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

Percentage Accurate: 41.5% → 90.4%

Time: 9.9s

Alternatives: 5

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: 41.5% 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: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* 0.5 (* im (sqrt (/ 1.0 re))))
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))

C

double code(double re, double im) {
	double tmp;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0)
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $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

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -3.9 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -8.5 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1.55 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* im 2.0))))
        (t_1 (* 0.5 (sqrt (* 2.0 (* re -2.0))))))
   (if (<= re -3.9e+152)
     t_1
     (if (<= re -8.5e-18)
       t_0
       (if (<= re -1e-52)
         t_1
         (if (<= re -1.55e-86)
           t_0
           (if (<= re -4.2e-103)
             t_1
             (if (<= re 3.5e-17) t_0 (* 0.5 (/ im (sqrt re)))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((im * 2.0));
	double t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -3.9e+152) {
		tmp = t_1;
	} else if (re <= -8.5e-18) {
		tmp = t_0;
	} else if (re <= -1e-52) {
		tmp = t_1;
	} else if (re <= -1.55e-86) {
		tmp = t_0;
	} else if (re <= -4.2e-103) {
		tmp = t_1;
	} else if (re <= 3.5e-17) {
		tmp = t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((im * 2.0d0))
    t_1 = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    if (re <= (-3.9d+152)) then
        tmp = t_1
    else if (re <= (-8.5d-18)) then
        tmp = t_0
    else if (re <= (-1d-52)) then
        tmp = t_1
    else if (re <= (-1.55d-86)) then
        tmp = t_0
    else if (re <= (-4.2d-103)) then
        tmp = t_1
    else if (re <= 3.5d-17) then
        tmp = t_0
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((im * 2.0));
	double t_1 = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -3.9e+152) {
		tmp = t_1;
	} else if (re <= -8.5e-18) {
		tmp = t_0;
	} else if (re <= -1e-52) {
		tmp = t_1;
	} else if (re <= -1.55e-86) {
		tmp = t_0;
	} else if (re <= -4.2e-103) {
		tmp = t_1;
	} else if (re <= 3.5e-17) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((im * 2.0))
	t_1 = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	tmp = 0
	if re <= -3.9e+152:
		tmp = t_1
	elif re <= -8.5e-18:
		tmp = t_0
	elif re <= -1e-52:
		tmp = t_1
	elif re <= -1.55e-86:
		tmp = t_0
	elif re <= -4.2e-103:
		tmp = t_1
	elif re <= 3.5e-17:
		tmp = t_0
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))))
	tmp = 0.0
	if (re <= -3.9e+152)
		tmp = t_1;
	elseif (re <= -8.5e-18)
		tmp = t_0;
	elseif (re <= -1e-52)
		tmp = t_1;
	elseif (re <= -1.55e-86)
		tmp = t_0;
	elseif (re <= -4.2e-103)
		tmp = t_1;
	elseif (re <= 3.5e-17)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((im * 2.0));
	t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	tmp = 0.0;
	if (re <= -3.9e+152)
		tmp = t_1;
	elseif (re <= -8.5e-18)
		tmp = t_0;
	elseif (re <= -1e-52)
		tmp = t_1;
	elseif (re <= -1.55e-86)
		tmp = t_0;
	elseif (re <= -4.2e-103)
		tmp = t_1;
	elseif (re <= 3.5e-17)
		tmp = t_0;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3.9e+152], t$95$1, If[LessEqual[re, -8.5e-18], t$95$0, If[LessEqual[re, -1e-52], t$95$1, If[LessEqual[re, -1.55e-86], t$95$0, If[LessEqual[re, -4.2e-103], t$95$1, If[LessEqual[re, 3.5e-17], t$95$0, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -9.6 \cdot 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -2.9 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- im re)))))
        (t_1 (* 0.5 (sqrt (* 2.0 (* re -2.0))))))
   (if (<= re -9.6e+163)
     t_1
     (if (<= re -4.2e-18)
       t_0
       (if (<= re -2.9e-51)
         t_1
         (if (<= re 8.5e-18) t_0 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (im - re)));
	double t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -9.6e+163) {
		tmp = t_1;
	} else if (re <= -4.2e-18) {
		tmp = t_0;
	} else if (re <= -2.9e-51) {
		tmp = t_1;
	} else if (re <= 8.5e-18) {
		tmp = t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    t_1 = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    if (re <= (-9.6d+163)) then
        tmp = t_1
    else if (re <= (-4.2d-18)) then
        tmp = t_0
    else if (re <= (-2.9d-51)) then
        tmp = t_1
    else if (re <= 8.5d-18) then
        tmp = t_0
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double t_1 = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -9.6e+163) {
		tmp = t_1;
	} else if (re <= -4.2e-18) {
		tmp = t_0;
	} else if (re <= -2.9e-51) {
		tmp = t_1;
	} else if (re <= 8.5e-18) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (im - re)))
	t_1 = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	tmp = 0
	if re <= -9.6e+163:
		tmp = t_1
	elif re <= -4.2e-18:
		tmp = t_0
	elif re <= -2.9e-51:
		tmp = t_1
	elif re <= 8.5e-18:
		tmp = t_0
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))))
	tmp = 0.0
	if (re <= -9.6e+163)
		tmp = t_1;
	elseif (re <= -4.2e-18)
		tmp = t_0;
	elseif (re <= -2.9e-51)
		tmp = t_1;
	elseif (re <= 8.5e-18)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (im - re)));
	t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	tmp = 0.0;
	if (re <= -9.6e+163)
		tmp = t_1;
	elseif (re <= -4.2e-18)
		tmp = t_0;
	elseif (re <= -2.9e-51)
		tmp = t_1;
	elseif (re <= 8.5e-18)
		tmp = t_0;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -9.6e+163], t$95$1, If[LessEqual[re, -4.2e-18], t$95$0, If[LessEqual[re, -2.9e-51], t$95$1, If[LessEqual[re, 8.5e-18], t$95$0, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 63.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.9e-13) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.9e-13) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= 2.9d-13) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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 <= 2.9e-13) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.9e-13:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.9e-13)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	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 <= 2.9e-13)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.9e-13], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 52.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2023342 
(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)))))