math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 9

Speedup: 1.0×

• 50.5% 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. Final simplification 100.0%

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

Alternative 2: 73.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.025:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.025)
   (cos re)
   (if (<= im 1.2e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (pow im 4.0) (* (cos re) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.025) {
		tmp = cos(re);
	} else if (im <= 1.2e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = pow(im, 4.0) * (cos(re) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.025d0) then
        tmp = cos(re)
    else if (im <= 1.2d+77) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = (im ** 4.0d0) * (cos(re) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.025) {
		tmp = Math.cos(re);
	} else if (im <= 1.2e+77) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.pow(im, 4.0) * (Math.cos(re) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.025:
		tmp = math.cos(re)
	elif im <= 1.2e+77:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.pow(im, 4.0) * (math.cos(re) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.025)
		tmp = cos(re);
	elseif (im <= 1.2e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64((im ^ 4.0) * Float64(cos(re) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.025)
		tmp = cos(re);
	elseif (im <= 1.2e+77)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = (im ^ 4.0) * (cos(re) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.025], 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[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.025000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 62.3%

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

   if 0.025000000000000001 < im < 1.1999999999999999e77

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 66.7%

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

   if 1.1999999999999999e77 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\cos re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]

   3. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.025:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 3: 86.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.094:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + {im}^{2}\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.094)
   (* (* 0.5 (cos re)) (+ 2.0 (pow im 2.0)))
   (if (<= im 1.2e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (pow im 4.0) (* (cos re) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.094) {
		tmp = (0.5 * cos(re)) * (2.0 + pow(im, 2.0));
	} else if (im <= 1.2e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = pow(im, 4.0) * (cos(re) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.094d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im ** 2.0d0))
    else if (im <= 1.2d+77) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = (im ** 4.0d0) * (cos(re) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.094) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + Math.pow(im, 2.0));
	} else if (im <= 1.2e+77) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.pow(im, 4.0) * (Math.cos(re) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.094:
		tmp = (0.5 * math.cos(re)) * (2.0 + math.pow(im, 2.0))
	elif im <= 1.2e+77:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.pow(im, 4.0) * (math.cos(re) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.094)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + (im ^ 2.0)));
	elseif (im <= 1.2e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64((im ^ 4.0) * Float64(cos(re) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.094)
		tmp = (0.5 * cos(re)) * (2.0 + (im ^ 2.0));
	elseif (im <= 1.2e+77)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = (im ^ 4.0) * (cos(re) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.094], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+77], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.094

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.6%

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

   if 0.094 < im < 1.1999999999999999e77

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 66.7%

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

   if 1.1999999999999999e77 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\cos re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]

   3. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.094:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + {im}^{2}\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 4: 68.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.037:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.037) (cos re) (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.037) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (exp(-im) + exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.037d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.037) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.037:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.037)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.037)
		tmp = cos(re);
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.037], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0369999999999999982

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 62.3%

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

   if 0.0369999999999999982 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 67.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.037:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 5: 64.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 12500:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+79}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 12500.0)
   (cos re)
   (if (<= im 1.1e+79)
     (+ 1.0 (* re (* re -0.5)))
     (* (pow im 4.0) 0.041666666666666664))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 12500.0) {
		tmp = cos(re);
	} else if (im <= 1.1e+79) {
		tmp = 1.0 + (re * (re * -0.5));
	} else {
		tmp = pow(im, 4.0) * 0.041666666666666664;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 12500.0d0) then
        tmp = cos(re)
    else if (im <= 1.1d+79) then
        tmp = 1.0d0 + (re * (re * (-0.5d0)))
    else
        tmp = (im ** 4.0d0) * 0.041666666666666664d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 12500.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.1e+79) {
		tmp = 1.0 + (re * (re * -0.5));
	} else {
		tmp = Math.pow(im, 4.0) * 0.041666666666666664;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 12500.0:
		tmp = math.cos(re)
	elif im <= 1.1e+79:
		tmp = 1.0 + (re * (re * -0.5))
	else:
		tmp = math.pow(im, 4.0) * 0.041666666666666664
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 12500.0)
		tmp = cos(re);
	elseif (im <= 1.1e+79)
		tmp = Float64(1.0 + Float64(re * Float64(re * -0.5)));
	else
		tmp = Float64((im ^ 4.0) * 0.041666666666666664);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 12500.0)
		tmp = cos(re);
	elseif (im <= 1.1e+79)
		tmp = 1.0 + (re * (re * -0.5));
	else
		tmp = (im ^ 4.0) * 0.041666666666666664;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 12500.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.1e+79], N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 12500

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 62.0%

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

   if 12500 < im < 1.0999999999999999e79

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.1%

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

   3. Taylor expanded in re around 0 35.4%

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

   5. Simplified 35.4%

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot -0.5} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 43.2%

         \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({re}^{2} \cdot -0.5\right)\right)} \]

   7. Applied egg-rr 43.2%

      \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({re}^{2} \cdot -0.5\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1 35.4%

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

      2. *-commutative 35.4%

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

      3. unpow2 35.4%

         \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)} \]

      4. associate-*r* 35.4%

         \[\leadsto 1 + \color{blue}{\left(-0.5 \cdot re\right) \cdot re} \]

   9. Applied egg-rr 35.4%

      \[\leadsto 1 + \color{blue}{\left(-0.5 \cdot re\right) \cdot re} \]

   if 1.0999999999999999e79 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\cos re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]

   3. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 68.5%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 12500:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+79}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \end{array} \]

Alternative 6: 53.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 10500:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 10500.0) (cos re) (+ 1.0 (* re (* re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 10500.0) {
		tmp = cos(re);
	} else {
		tmp = 1.0 + (re * (re * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 10500.0d0) then
        tmp = cos(re)
    else
        tmp = 1.0d0 + (re * (re * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 10500.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.0 + (re * (re * -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 10500.0:
		tmp = math.cos(re)
	else:
		tmp = 1.0 + (re * (re * -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 10500.0)
		tmp = cos(re);
	else
		tmp = Float64(1.0 + Float64(re * Float64(re * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 10500.0)
		tmp = cos(re);
	else
		tmp = 1.0 + (re * (re * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 10500.0], N[Cos[re], $MachinePrecision], N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 10500

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 62.0%

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

   if 10500 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.1%

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

   3. Taylor expanded in re around 0 16.4%

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

   5. Simplified 16.4%

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot -0.5} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 34.9%

         \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({re}^{2} \cdot -0.5\right)\right)} \]

   7. Applied egg-rr 34.9%

      \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({re}^{2} \cdot -0.5\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1 16.4%

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

      2. *-commutative 16.4%

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

      3. unpow2 16.4%

         \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)} \]

      4. associate-*r* 16.4%

         \[\leadsto 1 + \color{blue}{\left(-0.5 \cdot re\right) \cdot re} \]

   9. Applied egg-rr 16.4%

      \[\leadsto 1 + \color{blue}{\left(-0.5 \cdot re\right) \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10500:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \end{array} \]

Alternative 7: 30.9% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 10500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 10500.0) 1.0 (+ 1.0 (* re (* re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 10500.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (re * (re * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 10500.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + (re * (re * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 10500.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (re * (re * -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 10500.0:
		tmp = 1.0
	else:
		tmp = 1.0 + (re * (re * -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 10500.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(re * Float64(re * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 10500.0)
		tmp = 1.0;
	else
		tmp = 1.0 + (re * (re * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 10500.0], 1.0, N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 10500

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 62.0%

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

   3. Taylor expanded in re around 0 30.8%

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

   if 10500 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.1%

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

   3. Taylor expanded in re around 0 16.4%

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

   5. Simplified 16.4%

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot -0.5} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 34.9%

         \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({re}^{2} \cdot -0.5\right)\right)} \]

   7. Applied egg-rr 34.9%

      \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({re}^{2} \cdot -0.5\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1 16.4%

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

      2. *-commutative 16.4%

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

      3. unpow2 16.4%

         \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)} \]

      4. associate-*r* 16.4%

         \[\leadsto 1 + \color{blue}{\left(-0.5 \cdot re\right) \cdot re} \]

   9. Applied egg-rr 16.4%

      \[\leadsto 1 + \color{blue}{\left(-0.5 \cdot re\right) \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \end{array} \]

Alternative 8: 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%

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

3. Applied egg-rr 7.5%

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

4. Taylor expanded in re around 0 7.5%

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

5. Final simplification 7.5%

   \[\leadsto 0.25 \]

Alternative 9: 27.9% 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. Taylor expanded in im around 0 46.8%

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

3. Taylor expanded in re around 0 23.5%

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

4. Final simplification 23.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023326 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))