites 76.6%

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

   if 8.9000000000000005e33 < re

   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-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 32.8

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

   4. Applied rewrites 32.8%

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

   5. Taylor expanded in re around inf

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

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 82.0

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

   7. Applied rewrites 82.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.0

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

   9. Applied rewrites 82.8%

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

Alternative 5: 64.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e+14) (* 0.5 (sqrt (* -4.0 re))) (* (sqrt (+ im im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e+14) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = sqrt((im + im)) * 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 <= (-2.4d+14)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = sqrt((im + im)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.4e+14) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = Math.sqrt((im + im)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.4e+14:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = math.sqrt((im + im)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.4e+14)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.4e+14)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = sqrt((im + im)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.4e+14], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.4e14

   1. Initial program 37.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 75.7

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

   7. Applied rewrites 75.7%

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

   if -2.4e14 < re

   1. Initial program 39.3%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 57.3

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

   5. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.3

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

   7. Applied rewrites 57.6%

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

   9. Applied rewrites 57.6%

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

Alternative 6: 54.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.8 \cdot 10^{-223}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.8e-223) 0.0 (* (sqrt (+ im im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.8e-223) {
		tmp = 0.0;
	} else {
		tmp = sqrt((im + im)) * 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 (im <= 1.8d-223) then
        tmp = 0.0d0
    else
        tmp = sqrt((im + im)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.8e-223) {
		tmp = 0.0;
	} else {
		tmp = Math.sqrt((im + im)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.8e-223:
		tmp = 0.0
	else:
		tmp = math.sqrt((im + im)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.8e-223)
		tmp = 0.0;
	else
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.8e-223)
		tmp = 0.0;
	else
		tmp = sqrt((im + im)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.8e-223], 0.0, N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.8000000000000002e-223

   1. Initial program 33.6%

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

   4. Applied rewrites 19.6%

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

   if 1.8000000000000002e-223 < im

   1. Initial program 39.7%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 55.8

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

   5. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.8

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

   7. Applied rewrites 56.2%

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

   9. Applied rewrites 56.2%

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

Alternative 7: 5.9% 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 38.8%

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

4. Applied rewrites 5.9%

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

Reproduce

herbie shell --seed 2024305 
(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)))))