math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 11

Speedup: 1.0×

• 49.8% 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: 70.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+27}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8.8 \cdot 10^{+76}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.22e+27)
   (cos re)
   (if (<= im 8.8e+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 <= 1.22e+27) {
		tmp = cos(re);
	} else if (im <= 8.8e+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 <= 1.22d+27) then
        tmp = cos(re)
    else if (im <= 8.8d+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 <= 1.22e+27) {
		tmp = Math.cos(re);
	} else if (im <= 8.8e+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 <= 1.22e+27:
		tmp = math.cos(re)
	elif im <= 8.8e+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 <= 1.22e+27)
		tmp = cos(re);
	elseif (im <= 8.8e+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 <= 1.22e+27)
		tmp = cos(re);
	elseif (im <= 8.8e+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, 1.22e+27], N[Cos[re], $MachinePrecision], If[LessEqual[im, 8.8e+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 < 1.2200000000000001e27

   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 61.9%

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

   if 1.2200000000000001e27 < im < 8.8000000000000002e76

   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 2.1%

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

   5. Taylor expanded in re around 0 31.0%

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

      1. *-commutative 31.0%

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

   7. Simplified 31.0%

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

   if 8.8000000000000002e76 < 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 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+27}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8.8 \cdot 10^{+76}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.005:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.005)
   (cos re)
   (if (<= im 1.2e+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 <= 0.005) {
		tmp = cos(re);
	} else if (im <= 1.2e+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 <= 0.005d0) then
        tmp = cos(re)
    else if (im <= 1.2d+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 <= 0.005) {
		tmp = Math.cos(re);
	} else if (im <= 1.2e+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 <= 0.005:
		tmp = math.cos(re)
	elif im <= 1.2e+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 <= 0.005)
		tmp = cos(re);
	elseif (im <= 1.2e+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 <= 0.005)
		tmp = cos(re);
	elseif (im <= 1.2e+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, 0.005], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.2e+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 < 0.0050000000000000001

   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 0.0050000000000000001 < im < 1.1999999999999999e77

   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 55.2%

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

   if 1.1999999999999999e77 < 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 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.005:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.05:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.05)
   (* (* 0.5 (cos re)) (fma im im 2.0))
   (if (<= im 1.2e+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 <= 0.05) {
		tmp = (0.5 * cos(re)) * fma(im, im, 2.0);
	} else if (im <= 1.2e+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 <= 0.05)
		tmp = Float64(Float64(0.5 * cos(re)) * fma(im, im, 2.0));
	elseif (im <= 1.2e+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, 0.05], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+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 < 0.050000000000000003

   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 83.1%

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

      1. +-commutative 83.1%

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

      2. unpow2 83.1%

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

      3. fma-define 83.1%

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

   5. Simplified 83.1%

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

   if 0.050000000000000003 < im < 1.1999999999999999e77

   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 55.2%

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

   if 1.1999999999999999e77 < 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 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.05:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 80000000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+80} \lor \neg \left(im \leq 1.9 \cdot 10^{+265}\right) \land im \leq 3.7 \cdot 10^{+284}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 80000000000000.0)
   (cos re)
   (if (or (<= im 1.1e+80) (and (not (<= im 1.9e+265)) (<= im 3.7e+284)))
     (+ 1.0 (* (pow re 2.0) -0.5))
     (* 0.041666666666666664 (pow im 4.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 80000000000000.0) {
		tmp = cos(re);
	} else if ((im <= 1.1e+80) || (!(im <= 1.9e+265) && (im <= 3.7e+284))) {
		tmp = 1.0 + (pow(re, 2.0) * -0.5);
	} 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 <= 80000000000000.0d0) then
        tmp = cos(re)
    else if ((im <= 1.1d+80) .or. (.not. (im <= 1.9d+265)) .and. (im <= 3.7d+284)) then
        tmp = 1.0d0 + ((re ** 2.0d0) * (-0.5d0))
    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 <= 80000000000000.0) {
		tmp = Math.cos(re);
	} else if ((im <= 1.1e+80) || (!(im <= 1.9e+265) && (im <= 3.7e+284))) {
		tmp = 1.0 + (Math.pow(re, 2.0) * -0.5);
	} else {
		tmp = 0.041666666666666664 * Math.pow(im, 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 80000000000000.0:
		tmp = math.cos(re)
	elif (im <= 1.1e+80) or (not (im <= 1.9e+265) and (im <= 3.7e+284)):
		tmp = 1.0 + (math.pow(re, 2.0) * -0.5)
	else:
		tmp = 0.041666666666666664 * math.pow(im, 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 80000000000000.0)
		tmp = cos(re);
	elseif ((im <= 1.1e+80) || (!(im <= 1.9e+265) && (im <= 3.7e+284)))
		tmp = Float64(1.0 + Float64((re ^ 2.0) * -0.5));
	else
		tmp = Float64(0.041666666666666664 * (im ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 80000000000000.0)
		tmp = cos(re);
	elseif ((im <= 1.1e+80) || (~((im <= 1.9e+265)) && (im <= 3.7e+284)))
		tmp = 1.0 + ((re ^ 2.0) * -0.5);
	else
		tmp = 0.041666666666666664 * (im ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 80000000000000.0], N[Cos[re], $MachinePrecision], If[Or[LessEqual[im, 1.1e+80], And[N[Not[LessEqual[im, 1.9e+265]], $MachinePrecision], LessEqual[im, 3.7e+284]]], N[(1.0 + N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8e13

   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 64.0%

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

   if 8e13 < im < 1.10000000000000001e80 or 1.90000000000000007e265 < im < 3.69999999999999998e284

   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 18.2%

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

      1. +-commutative 18.2%

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

      2. unpow2 18.2%

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

      3. fma-define 18.2%

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

   5. Simplified 18.2%

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

   6. Taylor expanded in re around 0 22.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(2 + {im}^{2}\right)\right) + 0.5 \cdot \left(2 + {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 22.2%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(2 + {im}^{2}\right)} + 0.5 \cdot \left(2 + {im}^{2}\right) \]

      2. distribute-rgt-out 37.2%

         \[\leadsto \color{blue}{\left(2 + {im}^{2}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. +-commutative 37.2%

         \[\leadsto \color{blue}{\left({im}^{2} + 2\right)} \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. unpow2 37.2%

         \[\leadsto \left(\color{blue}{im \cdot im} + 2\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      5. fma-undefine 37.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      6. *-commutative 37.2%

         \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(\color{blue}{{re}^{2} \cdot -0.25} + 0.5\right) \]

   8. Simplified 37.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right) \cdot \left({re}^{2} \cdot -0.25 + 0.5\right)} \]

   9. Taylor expanded in im around 0 32.4%

      \[\leadsto \color{blue}{2 \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-in 32.4%

          \[\leadsto \color{blue}{0.5 \cdot 2 + \left(-0.25 \cdot {re}^{2}\right) \cdot 2} \]

       2. *-commutative 32.4%

          \[\leadsto 0.5 \cdot 2 + \color{blue}{\left({re}^{2} \cdot -0.25\right)} \cdot 2 \]

       3. metadata-eval 32.4%

          \[\leadsto \color{blue}{1} + \left({re}^{2} \cdot -0.25\right) \cdot 2 \]

       4. associate-*l* 32.4%

          \[\leadsto 1 + \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot 2\right)} \]

       5. metadata-eval 32.4%

          \[\leadsto 1 + {re}^{2} \cdot \color{blue}{-0.5} \]

   11. Simplified 32.4%

       \[\leadsto \color{blue}{1 + {re}^{2} \cdot -0.5} \]

   if 1.10000000000000001e80 < im < 1.90000000000000007e265 or 3.69999999999999998e284 < 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%

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

   7. Taylor expanded in re around 0 68.2%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 80000000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+80} \lor \neg \left(im \leq 1.9 \cdot 10^{+265}\right) \land im \leq 3.7 \cdot 10^{+284}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.06 \cdot 10^{+27}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.06e+27)
   (cos re)
   (if (<= im 1.2e+77)
     (+ 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 <= 1.06e+27) {
		tmp = cos(re);
	} else if (im <= 1.2e+77) {
		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 <= 1.06d+27) then
        tmp = cos(re)
    else if (im <= 1.2d+77) 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 <= 1.06e+27) {
		tmp = Math.cos(re);
	} else if (im <= 1.2e+77) {
		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 <= 1.06e+27:
		tmp = math.cos(re)
	elif im <= 1.2e+77:
		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 <= 1.06e+27)
		tmp = cos(re);
	elseif (im <= 1.2e+77)
		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 <= 1.06e+27)
		tmp = cos(re);
	elseif (im <= 1.2e+77)
		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, 1.06e+27], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.2e+77], 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 < 1.05999999999999994e27

   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 61.9%

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

   if 1.05999999999999994e27 < im < 1.1999999999999999e77

   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 2.1%

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

   5. Taylor expanded in re around 0 31.0%

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

      1. *-commutative 31.0%

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

   7. Simplified 31.0%

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

   if 1.1999999999999999e77 < 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%

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

   7. Taylor expanded in re around 0 63.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.06 \cdot 10^{+27}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{+62}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.5e+62) (cos re) (* 0.041666666666666664 (pow im 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.5e+62) {
		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 <= 7.5d+62) 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 <= 7.5e+62) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.041666666666666664 * Math.pow(im, 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.5e+62:
		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 <= 7.5e+62)
		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 <= 7.5e+62)
		tmp = cos(re);
	else
		tmp = 0.041666666666666664 * (im ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.5e+62], N[Cos[re], $MachinePrecision], N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.49999999999999998e62

   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 59.9%

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

   if 7.49999999999999998e62 < 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 94.6%

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

      1. +-commutative 94.6%

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

      2. fma-define 94.6%

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

      3. associate-*r* 94.6%

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

      4. distribute-rgt-out 94.6%

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

      5. *-commutative 94.6%

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

   5. Simplified 94.6%

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

   6. Taylor expanded in im around inf 94.6%

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

   7. Taylor expanded in re around 0 60.2%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{+62}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.8% 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 48.8%

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

4. Final simplification 48.8%

   \[\leadsto \cos re \]
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 2.3%

   \[\leadsto \color{blue}{\log \left({1}^{\cos re}\right)} \]
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 10: 8.1% 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 7.7%

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

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 11: 28.7% 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 26.3%

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

   1. +-inverses 26.3%

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

   2. +-rgt-identity 26.3%

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

   3. *-inverses 26.3%

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

6. Simplified 26.3%

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

7. Final simplification 26.3%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))