math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 10

Speedup: 1.0×

• 49.1% 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. Add Preprocessing

Alternative 2: 86.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.21:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.21)
   (* (* 0.5 (cos re)) (fma im im 2.0))
   (if (<= im 1.15e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.21) {
		tmp = (0.5 * cos(re)) * fma(im, im, 2.0);
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.21)
		tmp = Float64(Float64(0.5 * cos(re)) * fma(im, im, 2.0));
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.21], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.209999999999999992

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.1%

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

      1. +-commutative 86.1%

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

      2. unpow2 86.1%

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

      3. fma-define 86.1%

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

   5. Simplified 86.1%

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

   if 0.209999999999999992 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 76.5%

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

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.21:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.21:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.21)
   (cos re)
   (if (<= im 1.15e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.21) {
		tmp = cos(re);
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.21d0) then
        tmp = cos(re)
    else if (im <= 1.15d+77) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.21) {
		tmp = Math.cos(re);
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.21:
		tmp = math.cos(re)
	elif im <= 1.15e+77:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.21)
		tmp = cos(re);
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.21)
		tmp = cos(re);
	elseif (im <= 1.15e+77)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.21], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.209999999999999992

   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 60.9%

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

   if 0.209999999999999992 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 76.5%

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

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.21:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\sqrt[3]{{im}^{12} \cdot 7.233796296296296 \cdot 10^{-5}}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8.5e+26)
   (cos re)
   (if (<= im 1.15e+77)
     (cbrt (* (pow im 12.0) 7.233796296296296e-5))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.5e+26) {
		tmp = cos(re);
	} else if (im <= 1.15e+77) {
		tmp = cbrt((pow(im, 12.0) * 7.233796296296296e-5));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8.5e+26) {
		tmp = Math.cos(re);
	} else if (im <= 1.15e+77) {
		tmp = Math.cbrt((Math.pow(im, 12.0) * 7.233796296296296e-5));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8.5e+26)
		tmp = cos(re);
	elseif (im <= 1.15e+77)
		tmp = cbrt(Float64((im ^ 12.0) * 7.233796296296296e-5));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.5e+26], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.15e+77], N[Power[N[(N[Power[im, 12.0], $MachinePrecision] * 7.233796296296296e-5), $MachinePrecision], 1/3], $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 < 8.5e26

   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 59.2%

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

   if 8.5e26 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 5.6%

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

      1. *-commutative 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in im around inf 5.6%

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

      1. associate-*r* 5.6%

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

   8. Simplified 5.6%

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

   9. Taylor expanded in re around 0 4.6%

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

       1. add-cbrt-cube 80.0%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)}} \]

       2. pow1/3 80.0%

          \[\leadsto \color{blue}{{\left(\left(\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\right)}^{0.3333333333333333}} \]

       3. pow3 80.0%

          \[\leadsto {\color{blue}{\left({\left(0.041666666666666664 \cdot {im}^{4}\right)}^{3}\right)}}^{0.3333333333333333} \]

       4. *-commutative 80.0%

          \[\leadsto {\left({\color{blue}{\left({im}^{4} \cdot 0.041666666666666664\right)}}^{3}\right)}^{0.3333333333333333} \]

       5. unpow-prod-down 80.0%

          \[\leadsto {\color{blue}{\left({\left({im}^{4}\right)}^{3} \cdot {0.041666666666666664}^{3}\right)}}^{0.3333333333333333} \]

       6. pow-pow 80.0%

          \[\leadsto {\left(\color{blue}{{im}^{\left(4 \cdot 3\right)}} \cdot {0.041666666666666664}^{3}\right)}^{0.3333333333333333} \]

       7. metadata-eval 80.0%

          \[\leadsto {\left({im}^{\color{blue}{12}} \cdot {0.041666666666666664}^{3}\right)}^{0.3333333333333333} \]

       8. metadata-eval 80.0%

          \[\leadsto {\left({im}^{12} \cdot \color{blue}{7.233796296296296 \cdot 10^{-5}}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{{\left({im}^{12} \cdot 7.233796296296296 \cdot 10^{-5}\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 80.0%

          \[\leadsto \color{blue}{\sqrt[3]{{im}^{12} \cdot 7.233796296296296 \cdot 10^{-5}}} \]

   13. Simplified 80.0%

       \[\leadsto \color{blue}{\sqrt[3]{{im}^{12} \cdot 7.233796296296296 \cdot 10^{-5}}} \]

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\sqrt[3]{{im}^{12} \cdot 7.233796296296296 \cdot 10^{-5}}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{im}^{12} \cdot 7.233796296296296 \cdot 10^{-5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8.5e+26) (cos re) (cbrt (* (pow im 12.0) 7.233796296296296e-5))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.5e+26) {
		tmp = cos(re);
	} else {
		tmp = cbrt((pow(im, 12.0) * 7.233796296296296e-5));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8.5e+26) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.cbrt((Math.pow(im, 12.0) * 7.233796296296296e-5));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8.5e+26)
		tmp = cos(re);
	else
		tmp = cbrt(Float64((im ^ 12.0) * 7.233796296296296e-5));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.5e+26], N[Cos[re], $MachinePrecision], N[Power[N[(N[Power[im, 12.0], $MachinePrecision] * 7.233796296296296e-5), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.5e26

   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 59.2%

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

   if 8.5e26 < 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 84.0%

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

      1. *-commutative 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in im around inf 84.0%

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

      1. associate-*r* 84.0%

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

   8. Simplified 84.0%

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

   9. Taylor expanded in re around 0 58.4%

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

       1. add-cbrt-cube 71.2%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)}} \]

       2. pow1/3 71.2%

          \[\leadsto \color{blue}{{\left(\left(\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\right)}^{0.3333333333333333}} \]

       3. pow3 71.2%

          \[\leadsto {\color{blue}{\left({\left(0.041666666666666664 \cdot {im}^{4}\right)}^{3}\right)}}^{0.3333333333333333} \]

       4. *-commutative 71.2%

          \[\leadsto {\left({\color{blue}{\left({im}^{4} \cdot 0.041666666666666664\right)}}^{3}\right)}^{0.3333333333333333} \]

       5. unpow-prod-down 71.2%

          \[\leadsto {\color{blue}{\left({\left({im}^{4}\right)}^{3} \cdot {0.041666666666666664}^{3}\right)}}^{0.3333333333333333} \]

       6. pow-pow 71.2%

          \[\leadsto {\left(\color{blue}{{im}^{\left(4 \cdot 3\right)}} \cdot {0.041666666666666664}^{3}\right)}^{0.3333333333333333} \]

       7. metadata-eval 71.2%

          \[\leadsto {\left({im}^{\color{blue}{12}} \cdot {0.041666666666666664}^{3}\right)}^{0.3333333333333333} \]

       8. metadata-eval 71.2%

          \[\leadsto {\left({im}^{12} \cdot \color{blue}{7.233796296296296 \cdot 10^{-5}}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 71.2%

       \[\leadsto \color{blue}{{\left({im}^{12} \cdot 7.233796296296296 \cdot 10^{-5}\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 71.2%

          \[\leadsto \color{blue}{\sqrt[3]{{im}^{12} \cdot 7.233796296296296 \cdot 10^{-5}}} \]

   13. Simplified 71.2%

       \[\leadsto \color{blue}{\sqrt[3]{{im}^{12} \cdot 7.233796296296296 \cdot 10^{-5}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 65.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.5e+26) (cos re) (* 0.041666666666666664 (pow im 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.5e+26) {
		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 <= 9.5d+26) 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 <= 9.5e+26) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.041666666666666664 * Math.pow(im, 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.5e+26:
		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 <= 9.5e+26)
		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 <= 9.5e+26)
		tmp = cos(re);
	else
		tmp = 0.041666666666666664 * (im ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.5e+26], N[Cos[re], $MachinePrecision], N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.50000000000000054e26

   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 59.2%

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

   if 9.50000000000000054e26 < 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 84.0%

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

      1. *-commutative 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in im around inf 84.0%

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

      1. associate-*r* 84.0%

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

   8. Simplified 84.0%

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

   9. Taylor expanded in re around 0 58.4%

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

Alternative 7: 51.8% 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 46.3%

   \[\leadsto \color{blue}{\cos re} \]
4. Add Preprocessing

Alternative 8: 28.6% 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 25.5%

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

   1. +-inverses 25.5%

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

   2. +-rgt-identity 25.5%

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

   3. *-inverses 25.5%

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

6. Simplified 25.5%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Alternative 9: 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 7.5%

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

5. Taylor expanded in re around 0 7.5%

   \[\leadsto \color{blue}{0.25} \]
6. Add Preprocessing

Alternative 10: 2.4% 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. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))