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

Percentage Accurate: 42.3% → 89.8%

Time: 8.6s

Alternatives: 7

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

Java

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

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 42.3% 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: 89.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 12.7%

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

   3. Taylor expanded in re around inf 98.5%

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

      1. sqrt-unprod 99.5%

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

      2. metadata-eval 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-rgt-identity 99.5%

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

      5. sqrt-div 99.7%

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

      6. metadata-eval 99.7%

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

      7. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

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

   1. Initial program 47.0%

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

      1. sub-neg 47.0%

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

      2. sqr-neg 47.0%

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

      3. sub-neg 47.0%

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

      4. sqr-neg 47.0%

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

      5. hypot-define 89.2%

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

   3. Simplified 89.2%

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

Alternative 2: 76.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-79} \lor \neg \left(re \leq 8 \cdot 10^{+26}\right) \land re \leq 6.3 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \left(1 - \frac{re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1e-18)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (or (<= re 7e-79) (and (not (<= re 8e+26)) (<= re 6.3e+44)))
     (* 0.5 (sqrt (* 2.0 (* im (- 1.0 (/ re im))))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e-18) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 7e-79) || (!(re <= 8e+26) && (re <= 6.3e+44))) {
		tmp = 0.5 * sqrt((2.0 * (im * (1.0 - (re / im)))));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1d-18)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if ((re <= 7d-79) .or. (.not. (re <= 8d+26)) .and. (re <= 6.3d+44)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im * (1.0d0 - (re / im)))))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1e-18) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 7e-79) || (!(re <= 8e+26) && (re <= 6.3e+44))) {
		tmp = 0.5 * Math.sqrt((2.0 * (im * (1.0 - (re / im)))));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1e-18:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif (re <= 7e-79) or (not (re <= 8e+26) and (re <= 6.3e+44)):
		tmp = 0.5 * math.sqrt((2.0 * (im * (1.0 - (re / im)))))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e-18)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif ((re <= 7e-79) || (!(re <= 8e+26) && (re <= 6.3e+44)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im * Float64(1.0 - Float64(re / im))))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1e-18)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif ((re <= 7e-79) || (~((re <= 8e+26)) && (re <= 6.3e+44)))
		tmp = 0.5 * sqrt((2.0 * (im * (1.0 - (re / im)))));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e-18], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 7e-79], And[N[Not[LessEqual[re, 8e+26]], $MachinePrecision], LessEqual[re, 6.3e+44]]], N[(0.5 * N[Sqrt[N[(2.0 * N[(im * N[(1.0 - N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.0000000000000001e-18

   1. Initial program 49.1%

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

   3. Taylor expanded in re around -inf 76.7%

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

      1. *-commutative 76.7%

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

   5. Simplified 76.7%

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

   if -1.0000000000000001e-18 < re < 7.00000000000000059e-79 or 8.00000000000000038e26 < re < 6.3e44

   1. Initial program 66.2%

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

   3. Taylor expanded in im around inf 87.8%

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

      1. mul-1-neg 87.8%

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

      2. unsub-neg 87.8%

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

   5. Simplified 87.8%

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

   if 7.00000000000000059e-79 < re < 8.00000000000000038e26 or 6.3e44 < 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. Taylor expanded in re around inf 73.9%

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

      1. sqrt-unprod 74.5%

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

      2. metadata-eval 74.5%

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

      3. metadata-eval 74.5%

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

      4. *-rgt-identity 74.5%

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

      5. sqrt-div 74.5%

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

      6. metadata-eval 74.5%

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

      7. un-div-inv 74.6%

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

   5. Applied egg-rr 74.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-79} \lor \neg \left(re \leq 8 \cdot 10^{+26}\right) \land re \leq 6.3 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \left(1 - \frac{re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-66} \lor \neg \left(re \leq 1.75 \cdot 10^{+28}\right) \land re \leq 1.45 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.2e-19)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (or (<= re 2.9e-66) (and (not (<= re 1.75e+28)) (<= re 1.45e+44)))
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.2e-19) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 2.9e-66) || (!(re <= 1.75e+28) && (re <= 1.45e+44))) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.2d-19)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if ((re <= 2.9d-66) .or. (.not. (re <= 1.75d+28)) .and. (re <= 1.45d+44)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.2e-19) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 2.9e-66) || (!(re <= 1.75e+28) && (re <= 1.45e+44))) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.2e-19:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif (re <= 2.9e-66) or (not (re <= 1.75e+28) and (re <= 1.45e+44)):
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.2e-19)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif ((re <= 2.9e-66) || (!(re <= 1.75e+28) && (re <= 1.45e+44)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.2e-19)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif ((re <= 2.9e-66) || (~((re <= 1.75e+28)) && (re <= 1.45e+44)))
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.2e-19], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 2.9e-66], And[N[Not[LessEqual[re, 1.75e+28]], $MachinePrecision], LessEqual[re, 1.45e+44]]], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.1999999999999998e-19

   1. Initial program 49.1%

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

   3. Taylor expanded in re around -inf 76.7%

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

      1. *-commutative 76.7%

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

   5. Simplified 76.7%

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

   if -2.1999999999999998e-19 < re < 2.90000000000000011e-66 or 1.75e28 < re < 1.4500000000000001e44

   1. Initial program 66.2%

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

   3. Taylor expanded in re around 0 87.8%

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

   if 2.90000000000000011e-66 < re < 1.75e28 or 1.4500000000000001e44 < 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. Taylor expanded in re around inf 73.9%

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

      1. sqrt-unprod 74.5%

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

      2. metadata-eval 74.5%

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

      3. metadata-eval 74.5%

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

      4. *-rgt-identity 74.5%

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

      5. sqrt-div 74.5%

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

      6. metadata-eval 74.5%

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

      7. un-div-inv 74.6%

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

   5. Applied egg-rr 74.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-66} \lor \neg \left(re \leq 1.75 \cdot 10^{+28}\right) \land re \leq 1.45 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re \cdot \left(0.5 \cdot \frac{re}{im} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.5e-20)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 3.3e-74)
     (* 0.5 (sqrt (* 2.0 (+ im (* re (+ (* 0.5 (/ re im)) -1.0))))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.5e-20) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.3e-74) {
		tmp = 0.5 * sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.5d-20)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 3.3d-74) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im + (re * ((0.5d0 * (re / im)) + (-1.0d0))))))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.5e-20) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.3e-74) {
		tmp = 0.5 * Math.sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.5e-20:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 3.3e-74:
		tmp = 0.5 * math.sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.5e-20)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 3.3e-74)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im + Float64(re * Float64(Float64(0.5 * Float64(re / im)) + -1.0))))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.5e-20)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 3.3e-74)
		tmp = 0.5 * sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.5e-20], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.3e-74], N[(0.5 * N[Sqrt[N[(2.0 * N[(im + N[(re * N[(N[(0.5 * N[(re / im), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.4999999999999999e-20

   1. Initial program 49.1%

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

   3. Taylor expanded in re around -inf 76.7%

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

      1. *-commutative 76.7%

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

   5. Simplified 76.7%

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

   if -2.4999999999999999e-20 < re < 3.29999999999999996e-74

   1. Initial program 65.1%

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

   3. Taylor expanded in re around 0 87.3%

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

   if 3.29999999999999996e-74 < re

   1. Initial program 14.5%

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

   3. Taylor expanded in re around inf 69.6%

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

      1. sqrt-unprod 70.1%

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

      2. metadata-eval 70.1%

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

      3. metadata-eval 70.1%

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

      4. *-rgt-identity 70.1%

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

      5. sqrt-div 70.1%

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

      6. metadata-eval 70.1%

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

      7. un-div-inv 70.2%

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

   5. Applied egg-rr 70.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re \cdot \left(0.5 \cdot \frac{re}{im} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.15e-23)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 3.1e-55) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.15e-23) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.1e-55) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.15d-23)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 3.1d-55) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.15e-23) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.1e-55) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.15e-23:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 3.1e-55:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.15e-23)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 3.1e-55)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.15e-23)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 3.1e-55)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.15e-23], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.1e-55], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.15000000000000001e-23

   1. Initial program 49.1%

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

   3. Taylor expanded in re around -inf 76.7%

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

      1. *-commutative 76.7%

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

   5. Simplified 76.7%

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

   if -2.15000000000000001e-23 < re < 3.09999999999999997e-55

   1. Initial program 63.8%

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

      1. hypot-define 94.6%

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

      2. expm1-log1p-u 90.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

   4. Applied egg-rr 90.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

   5. Taylor expanded in re around 0 84.2%

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

   if 3.09999999999999997e-55 < re

   1. Initial program 13.9%

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

   3. Taylor expanded in re around inf 70.4%

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

      1. sqrt-unprod 70.9%

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

      2. metadata-eval 70.9%

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

      3. metadata-eval 70.9%

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

      4. *-rgt-identity 70.9%

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

      5. sqrt-div 70.9%

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

      6. metadata-eval 70.9%

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

      7. un-div-inv 71.0%

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

   5. Applied egg-rr 71.0%

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

Alternative 6: 63.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.3 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.3e-55) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.3e-55) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 4.3d-55) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.3e-55) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.3e-55:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.3e-55)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.3e-55)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.3e-55], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.3000000000000001e-55

   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. Step-by-step derivation

      1. hypot-define 96.6%

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

      2. expm1-log1p-u 91.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

   4. Applied egg-rr 91.3%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

   5. Taylor expanded in re around 0 63.6%

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

   if 4.3000000000000001e-55 < re

   1. Initial program 13.9%

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

   3. Taylor expanded in re around inf 70.4%

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

      1. sqrt-unprod 70.9%

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

      2. metadata-eval 70.9%

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

      3. metadata-eval 70.9%

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

      4. *-rgt-identity 70.9%

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

      5. sqrt-div 70.9%

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

      6. metadata-eval 70.9%

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

      7. un-div-inv 71.0%

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

   5. Applied egg-rr 71.0%

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

Alternative 7: 52.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot im} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * im))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * im));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * im))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * im));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.2%

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

   1. hypot-define 80.8%

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

   2. expm1-log1p-u 76.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

4. Applied egg-rr 76.4%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

5. Taylor expanded in re around 0 53.4%

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

Reproduce

herbie shell --seed 2024103 
(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)))))