math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 10

Speedup: 1.0×

• 50.7% 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: 69.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 23000000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+76}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 23000000000000.0)
   (cos re)
   (if (<= im 2.15e+76)
     (+ 0.25 (* 0.25 (pow re 2.0)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 23000000000000.0) {
		tmp = cos(re);
	} else if (im <= 2.15e+76) {
		tmp = 0.25 + (0.25 * pow(re, 2.0));
	} 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 <= 23000000000000.0d0) then
        tmp = cos(re)
    else if (im <= 2.15d+76) then
        tmp = 0.25d0 + (0.25d0 * (re ** 2.0d0))
    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 <= 23000000000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 2.15e+76) {
		tmp = 0.25 + (0.25 * Math.pow(re, 2.0));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 23000000000000.0:
		tmp = math.cos(re)
	elif im <= 2.15e+76:
		tmp = 0.25 + (0.25 * math.pow(re, 2.0))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 23000000000000.0)
		tmp = cos(re);
	elseif (im <= 2.15e+76)
		tmp = Float64(0.25 + Float64(0.25 * (re ^ 2.0)));
	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 <= 23000000000000.0)
		tmp = cos(re);
	elseif (im <= 2.15e+76)
		tmp = 0.25 + (0.25 * (re ^ 2.0));
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 23000000000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.15e+76], N[(0.25 + N[(0.25 * N[Power[re, 2.0], $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 < 2.3e13

   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 63.2%

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

   if 2.3e13 < im < 2.14999999999999989e76

   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%

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

   5. Taylor expanded in re around 0 24.6%

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

      1. *-commutative 24.6%

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

   7. Simplified 24.6%

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

   if 2.14999999999999989e76 < 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 98.1%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 98.1%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 98.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 98.1%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 98.1%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 98.1%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   6. Taylor expanded in im around inf 98.1%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 23000000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+76}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00048:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00048)
   (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 <= 0.00048) {
		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 <= 0.00048d0) 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 <= 0.00048) {
		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 <= 0.00048:
		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 <= 0.00048)
		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 <= 0.00048)
		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, 0.00048], 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 < 4.80000000000000012e-4

   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 64.1%

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

   if 4.80000000000000012e-4 < 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 76.9%

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

   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%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   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 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00048:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.042:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.042)
   (* (* 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 <= 0.042) {
		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 <= 0.042)
		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, 0.042], 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 < 0.0420000000000000026

   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 85.2%

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

      1. +-commutative 85.2%

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

      2. unpow2 85.2%

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

      3. fma-define 85.2%

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

   5. Simplified 85.2%

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

   if 0.0420000000000000026 < 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 76.9%

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

   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%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   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 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.042:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4800000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+76}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4800000000000.0)
   (cos re)
   (if (<= im 1.16e+76)
     (+ 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 <= 4800000000000.0) {
		tmp = cos(re);
	} else if (im <= 1.16e+76) {
		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 <= 4800000000000.0d0) then
        tmp = cos(re)
    else if (im <= 1.16d+76) 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 <= 4800000000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.16e+76) {
		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 <= 4800000000000.0:
		tmp = math.cos(re)
	elif im <= 1.16e+76:
		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 <= 4800000000000.0)
		tmp = cos(re);
	elseif (im <= 1.16e+76)
		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 <= 4800000000000.0)
		tmp = cos(re);
	elseif (im <= 1.16e+76)
		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, 4800000000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.16e+76], 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 < 4.8e12

   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 63.2%

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

   if 4.8e12 < im < 1.1599999999999999e76

   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%

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

   5. Taylor expanded in re around 0 24.6%

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

      1. *-commutative 24.6%

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

   7. Simplified 24.6%

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

   if 1.1599999999999999e76 < 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 98.1%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 98.1%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 98.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 98.1%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 98.1%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 98.1%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   6. Taylor expanded in im around inf 98.1%

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

   7. Taylor expanded in re around 0 74.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4800000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+76}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.4) (cos re) (* 0.041666666666666664 (pow im 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.4) {
		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 <= 4.4d0) 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 <= 4.4) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.041666666666666664 * Math.pow(im, 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.4:
		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 <= 4.4)
		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 <= 4.4)
		tmp = cos(re);
	else
		tmp = 0.041666666666666664 * (im ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.4], N[Cos[re], $MachinePrecision], N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.4000000000000004

   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 64.1%

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

   if 4.4000000000000004 < 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 65.7%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 65.7%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 65.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 65.7%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 65.7%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 65.7%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 65.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   6. Taylor expanded in im around inf 65.7%

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

   7. Taylor expanded in re around 0 49.8%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.0% 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 47.2%

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

4. Final simplification 47.2%

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

Alternative 8: 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 2.3%

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

   1. pow-base-1 2.3%

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

   2. metadata-eval 2.3%

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

6. Simplified 2.3%

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

7. Final simplification 2.3%

   \[\leadsto 0 \]
8. Add Preprocessing

Alternative 9: 7.9% 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 7.8%

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

5. Taylor expanded in re around 0 7.8%

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

6. Final simplification 7.8%

   \[\leadsto 0.25 \]
7. Add Preprocessing

Alternative 10: 28.0% 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

4. Applied egg-rr 28.3%

   \[\leadsto \color{blue}{\frac{\cos re \cdot -2}{\cos re \cdot -2 + \left(\cos re \cdot -2 - \cos re \cdot -2\right)}} \]
5. Step-by-step derivation

   1. +-inverses 28.3%

      \[\leadsto \frac{\cos re \cdot -2}{\cos re \cdot -2 + \color{blue}{0}} \]

   2. +-rgt-identity 28.3%

      \[\leadsto \frac{\cos re \cdot -2}{\color{blue}{\cos re \cdot -2}} \]

   3. *-inverses 28.3%

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

6. Simplified 28.3%

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

7. Final simplification 28.3%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))