math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 11

Speedup: 1.0×

• 49.4% 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: 85.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8 \lor \neg \left(im \leq 2.15 \cdot 10^{+151}\right):\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(0.5 \cdot im, im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 8.0) (not (<= im 2.15e+151)))
   (* (cos re) (fma (* 0.5 im) im 1.0))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 8.0) || !(im <= 2.15e+151)) {
		tmp = cos(re) * fma((0.5 * im), im, 1.0);
	} else {
		tmp = 0.5 * (exp(-im) + exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 8.0) || !(im <= 2.15e+151))
		tmp = Float64(cos(re) * fma(Float64(0.5 * im), im, 1.0));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 8.0], N[Not[LessEqual[im, 2.15e+151]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision]), $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 < 8 or 2.14999999999999991e151 < 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 88.0%

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

      1. *-lft-identity 88.0%

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

      2. associate-*r* 88.0%

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

      3. distribute-rgt-out 88.0%

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

   5. Simplified 88.0%

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

      1. +-commutative 88.0%

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

      2. unpow2 88.0%

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

      3. associate-*r* 88.0%

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

      4. fma-define 88.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right)} \]

   7. Applied egg-rr 88.0%

      \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right)} \]

   if 8 < im < 2.14999999999999991e151

   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 75.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8 \lor \neg \left(im \leq 2.15 \cdot 10^{+151}\right):\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(0.5 \cdot im, im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8.0)
   (cos re)
   (if (<= im 2.15e+151)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.0) {
		tmp = cos(re);
	} else if (im <= 2.15e+151) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 8.0d0) then
        tmp = cos(re)
    else if (im <= 2.15d+151) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8.0) {
		tmp = Math.cos(re);
	} else if (im <= 2.15e+151) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8.0:
		tmp = math.cos(re)
	elif im <= 2.15e+151:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8.0)
		tmp = cos(re);
	elseif (im <= 2.15e+151)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8.0)
		tmp = cos(re);
	elseif (im <= 2.15e+151)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.15e+151], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8

   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 67.0%

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

   if 8 < im < 2.14999999999999991e151

   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 75.7%

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

   if 2.14999999999999991e151 < 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 97.4%

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

      1. *-lft-identity 97.4%

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

      2. associate-*r* 97.4%

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

      3. distribute-rgt-out 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in im around inf 97.4%

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

      1. associate-*r* 97.4%

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

   8. Simplified 97.4%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8.0) (cos re) (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.0) {
		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 <= 8.0d0) 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 <= 8.0) {
		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 <= 8.0:
		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 <= 8.0)
		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 <= 8.0)
		tmp = cos(re);
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.0], 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 < 8

   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 67.0%

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

   if 8 < 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 re around 0 70.4%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+39}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+239}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7e+39)
   (cos re)
   (if (<= im 8.5e+239)
     (+ 0.25 (* 0.25 (pow re 2.0)))
     (+ 1.0 (* (pow re 2.0) -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7e+39) {
		tmp = cos(re);
	} else if (im <= 8.5e+239) {
		tmp = 0.25 + (0.25 * pow(re, 2.0));
	} else {
		tmp = 1.0 + (pow(re, 2.0) * -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 <= 7d+39) then
        tmp = cos(re)
    else if (im <= 8.5d+239) then
        tmp = 0.25d0 + (0.25d0 * (re ** 2.0d0))
    else
        tmp = 1.0d0 + ((re ** 2.0d0) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7e+39) {
		tmp = Math.cos(re);
	} else if (im <= 8.5e+239) {
		tmp = 0.25 + (0.25 * Math.pow(re, 2.0));
	} else {
		tmp = 1.0 + (Math.pow(re, 2.0) * -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7e+39:
		tmp = math.cos(re)
	elif im <= 8.5e+239:
		tmp = 0.25 + (0.25 * math.pow(re, 2.0))
	else:
		tmp = 1.0 + (math.pow(re, 2.0) * -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7e+39)
		tmp = cos(re);
	elseif (im <= 8.5e+239)
		tmp = Float64(0.25 + Float64(0.25 * (re ^ 2.0)));
	else
		tmp = Float64(1.0 + Float64((re ^ 2.0) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7e+39)
		tmp = cos(re);
	elseif (im <= 8.5e+239)
		tmp = 0.25 + (0.25 * (re ^ 2.0));
	else
		tmp = 1.0 + ((re ^ 2.0) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7e+39], N[Cos[re], $MachinePrecision], If[LessEqual[im, 8.5e+239], N[(0.25 + N[(0.25 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 7.0000000000000003e39

   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 7.0000000000000003e39 < im < 8.50000000000000021e239

   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 2.5%

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

   5. Taylor expanded in re around 0 14.0%

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

      1. *-commutative 14.0%

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

   7. Simplified 14.0%

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

   if 8.50000000000000021e239 < 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 3.1%

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

   4. Taylor expanded in re around 0 47.2%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+39}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+239}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5 \cdot 10^{+39}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+126}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5e+39)
   (cos re)
   (if (<= im 2e+126)
     (+ 0.25 (* 0.25 (pow re 2.0)))
     (+ 1.0 (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5e+39) {
		tmp = cos(re);
	} else if (im <= 2e+126) {
		tmp = 0.25 + (0.25 * pow(re, 2.0));
	} else {
		tmp = 1.0 + (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 5d+39) then
        tmp = cos(re)
    else if (im <= 2d+126) then
        tmp = 0.25d0 + (0.25d0 * (re ** 2.0d0))
    else
        tmp = 1.0d0 + (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5e+39) {
		tmp = Math.cos(re);
	} else if (im <= 2e+126) {
		tmp = 0.25 + (0.25 * Math.pow(re, 2.0));
	} else {
		tmp = 1.0 + (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5e+39:
		tmp = math.cos(re)
	elif im <= 2e+126:
		tmp = 0.25 + (0.25 * math.pow(re, 2.0))
	else:
		tmp = 1.0 + (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5e+39)
		tmp = cos(re);
	elseif (im <= 2e+126)
		tmp = Float64(0.25 + Float64(0.25 * (re ^ 2.0)));
	else
		tmp = Float64(1.0 + Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5e+39)
		tmp = cos(re);
	elseif (im <= 2e+126)
		tmp = 0.25 + (0.25 * (re ^ 2.0));
	else
		tmp = 1.0 + (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5e+39], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2e+126], N[(0.25 + N[(0.25 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.00000000000000015e39

   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 5.00000000000000015e39 < im < 1.99999999999999985e126

   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 2.6%

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

   5. Taylor expanded in re around 0 14.9%

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

      1. *-commutative 14.9%

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

   7. Simplified 14.9%

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

   if 1.99999999999999985e126 < 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 88.2%

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

      1. *-lft-identity 88.2%

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

      2. associate-*r* 88.2%

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

      3. distribute-rgt-out 88.2%

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

   5. Simplified 88.2%

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

   6. Taylor expanded in re around 0 58.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {im}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5 \cdot 10^{+39}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+126}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+40}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.5e+40) (cos re) (+ 0.25 (* 0.25 (pow re 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+40) {
		tmp = cos(re);
	} else {
		tmp = 0.25 + (0.25 * pow(re, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.5d+40) then
        tmp = cos(re)
    else
        tmp = 0.25d0 + (0.25d0 * (re ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+40) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.25 + (0.25 * Math.pow(re, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.5e+40:
		tmp = math.cos(re)
	else:
		tmp = 0.25 + (0.25 * math.pow(re, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.5e+40)
		tmp = cos(re);
	else
		tmp = Float64(0.25 + Float64(0.25 * (re ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.5e+40)
		tmp = cos(re);
	else
		tmp = 0.25 + (0.25 * (re ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.5e+40], N[Cos[re], $MachinePrecision], N[(0.25 + N[(0.25 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.5000000000000001e40

   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 1.5000000000000001e40 < im

   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 11.8%

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

      1. *-commutative 11.8%

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

   7. Simplified 11.8%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+40}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.3% 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 49.3%

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

4. Final simplification 49.3%

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

Alternative 9: 3.4% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -4.0)

C

double code(double re, double im) {
	return -4.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -4.0d0
end function

Java

public static double code(double re, double im) {
	return -4.0;
}

Python

def code(re, im):
	return -4.0

Julia

function code(re, im)
	return -4.0
end

MATLAB

function tmp = code(re, im)
	tmp = -4.0;
end

Wolfram

code[re_, im_] := -4.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 49.3%

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

5. Applied egg-rr 4.1%

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

   1. *-commutative 4.1%

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

7. Simplified 4.1%

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

8. Taylor expanded in re around 0 3.8%

   \[\leadsto \color{blue}{-4} \]

9. Final simplification 3.8%

   \[\leadsto -4 \]
10. Add Preprocessing

Alternative 10: 8.0% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.25)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

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

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 7.1%

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

5. Taylor expanded in re around 0 7.2%

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

6. Final simplification 7.2%

   \[\leadsto 0.25 \]
7. Add Preprocessing

Alternative 11: 28.3% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 49.3%

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

4. Taylor expanded in re around 0 26.4%

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

5. Final simplification 26.4%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))