math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 11

Speedup: 1.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

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

Java

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

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

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

Java

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

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

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

Java

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

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
4. Add Preprocessing

Alternative 2: 71.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.2e+18)
   (cos re)
   (if (<= im 1.15e+77)
     (sqrt (* (pow im 8.0) 0.001736111111111111))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.2e+18) {
		tmp = cos(re);
	} else if (im <= 1.15e+77) {
		tmp = sqrt((pow(im, 8.0) * 0.001736111111111111));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 5.2d+18) then
        tmp = cos(re)
    else if (im <= 1.15d+77) then
        tmp = sqrt(((im ** 8.0d0) * 0.001736111111111111d0))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.2e+18) {
		tmp = Math.cos(re);
	} else if (im <= 1.15e+77) {
		tmp = Math.sqrt((Math.pow(im, 8.0) * 0.001736111111111111));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.2e+18:
		tmp = math.cos(re)
	elif im <= 1.15e+77:
		tmp = math.sqrt((math.pow(im, 8.0) * 0.001736111111111111))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.2e+18)
		tmp = cos(re);
	elseif (im <= 1.15e+77)
		tmp = sqrt(Float64((im ^ 8.0) * 0.001736111111111111));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.2e+18)
		tmp = cos(re);
	elseif (im <= 1.15e+77)
		tmp = sqrt(((im ^ 8.0) * 0.001736111111111111));
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.2e+18], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.15e+77], N[Sqrt[N[(N[Power[im, 8.0], $MachinePrecision] * 0.001736111111111111), $MachinePrecision]], $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.2e18

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 66.5%

      \[\leadsto \color{blue}{\cos re} \]

   if 5.2e18 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 5.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 5.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + \color{blue}{{im}^{2} \cdot 0.08333333333333333}\right)\right) \]

   5. Simplified 5.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   6. Taylor expanded in im around inf 5.4%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]

   7. Taylor expanded in re around 0 5.2%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 5.2%

         \[\leadsto \color{blue}{\sqrt{0.041666666666666664 \cdot {im}^{4}} \cdot \sqrt{0.041666666666666664 \cdot {im}^{4}}} \]

      2. sqrt-unprod 54.7%

         \[\leadsto \color{blue}{\sqrt{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)}} \]

      3. *-commutative 54.7%

         \[\leadsto \sqrt{\color{blue}{\left({im}^{4} \cdot 0.041666666666666664\right)} \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} \]

      4. *-commutative 54.7%

         \[\leadsto \sqrt{\left({im}^{4} \cdot 0.041666666666666664\right) \cdot \color{blue}{\left({im}^{4} \cdot 0.041666666666666664\right)}} \]

      5. swap-sqr 54.7%

         \[\leadsto \sqrt{\color{blue}{\left({im}^{4} \cdot {im}^{4}\right) \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)}} \]

      6. pow-prod-up 54.7%

         \[\leadsto \sqrt{\color{blue}{{im}^{\left(4 + 4\right)}} \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)} \]

      7. metadata-eval 54.7%

         \[\leadsto \sqrt{{im}^{\color{blue}{8}} \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)} \]

      8. metadata-eval 54.7%

         \[\leadsto \sqrt{{im}^{8} \cdot \color{blue}{0.001736111111111111}} \]

   9. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\sqrt{{im}^{8} \cdot 0.001736111111111111}} \]

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + \color{blue}{{im}^{2} \cdot 0.08333333333333333}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0035:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0035)
   (cos re)
   (if (<= im 1.15e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0035) {
		tmp = cos(re);
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0035d0) then
        tmp = cos(re)
    else if (im <= 1.15d+77) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0035) {
		tmp = Math.cos(re);
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0035:
		tmp = math.cos(re)
	elif im <= 1.15e+77:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0035)
		tmp = cos(re);
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0035)
		tmp = cos(re);
	elseif (im <= 1.15e+77)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0035], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00350000000000000007

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 68.3%

      \[\leadsto \color{blue}{\cos re} \]

   if 0.00350000000000000007 < im < 1.14999999999999997e77

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 78.4%

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

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + \color{blue}{{im}^{2} \cdot 0.08333333333333333}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0035:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.88:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.88)
   (* (* 0.5 (cos re)) (fma im im 2.0))
   (if (<= im 1.15e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.88) {
		tmp = (0.5 * cos(re)) * fma(im, im, 2.0);
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.88)
		tmp = Float64(Float64(0.5 * cos(re)) * fma(im, im, 2.0));
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.88], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.880000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.9%

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

      1. +-commutative 86.9%

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

      2. unpow2 86.9%

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

      3. fma-define 86.9%

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

   5. Simplified 86.9%

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

   if 0.880000000000000004 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 85.7%

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

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + \color{blue}{{im}^{2} \cdot 0.08333333333333333}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.88:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.5e+18) (cos re) (sqrt (* (pow im 8.0) 0.001736111111111111))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.5e+18) {
		tmp = cos(re);
	} else {
		tmp = sqrt((pow(im, 8.0) * 0.001736111111111111));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.5d+18) then
        tmp = cos(re)
    else
        tmp = sqrt(((im ** 8.0d0) * 0.001736111111111111d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.5e+18) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.sqrt((Math.pow(im, 8.0) * 0.001736111111111111));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.5e+18:
		tmp = math.cos(re)
	else:
		tmp = math.sqrt((math.pow(im, 8.0) * 0.001736111111111111))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.5e+18)
		tmp = cos(re);
	else
		tmp = sqrt(Float64((im ^ 8.0) * 0.001736111111111111));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.5e+18)
		tmp = cos(re);
	else
		tmp = sqrt(((im ^ 8.0) * 0.001736111111111111));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.5e+18], N[Cos[re], $MachinePrecision], N[Sqrt[N[(N[Power[im, 8.0], $MachinePrecision] * 0.001736111111111111), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.5e18

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 66.5%

      \[\leadsto \color{blue}{\cos re} \]

   if 6.5e18 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 74.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + \color{blue}{{im}^{2} \cdot 0.08333333333333333}\right)\right) \]

   5. Simplified 74.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   6. Taylor expanded in im around inf 74.9%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]

   7. Taylor expanded in re around 0 60.7%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 60.7%

         \[\leadsto \color{blue}{\sqrt{0.041666666666666664 \cdot {im}^{4}} \cdot \sqrt{0.041666666666666664 \cdot {im}^{4}}} \]

      2. sqrt-unprod 73.9%

         \[\leadsto \color{blue}{\sqrt{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)}} \]

      3. *-commutative 73.9%

         \[\leadsto \sqrt{\color{blue}{\left({im}^{4} \cdot 0.041666666666666664\right)} \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} \]

      4. *-commutative 73.9%

         \[\leadsto \sqrt{\left({im}^{4} \cdot 0.041666666666666664\right) \cdot \color{blue}{\left({im}^{4} \cdot 0.041666666666666664\right)}} \]

      5. swap-sqr 73.9%

         \[\leadsto \sqrt{\color{blue}{\left({im}^{4} \cdot {im}^{4}\right) \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)}} \]

      6. pow-prod-up 73.9%

         \[\leadsto \sqrt{\color{blue}{{im}^{\left(4 + 4\right)}} \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)} \]

      7. metadata-eval 73.9%

         \[\leadsto \sqrt{{im}^{\color{blue}{8}} \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)} \]

      8. metadata-eval 73.9%

         \[\leadsto \sqrt{{im}^{8} \cdot \color{blue}{0.001736111111111111}} \]

   9. Applied egg-rr 73.9%

      \[\leadsto \color{blue}{\sqrt{{im}^{8} \cdot 0.001736111111111111}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+67}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.5e+23)
   (cos re)
   (if (<= im 3.2e+67)
     (+ 0.25 (* 0.25 (pow re 2.0)))
     (* 0.041666666666666664 (pow im 4.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+23) {
		tmp = cos(re);
	} else if (im <= 3.2e+67) {
		tmp = 0.25 + (0.25 * pow(re, 2.0));
	} else {
		tmp = 0.041666666666666664 * pow(im, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.5d+23) then
        tmp = cos(re)
    else if (im <= 3.2d+67) then
        tmp = 0.25d0 + (0.25d0 * (re ** 2.0d0))
    else
        tmp = 0.041666666666666664d0 * (im ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+23) {
		tmp = Math.cos(re);
	} else if (im <= 3.2e+67) {
		tmp = 0.25 + (0.25 * Math.pow(re, 2.0));
	} else {
		tmp = 0.041666666666666664 * Math.pow(im, 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.5e+23:
		tmp = math.cos(re)
	elif im <= 3.2e+67:
		tmp = 0.25 + (0.25 * math.pow(re, 2.0))
	else:
		tmp = 0.041666666666666664 * math.pow(im, 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.5e+23)
		tmp = cos(re);
	elseif (im <= 3.2e+67)
		tmp = Float64(0.25 + Float64(0.25 * (re ^ 2.0)));
	else
		tmp = Float64(0.041666666666666664 * (im ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.5e+23)
		tmp = cos(re);
	elseif (im <= 3.2e+67)
		tmp = 0.25 + (0.25 * (re ^ 2.0));
	else
		tmp = 0.041666666666666664 * (im ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.5e+23], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.2e+67], N[(0.25 + N[(0.25 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.5e23

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 66.2%

      \[\leadsto \color{blue}{\cos re} \]

   if 1.5e23 < im < 3.19999999999999983e67

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.9%

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

   5. Taylor expanded in re around 0 18.3%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 18.3%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 18.3%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   if 3.19999999999999983e67 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 94.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 94.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + \color{blue}{{im}^{2} \cdot 0.08333333333333333}\right)\right) \]

   5. Simplified 94.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   6. Taylor expanded in im around inf 94.5%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]

   7. Taylor expanded in re around 0 76.5%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+67}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.25e+20) (cos re) (* 0.041666666666666664 (pow im 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.25e+20) {
		tmp = cos(re);
	} else {
		tmp = 0.041666666666666664 * pow(im, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.25d+20) then
        tmp = cos(re)
    else
        tmp = 0.041666666666666664d0 * (im ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.25e+20) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.041666666666666664 * Math.pow(im, 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.25e+20:
		tmp = math.cos(re)
	else:
		tmp = 0.041666666666666664 * math.pow(im, 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.25e+20)
		tmp = cos(re);
	else
		tmp = Float64(0.041666666666666664 * (im ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.25e+20)
		tmp = cos(re);
	else
		tmp = 0.041666666666666664 * (im ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.25e+20], N[Cos[re], $MachinePrecision], N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.25e20

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 66.5%

      \[\leadsto \color{blue}{\cos re} \]

   if 1.25e20 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 74.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + \color{blue}{{im}^{2} \cdot 0.08333333333333333}\right)\right) \]

   5. Simplified 74.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   6. Taylor expanded in im around inf 74.9%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]

   7. Taylor expanded in re around 0 60.7%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \cos re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (cos re))

C

double code(double re, double im) {
	return cos(re);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(re)
end function

Java

public static double code(double re, double im) {
	return Math.cos(re);
}

Python

def code(re, im):
	return math.cos(re)

Julia

function code(re, im)
	return cos(re)
end

MATLAB

function tmp = code(re, im)
	tmp = cos(re);
end

Wolfram

code[re_, im_] := N[Cos[re], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 50.7%

   \[\leadsto \color{blue}{\cos re} \]

4. Final simplification 50.7%

   \[\leadsto \cos re \]
5. Add Preprocessing

Alternative 9: 2.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

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

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 2.4%

   \[\leadsto \color{blue}{\log \left({1}^{\cos re}\right)} \]
5. Step-by-step derivation

   1. pow-base-1 2.4%

      \[\leadsto \log \color{blue}{1} \]

   2. metadata-eval 2.4%

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

6. Simplified 2.4%

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

7. Final simplification 2.4%

   \[\leadsto 0 \]
8. Add Preprocessing

Alternative 10: 8.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 8.0%

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

5. Taylor expanded in re around 0 8.3%

   \[\leadsto \color{blue}{0.25} \]

6. Final simplification 8.3%

   \[\leadsto 0.25 \]
7. Add Preprocessing

Alternative 11: 28.5% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 50.7%

   \[\leadsto \color{blue}{\cos re} \]

4. Taylor expanded in re around 0 25.7%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 25.7%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))