math.sqrt on complex, real part

Percentage Accurate: 41.3% → 89.2%

Time: 16.3s

Alternatives: 8

Speedup: 1.9×

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: 89.2% accurate, 0.7× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ (sqrt (+ (* re re) (* im_m im_m))) re) 0.0)
   (* 0.5 (* (sqrt im_m) (sqrt (- (/ im_m 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(((re * re) + (im_m * im_m))) + re) <= 0.0) {
		tmp = 0.5 * (sqrt(im_m) * sqrt(-(im_m / re)));
	} 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(((re * re) + (im_m * im_m))) + re) <= 0.0) {
		tmp = 0.5 * (Math.sqrt(im_m) * Math.sqrt(-(im_m / re)));
	} 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(((re * re) + (im_m * im_m))) + re) <= 0.0:
		tmp = 0.5 * (math.sqrt(im_m) * math.sqrt(-(im_m / re)))
	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 (Float64(sqrt(Float64(Float64(re * re) + Float64(im_m * im_m))) + re) <= 0.0)
		tmp = Float64(0.5 * Float64(sqrt(im_m) * sqrt(Float64(-Float64(im_m / re)))));
	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(((re * re) + (im_m * im_m))) + re) <= 0.0)
		tmp = 0.5 * (sqrt(im_m) * sqrt(-(im_m / re)));
	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[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision], 0.0], N[(0.5 * N[(N[Sqrt[im$95$m], $MachinePrecision] * N[Sqrt[(-N[(im$95$m / re), $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 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 6.4%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in re around -inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

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

   1. Initial program 45.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -8.6 \cdot 10^{+159}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{im\_m}{\frac{re}{im\_m}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -8.6e+159)
   (* 0.5 (sqrt (- (/ im_m (/ re im_m)))))
   (* 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 (re <= -8.6e+159) {
		tmp = 0.5 * sqrt(-(im_m / (re / im_m)));
	} 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 (re <= -8.6e+159) {
		tmp = 0.5 * Math.sqrt(-(im_m / (re / im_m)));
	} 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 re <= -8.6e+159:
		tmp = 0.5 * math.sqrt(-(im_m / (re / im_m)))
	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 (re <= -8.6e+159)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(im_m / Float64(re / im_m)))));
	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 (re <= -8.6e+159)
		tmp = 0.5 * sqrt(-(im_m / (re / im_m)));
	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[re, -8.6e+159], N[(0.5 * N[Sqrt[(-N[(im$95$m / N[(re / im$95$m), $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 re < -8.6000000000000004e159

   1. Initial program 2.7%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in re around -inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   if -8.6000000000000004e159 < re

   1. Initial program 45.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.05 \cdot 10^{+173}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{im\_m}{\frac{re}{im\_m}}}\\ \mathbf{elif}\;re \leq 9.6 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{2}{\frac{1 - \frac{re}{im\_m}}{im\_m}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im\_m}{re} \cdot im\_m + re \cdot 4}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.05e+173)
   (* 0.5 (sqrt (- (/ im_m (/ re im_m)))))
   (if (<= re 9.6e-8)
     (* 0.5 (sqrt (/ 2.0 (/ (- 1.0 (/ re im_m)) im_m))))
     (* 0.5 (sqrt (+ (* (/ im_m re) im_m) (* re 4.0)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.05e+173) {
		tmp = 0.5 * sqrt(-(im_m / (re / im_m)));
	} else if (re <= 9.6e-8) {
		tmp = 0.5 * sqrt((2.0 / ((1.0 - (re / im_m)) / im_m)));
	} else {
		tmp = 0.5 * sqrt((((im_m / re) * im_m) + (re * 4.0)));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-3.05d+173)) then
        tmp = 0.5d0 * sqrt(-(im_m / (re / im_m)))
    else if (re <= 9.6d-8) then
        tmp = 0.5d0 * sqrt((2.0d0 / ((1.0d0 - (re / im_m)) / im_m)))
    else
        tmp = 0.5d0 * sqrt((((im_m / re) * im_m) + (re * 4.0d0)))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -3.05e+173) {
		tmp = 0.5 * Math.sqrt(-(im_m / (re / im_m)));
	} else if (re <= 9.6e-8) {
		tmp = 0.5 * Math.sqrt((2.0 / ((1.0 - (re / im_m)) / im_m)));
	} else {
		tmp = 0.5 * Math.sqrt((((im_m / re) * im_m) + (re * 4.0)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3.05e+173:
		tmp = 0.5 * math.sqrt(-(im_m / (re / im_m)))
	elif re <= 9.6e-8:
		tmp = 0.5 * math.sqrt((2.0 / ((1.0 - (re / im_m)) / im_m)))
	else:
		tmp = 0.5 * math.sqrt((((im_m / re) * im_m) + (re * 4.0)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.05e+173)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(im_m / Float64(re / im_m)))));
	elseif (re <= 9.6e-8)
		tmp = Float64(0.5 * sqrt(Float64(2.0 / Float64(Float64(1.0 - Float64(re / im_m)) / im_m))));
	else
		tmp = Float64(0.5 * sqrt(Float64(Float64(Float64(im_m / re) * im_m) + Float64(re * 4.0))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -3.05e+173)
		tmp = 0.5 * sqrt(-(im_m / (re / im_m)));
	elseif (re <= 9.6e-8)
		tmp = 0.5 * sqrt((2.0 / ((1.0 - (re / im_m)) / im_m)));
	else
		tmp = 0.5 * sqrt((((im_m / re) * im_m) + (re * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.05e+173], N[(0.5 * N[Sqrt[(-N[(im$95$m / N[(re / im$95$m), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.6e-8], N[(0.5 * N[Sqrt[N[(2.0 / N[(N[(1.0 - N[(re / im$95$m), $MachinePrecision]), $MachinePrecision] / im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(N[(N[(im$95$m / re), $MachinePrecision] * im$95$m), $MachinePrecision] + N[(re * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.04999999999999989e173

   1. Initial program 2.7%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in re around -inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   if -3.04999999999999989e173 < re < 9.59999999999999994e-8

   1. Initial program 46.4%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in im around inf 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   if 9.59999999999999994e-8 < re

   1. Initial program 42.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 70.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{+157}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{im\_m}{\frac{re}{im\_m}}}\\ \mathbf{elif}\;re \leq 1.42 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im\_m}{re} \cdot im\_m + re \cdot 4}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -6.8e+157)
   (* 0.5 (sqrt (- (/ im_m (/ re im_m)))))
   (if (<= re 1.42e-8)
     (* 0.5 (sqrt (* im_m 2.0)))
     (* 0.5 (sqrt (+ (* (/ im_m re) im_m) (* re 4.0)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -6.8e+157) {
		tmp = 0.5 * sqrt(-(im_m / (re / im_m)));
	} else if (re <= 1.42e-8) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else {
		tmp = 0.5 * sqrt((((im_m / re) * im_m) + (re * 4.0)));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-6.8d+157)) then
        tmp = 0.5d0 * sqrt(-(im_m / (re / im_m)))
    else if (re <= 1.42d-8) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else
        tmp = 0.5d0 * sqrt((((im_m / re) * im_m) + (re * 4.0d0)))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -6.8e+157) {
		tmp = 0.5 * Math.sqrt(-(im_m / (re / im_m)));
	} else if (re <= 1.42e-8) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else {
		tmp = 0.5 * Math.sqrt((((im_m / re) * im_m) + (re * 4.0)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -6.8e+157:
		tmp = 0.5 * math.sqrt(-(im_m / (re / im_m)))
	elif re <= 1.42e-8:
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	else:
		tmp = 0.5 * math.sqrt((((im_m / re) * im_m) + (re * 4.0)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -6.8e+157)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(im_m / Float64(re / im_m)))));
	elseif (re <= 1.42e-8)
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(Float64(Float64(im_m / re) * im_m) + Float64(re * 4.0))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -6.8e+157)
		tmp = 0.5 * sqrt(-(im_m / (re / im_m)));
	elseif (re <= 1.42e-8)
		tmp = 0.5 * sqrt((im_m * 2.0));
	else
		tmp = 0.5 * sqrt((((im_m / re) * im_m) + (re * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -6.8e+157], N[(0.5 * N[Sqrt[(-N[(im$95$m / N[(re / im$95$m), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.42e-8], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(N[(N[(im$95$m / re), $MachinePrecision] * im$95$m), $MachinePrecision] + N[(re * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.79999999999999958e157

   1. Initial program 2.7%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in re around -inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   if -6.79999999999999958e157 < re < 1.41999999999999998e-8

   1. Initial program 47.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in re around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 1.41999999999999998e-8 < re

   1. Initial program 42.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 70.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{+157}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{im\_m}{\frac{re}{im\_m}}}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -6.8e+157)
   (* 0.5 (sqrt (- (/ im_m (/ re im_m)))))
   (if (<= re 3.4e-7) (* 0.5 (sqrt (* im_m 2.0))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -6.8e+157) {
		tmp = 0.5 * sqrt(-(im_m / (re / im_m)));
	} else if (re <= 3.4e-7) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-6.8d+157)) then
        tmp = 0.5d0 * sqrt(-(im_m / (re / im_m)))
    else if (re <= 3.4d-7) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -6.8e+157) {
		tmp = 0.5 * Math.sqrt(-(im_m / (re / im_m)));
	} else if (re <= 3.4e-7) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -6.8e+157:
		tmp = 0.5 * math.sqrt(-(im_m / (re / im_m)))
	elif re <= 3.4e-7:
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -6.8e+157)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(im_m / Float64(re / im_m)))));
	elseif (re <= 3.4e-7)
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -6.8e+157)
		tmp = 0.5 * sqrt(-(im_m / (re / im_m)));
	elseif (re <= 3.4e-7)
		tmp = 0.5 * sqrt((im_m * 2.0));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -6.8e+157], N[(0.5 * N[Sqrt[(-N[(im$95$m / N[(re / im$95$m), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.4e-7], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.79999999999999958e157

   1. Initial program 2.7%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in re around -inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   if -6.79999999999999958e157 < re < 3.39999999999999974e-7

   1. Initial program 47.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in re around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 3.39999999999999974e-7 < re

   1. Initial program 42.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in re around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 64.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 2.8 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 2.8e-6) (* 0.5 (sqrt (* im_m 2.0))) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 2.8e-6) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 2.8d-6) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 2.8e-6) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 2.8e-6:
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 2.8e-6)
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 2.8e-6)
		tmp = 0.5 * sqrt((im_m * 2.0));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 2.8e-6], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.79999999999999987e-6

   1. Initial program 39.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in re around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 2.79999999999999987e-6 < re

   1. Initial program 42.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in re around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 30.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (if (<= re -4e-310) 0.0 (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -4e-310) {
		tmp = 0.0;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-4d-310)) then
        tmp = 0.0d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -4e-310) {
		tmp = 0.0;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -4e-310:
		tmp = 0.0
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -4e-310)
		tmp = 0.0;
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -4e-310)
		tmp = 0.0;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -4e-310], 0.0, N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.999999999999988e-310

   1. Initial program 30.0%

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

   3. Taylor expanded in re around -inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in im around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   10. Applied egg-rr 0

       \[\leadsto expr\]

   if -3.999999999999988e-310 < re

   1. Initial program 49.5%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in re around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 6.0% accurate, 213.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 0.0)

C

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

Fortran

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

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.0

Julia

im_m = abs(im)
function code(re, im_m)
	return 0.0
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 0.0

Derivation

1. Initial program 40.3%

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

3. Taylor expanded in re around -inf 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

6. Taylor expanded in im around 0 0

   \[\leadsto expr\]

8. Simplified 0

   \[\leadsto expr\]

10. Applied egg-rr 0

    \[\leadsto expr\]
11. Add Preprocessing

Developer target: 48.3% accurate, 0.7× 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 2024110 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

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