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

Percentage Accurate: 40.5% → 87.9%

Time: 6.3s

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: 40.5% 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: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.55 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.55e+31)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.55e+31) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.55e+31) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.55e+31:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.55e+31)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.55e+31)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.55e+31], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.5500000000000001e31

   1. Initial program 45.2%

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

      1. hypot-def 92.6%

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

   3. Simplified 92.6%

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

   if 1.5500000000000001e31 < re

   1. Initial program 7.2%

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

   2. Taylor expanded in re around inf 57.5%

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

      1. unpow2 57.5%

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

   4. Simplified 57.5%

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

   5. Taylor expanded in im around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. unpow1/2 83.4%

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

      3. sqr-pow 83.0%

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

      4. sqr-pow 83.4%

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

      5. rem-exp-log 78.4%

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

      6. exp-neg 78.4%

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

      7. exp-prod 78.4%

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

      8. distribute-lft-neg-out 78.4%

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

      9. distribute-rgt-neg-in 78.4%

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

      10. metadata-eval 78.4%

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

      11. exp-to-pow 83.4%

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

   7. Simplified 83.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.55 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 2: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{+141}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.32 \cdot 10^{-30} \lor \neg \left(re \leq 4.8 \cdot 10^{+14}\right) \land re \leq 1.55 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.9e+141)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (or (<= re 1.32e-30) (and (not (<= re 4.8e+14)) (<= re 1.55e+31)))
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.9e+141) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 1.32e-30) || (!(re <= 4.8e+14) && (re <= 1.55e+31))) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * pow(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 <= (-2.9d+141)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if ((re <= 1.32d-30) .or. (.not. (re <= 4.8d+14)) .and. (re <= 1.55d+31)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.9e+141) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 1.32e-30) || (!(re <= 4.8e+14) && (re <= 1.55e+31))) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.9e+141:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif (re <= 1.32e-30) or (not (re <= 4.8e+14) and (re <= 1.55e+31)):
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.9e+141)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif ((re <= 1.32e-30) || (!(re <= 4.8e+14) && (re <= 1.55e+31)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.9e+141)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif ((re <= 1.32e-30) || (~((re <= 4.8e+14)) && (re <= 1.55e+31)))
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.9e+141], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.32e-30], And[N[Not[LessEqual[re, 4.8e+14]], $MachinePrecision], LessEqual[re, 1.55e+31]]], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.90000000000000007e141

   1. Initial program 6.4%

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

   2. Taylor expanded in re around -inf 95.9%

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

      1. *-commutative 95.9%

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

   4. Simplified 95.9%

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

   if -2.90000000000000007e141 < re < 1.32e-30 or 4.8e14 < re < 1.5500000000000001e31

   1. Initial program 58.0%

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

   2. Taylor expanded in re around 0 76.8%

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

   if 1.32e-30 < re < 4.8e14 or 1.5500000000000001e31 < re

   1. Initial program 8.6%

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

   2. Taylor expanded in re around inf 55.3%

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

      1. unpow2 55.3%

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

   4. Simplified 55.3%

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

   5. Taylor expanded in im around 0 82.4%

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

      1. *-commutative 82.4%

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

      2. unpow1/2 82.4%

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

      3. sqr-pow 82.0%

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

      4. sqr-pow 82.4%

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

      5. rem-exp-log 77.9%

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

      6. exp-neg 77.9%

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

      7. exp-prod 77.9%

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

      8. distribute-lft-neg-out 77.9%

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

      9. distribute-rgt-neg-in 77.9%

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

      10. metadata-eval 77.9%

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

      11. exp-to-pow 82.4%

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

   7. Simplified 82.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{+141}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.32 \cdot 10^{-30} \lor \neg \left(re \leq 4.8 \cdot 10^{+14}\right) \land re \leq 1.55 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3: 52.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e-310)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * (im * pow(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 <= (-5d-310)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-310:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e-310)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-310], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 47.3%

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

   2. Taylor expanded in re around -inf 55.5%

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

      1. *-commutative 55.5%

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

   4. Simplified 55.5%

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

   if -4.999999999999985e-310 < re

   1. Initial program 26.6%

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

   2. Taylor expanded in re around inf 31.4%

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

      1. unpow2 31.4%

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

   4. Simplified 31.4%

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

   5. Taylor expanded in im around 0 52.0%

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

      1. *-commutative 52.0%

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

      2. unpow1/2 52.0%

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

      3. sqr-pow 51.8%

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

      4. sqr-pow 52.0%

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

      5. rem-exp-log 49.2%

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

      6. exp-neg 49.2%

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

      7. exp-prod 49.2%

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

      8. distribute-lft-neg-out 49.2%

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

      9. distribute-rgt-neg-in 49.2%

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

      10. metadata-eval 49.2%

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

      11. exp-to-pow 52.0%

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

   7. Simplified 52.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 26.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* im (pow re -0.5))))

C

double code(double re, double im) {
	return 0.5 * (im * pow(re, -0.5));
}

Fortran

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

Java

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

Python

def code(re, im):
	return 0.5 * (im * math.pow(re, -0.5))

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * (re ^ -0.5)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * (re ^ -0.5));
end

Wolfram

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

Derivation

1. Initial program 37.7%

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

2. Taylor expanded in re around inf 15.0%

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

   1. unpow2 15.0%

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

4. Simplified 15.0%

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

5. Taylor expanded in im around 0 24.0%

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

   1. *-commutative 24.0%

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

   2. unpow1/2 24.0%

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

   3. sqr-pow 23.9%

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

   4. sqr-pow 24.0%

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

   5. rem-exp-log 22.7%

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

   6. exp-neg 22.7%

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

   7. exp-prod 22.7%

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

   8. distribute-lft-neg-out 22.7%

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

   9. distribute-rgt-neg-in 22.7%

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

   10. metadata-eval 22.7%

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

   11. exp-to-pow 24.0%

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

7. Simplified 24.0%

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

8. Final simplification 24.0%

   \[\leadsto 0.5 \cdot \left(im \cdot {re}^{-0.5}\right) \]

Alternative 5: 3.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.7%

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

2. Taylor expanded in re around inf 15.0%

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

   1. unpow2 15.0%

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

4. Simplified 15.0%

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

5. Taylor expanded in im around 0 24.0%

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

   1. *-commutative 24.0%

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

   2. unpow1/2 24.0%

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

   3. sqr-pow 23.9%

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

   4. sqr-pow 24.0%

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

   5. rem-exp-log 22.7%

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

   6. exp-neg 22.7%

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

   7. exp-prod 22.7%

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

   8. distribute-lft-neg-out 22.7%

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

   9. distribute-rgt-neg-in 22.7%

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

   10. metadata-eval 22.7%

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

   11. exp-to-pow 24.0%

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

7. Simplified 24.0%

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

   1. expm1-log1p-u 23.7%

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

   2. expm1-udef 10.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot {re}^{-0.5}\right)} - 1\right)} \]

   3. pow-to-exp 10.5%

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \color{blue}{e^{\log re \cdot -0.5}}\right)} - 1\right) \]

   4. metadata-eval 10.5%

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot e^{\log re \cdot \color{blue}{\left(-0.5\right)}}\right)} - 1\right) \]

   5. distribute-rgt-neg-in 10.5%

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right)} - 1\right) \]

   6. distribute-lft-neg-out 10.5%

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot e^{\color{blue}{\left(-\log re\right) \cdot 0.5}}\right)} - 1\right) \]

   7. exp-prod 10.5%

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \color{blue}{{\left(e^{-\log re}\right)}^{0.5}}\right)} - 1\right) \]

   8. unpow1/2 10.5%

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \color{blue}{\sqrt{e^{-\log re}}}\right)} - 1\right) \]

   9. add-sqr-sqrt 1.2%

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \sqrt{e^{\color{blue}{\sqrt{-\log re} \cdot \sqrt{-\log re}}}}\right)} - 1\right) \]

   10. sqrt-unprod 6.3%

       \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \sqrt{e^{\color{blue}{\sqrt{\left(-\log re\right) \cdot \left(-\log re\right)}}}}\right)} - 1\right) \]

   11. sqr-neg 6.3%

       \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \sqrt{e^{\sqrt{\color{blue}{\log re \cdot \log re}}}}\right)} - 1\right) \]

   12. sqrt-unprod 5.1%

       \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \sqrt{e^{\color{blue}{\sqrt{\log re} \cdot \sqrt{\log re}}}}\right)} - 1\right) \]

   13. add-sqr-sqrt 6.5%

       \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \sqrt{e^{\color{blue}{\log re}}}\right)} - 1\right) \]

   14. add-exp-log 6.5%

       \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \sqrt{\color{blue}{re}}\right)} - 1\right) \]

9. Applied egg-rr 6.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \sqrt{re}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 2.9%

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

    2. expm1-log1p 2.9%

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

11. Simplified 2.9%

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

12. Final simplification 2.9%

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

Reproduce

herbie shell --seed 2023199 
(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)))))