math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.1s

Alternatives: 16

Speedup: 1.0×

• 48.6% 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: 72.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.02e-8)
   (cos re)
   (if (<= im 1e+103)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (*
      0.5
      (*
       (cos re)
       (- (+ 2.0 (* im (+ 1.0 (* im (* im 0.16666666666666666))))) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = cos(re);
	} else if (im <= 1e+103) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - 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 <= 1.02d-8) then
        tmp = cos(re)
    else if (im <= 1d+103) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = 0.5d0 * (cos(re) * ((2.0d0 + (im * (1.0d0 + (im * (im * 0.16666666666666666d0))))) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = Math.cos(re);
	} else if (im <= 1e+103) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.5 * (Math.cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.02e-8:
		tmp = math.cos(re)
	elif im <= 1e+103:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.5 * (math.cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.02e-8)
		tmp = cos(re);
	elseif (im <= 1e+103)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(im * 0.16666666666666666))))) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.02e-8)
		tmp = cos(re);
	elseif (im <= 1e+103)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.02e-8], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1e+103], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(2.0 + N[(im * N[(1.0 + N[(im * N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.02000000000000003e-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 72.1%

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

   4. Taylor expanded in re around inf 72.1%

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

   if 1.02000000000000003e-8 < im < 1e103

   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 81.8%

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

   if 1e103 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   9. Taylor expanded in re around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) - im\right)\right)} \]

   10. Taylor expanded in im around inf 100.0%

       \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(0.16666666666666666 \cdot im\right)}\right)\right) - im\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(im \cdot 0.16666666666666666\right)}\right)\right) - im\right)\right) \]

   12. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(im \cdot 0.16666666666666666\right)}\right)\right) - im\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + \left(1 - im\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp im) (- 1.0 im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(im) + (1.0 - 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) + (1.0d0 - im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(im) + (1.0 - im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(im) + (1.0 - im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(im) + Float64(1.0 - im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(im) + (1.0 - im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[(1.0 - 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. Taylor expanded in im around 0 79.7%

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

   1. neg-mul-1 79.7%

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

   2. unsub-neg 79.7%

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

5. Simplified 79.7%

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

6. Final simplification 79.7%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + \left(1 - im\right)\right) \]
7. Add Preprocessing

Alternative 4: 72.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.02e-8)
   (cos re)
   (if (<= im 1e+103)
     (*
      0.5
      (+
       (exp im)
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
     (*
      0.5
      (*
       (cos re)
       (- (+ 2.0 (* im (+ 1.0 (* im (* im 0.16666666666666666))))) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = cos(re);
	} else if (im <= 1e+103) {
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - 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 <= 1.02d-8) then
        tmp = cos(re)
    else if (im <= 1d+103) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = 0.5d0 * (cos(re) * ((2.0d0 + (im * (1.0d0 + (im * (im * 0.16666666666666666d0))))) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = Math.cos(re);
	} else if (im <= 1e+103) {
		tmp = 0.5 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = 0.5 * (Math.cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.02e-8:
		tmp = math.cos(re)
	elif im <= 1e+103:
		tmp = 0.5 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = 0.5 * (math.cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.02e-8)
		tmp = cos(re);
	elseif (im <= 1e+103)
		tmp = Float64(0.5 * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(im * 0.16666666666666666))))) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.02e-8)
		tmp = cos(re);
	elseif (im <= 1e+103)
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.02e-8], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1e+103], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(2.0 + N[(im * N[(1.0 + N[(im * N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.02000000000000003e-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 72.1%

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

   4. Taylor expanded in re around inf 72.1%

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

   if 1.02000000000000003e-8 < im < 1e103

   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 81.8%

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

   4. Taylor expanded in im around 0 81.1%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + -0.16666666666666666 \cdot im\right) - 1\right)\right)} + e^{im}\right) \]

   if 1e103 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   9. Taylor expanded in re around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) - im\right)\right)} \]

   10. Taylor expanded in im around inf 100.0%

       \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(0.16666666666666666 \cdot im\right)}\right)\right) - im\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(im \cdot 0.16666666666666666\right)}\right)\right) - im\right)\right) \]

   12. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(im \cdot 0.16666666666666666\right)}\right)\right) - im\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 + 0.5 \cdot \left(e^{im} - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.02e-8)
   (cos re)
   (if (<= im 1e+103)
     (+ 0.5 (* 0.5 (- (exp im) im)))
     (*
      0.5
      (*
       (cos re)
       (- (+ 2.0 (* im (+ 1.0 (* im (* im 0.16666666666666666))))) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = cos(re);
	} else if (im <= 1e+103) {
		tmp = 0.5 + (0.5 * (exp(im) - im));
	} else {
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - 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 <= 1.02d-8) then
        tmp = cos(re)
    else if (im <= 1d+103) then
        tmp = 0.5d0 + (0.5d0 * (exp(im) - im))
    else
        tmp = 0.5d0 * (cos(re) * ((2.0d0 + (im * (1.0d0 + (im * (im * 0.16666666666666666d0))))) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = Math.cos(re);
	} else if (im <= 1e+103) {
		tmp = 0.5 + (0.5 * (Math.exp(im) - im));
	} else {
		tmp = 0.5 * (Math.cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.02e-8:
		tmp = math.cos(re)
	elif im <= 1e+103:
		tmp = 0.5 + (0.5 * (math.exp(im) - im))
	else:
		tmp = 0.5 * (math.cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.02e-8)
		tmp = cos(re);
	elseif (im <= 1e+103)
		tmp = Float64(0.5 + Float64(0.5 * Float64(exp(im) - im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(im * 0.16666666666666666))))) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.02e-8)
		tmp = cos(re);
	elseif (im <= 1e+103)
		tmp = 0.5 + (0.5 * (exp(im) - im));
	else
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.02e-8], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1e+103], N[(0.5 + N[(0.5 * N[(N[Exp[im], $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(2.0 + N[(im * N[(1.0 + N[(im * N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.02000000000000003e-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 72.1%

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

   4. Taylor expanded in re around inf 72.1%

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

   if 1.02000000000000003e-8 < im < 1e103

   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 97.7%

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

      1. neg-mul-1 97.7%

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

      2. unsub-neg 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in re around 0 79.6%

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

      1. associate--l+ 79.6%

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

      2. distribute-rgt-in 79.6%

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

      3. metadata-eval 79.6%

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

   8. Simplified 79.6%

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

   if 1e103 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   9. Taylor expanded in re around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) - im\right)\right)} \]

   10. Taylor expanded in im around inf 100.0%

       \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(0.16666666666666666 \cdot im\right)}\right)\right) - im\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(im \cdot 0.16666666666666666\right)}\right)\right) - im\right)\right) \]

   12. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(im \cdot 0.16666666666666666\right)}\right)\right) - im\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 + 0.5 \cdot \left(e^{im} - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 + 0.5 \cdot \left(e^{im} - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.02e-8)
   (cos re)
   (if (<= im 1.9e+154)
     (+ 0.5 (* 0.5 (- (exp im) im)))
     (* 0.5 (* (cos re) (- (+ 2.0 (* im (+ 1.0 (* 0.5 im)))) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = cos(re);
	} else if (im <= 1.9e+154) {
		tmp = 0.5 + (0.5 * (exp(im) - im));
	} else {
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - 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 <= 1.02d-8) then
        tmp = cos(re)
    else if (im <= 1.9d+154) then
        tmp = 0.5d0 + (0.5d0 * (exp(im) - im))
    else
        tmp = 0.5d0 * (cos(re) * ((2.0d0 + (im * (1.0d0 + (0.5d0 * im)))) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = Math.cos(re);
	} else if (im <= 1.9e+154) {
		tmp = 0.5 + (0.5 * (Math.exp(im) - im));
	} else {
		tmp = 0.5 * (Math.cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.02e-8:
		tmp = math.cos(re)
	elif im <= 1.9e+154:
		tmp = 0.5 + (0.5 * (math.exp(im) - im))
	else:
		tmp = 0.5 * (math.cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.02e-8)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = Float64(0.5 + Float64(0.5 * Float64(exp(im) - im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.02e-8)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = 0.5 + (0.5 * (exp(im) - im));
	else
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.02e-8], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(0.5 + N[(0.5 * N[(N[Exp[im], $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.02000000000000003e-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 72.1%

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

   4. Taylor expanded in re around inf 72.1%

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

   if 1.02000000000000003e-8 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.4%

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

      1. neg-mul-1 98.4%

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

      2. unsub-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in re around 0 79.0%

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

      1. associate--l+ 79.0%

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

      2. distribute-rgt-in 79.0%

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

      3. metadata-eval 79.0%

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

   8. Simplified 79.0%

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

   if 1.8999999999999999e154 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   9. Taylor expanded in re around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) - im\right)\right)} \]

   10. Taylor expanded in im around 0 100.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 + 0.5 \cdot \left(e^{im} - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot \left(e^{im} - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.02e-8) (cos re) (+ 0.5 (* 0.5 (- (exp im) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = cos(re);
	} else {
		tmp = 0.5 + (0.5 * (exp(im) - 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 <= 1.02d-8) then
        tmp = cos(re)
    else
        tmp = 0.5d0 + (0.5d0 * (exp(im) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 + (0.5 * (Math.exp(im) - im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.02e-8:
		tmp = math.cos(re)
	else:
		tmp = 0.5 + (0.5 * (math.exp(im) - im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.02e-8)
		tmp = cos(re);
	else
		tmp = Float64(0.5 + Float64(0.5 * Float64(exp(im) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.02e-8)
		tmp = cos(re);
	else
		tmp = 0.5 + (0.5 * (exp(im) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.02e-8], N[Cos[re], $MachinePrecision], N[(0.5 + N[(0.5 * N[(N[Exp[im], $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.02000000000000003e-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 72.1%

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

   4. Taylor expanded in re around inf 72.1%

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

   if 1.02000000000000003e-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 im around 0 99.3%

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

      1. neg-mul-1 99.3%

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

      2. unsub-neg 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in re around 0 76.9%

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

      1. associate--l+ 76.9%

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

      2. distribute-rgt-in 76.9%

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

      3. metadata-eval 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot \left(e^{im} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{+28}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.8e+28)
   (cos re)
   (*
    0.5
    (+
     (- 1.0 im)
     (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.8e+28) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.8d+28) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.8e+28) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.8e+28:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.8e+28)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.8e+28)
		tmp = cos(re);
	else
		tmp = 0.5 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.8e+28], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.8e28

   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 69.8%

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

   4. Taylor expanded in re around inf 69.8%

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

   if 6.8e28 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 76.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 76.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   8. Simplified 76.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   9. Taylor expanded in re around 0 57.8%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 44.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.8e+166)
   (*
    0.5
    (+
     (- 1.0 im)
     (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))
   (+ 1.0 (* -0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+166) {
		tmp = 0.5 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else {
		tmp = 1.0 + (-0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.8d+166) then
        tmp = 0.5d0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    else
        tmp = 1.0d0 + ((-0.5d0) * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+166) {
		tmp = 0.5 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else {
		tmp = 1.0 + (-0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.8e+166:
		tmp = 0.5 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))
	else:
		tmp = 1.0 + (-0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.8e+166)
		tmp = Float64(0.5 * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.8e+166)
		tmp = 0.5 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	else
		tmp = 1.0 + (-0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.8e+166], N[(0.5 * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.8000000000000003e166

   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 79.2%

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

      1. neg-mul-1 79.2%

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

      2. unsub-neg 79.2%

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

   5. Simplified 79.2%

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

   6. Taylor expanded in im around 0 70.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 70.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   8. Simplified 70.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   9. Taylor expanded in re around 0 46.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right) \]

   if 5.8000000000000003e166 < re

   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%

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

   4. Taylor expanded in re around 0 25.4%

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

      1. unpow2 25.4%

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

   6. Applied egg-rr 25.4%

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

Alternative 10: 44.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.8e+166)
   (*
    0.5
    (- (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))) im))
   (+ 1.0 (* -0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+166) {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
	} else {
		tmp = 1.0 + (-0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.8d+166) then
        tmp = 0.5d0 * ((2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) - im)
    else
        tmp = 1.0d0 + ((-0.5d0) * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+166) {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
	} else {
		tmp = 1.0 + (-0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.8e+166:
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im)
	else:
		tmp = 1.0 + (-0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.8e+166)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))) - im));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.8e+166)
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
	else
		tmp = 1.0 + (-0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.8e+166], N[(0.5 * N[(N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.8000000000000003e166

   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 79.2%

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

      1. neg-mul-1 79.2%

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

      2. unsub-neg 79.2%

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

   5. Simplified 79.2%

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

   6. Taylor expanded in im around 0 70.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 70.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   8. Simplified 70.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   9. Taylor expanded in re around 0 46.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) - im\right)} \]

   if 5.8000000000000003e166 < re

   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%

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

   4. Taylor expanded in re around 0 25.4%

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

      1. unpow2 25.4%

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

   6. Applied egg-rr 25.4%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.5% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(im \cdot 0.16666666666666666\right)\right)\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.8e+166)
   (* 0.5 (- (+ 2.0 (* im (+ 1.0 (* im (* im 0.16666666666666666))))) im))
   (+ 1.0 (* -0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+166) {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im);
	} else {
		tmp = 1.0 + (-0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.8d+166) then
        tmp = 0.5d0 * ((2.0d0 + (im * (1.0d0 + (im * (im * 0.16666666666666666d0))))) - im)
    else
        tmp = 1.0d0 + ((-0.5d0) * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+166) {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im);
	} else {
		tmp = 1.0 + (-0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.8e+166:
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im)
	else:
		tmp = 1.0 + (-0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.8e+166)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(im * 0.16666666666666666))))) - im));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.8e+166)
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (im * 0.16666666666666666))))) - im);
	else
		tmp = 1.0 + (-0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.8e+166], N[(0.5 * N[(N[(2.0 + N[(im * N[(1.0 + N[(im * N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.8000000000000003e166

   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 79.2%

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

      1. neg-mul-1 79.2%

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

      2. unsub-neg 79.2%

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

   5. Simplified 79.2%

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

   6. Taylor expanded in im around 0 70.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 70.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   8. Simplified 70.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   9. Taylor expanded in re around 0 46.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) - im\right)} \]

   10. Taylor expanded in im around inf 46.6%

       \[\leadsto 0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(0.16666666666666666 \cdot im\right)}\right)\right) - im\right) \]
   11. Step-by-step derivation

       1. *-commutative 70.9%

          \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(im \cdot 0.16666666666666666\right)}\right)\right) - im\right)\right) \]

   12. Simplified 46.6%

       \[\leadsto 0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \color{blue}{\left(im \cdot 0.16666666666666666\right)}\right)\right) - im\right) \]

   if 5.8000000000000003e166 < re

   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%

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

   4. Taylor expanded in re around 0 25.4%

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

      1. unpow2 25.4%

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

   6. Applied egg-rr 25.4%

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

Alternative 12: 47.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.8e+166)
   (* 0.5 (- (+ 2.0 (* im (+ 1.0 (* 0.5 im)))) im))
   (+ 1.0 (* -0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+166) {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im);
	} else {
		tmp = 1.0 + (-0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.8d+166) then
        tmp = 0.5d0 * ((2.0d0 + (im * (1.0d0 + (0.5d0 * im)))) - im)
    else
        tmp = 1.0d0 + ((-0.5d0) * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+166) {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im);
	} else {
		tmp = 1.0 + (-0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.8e+166:
		tmp = 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im)
	else:
		tmp = 1.0 + (-0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.8e+166)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))) - im));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.8e+166)
		tmp = 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im);
	else
		tmp = 1.0 + (-0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.8e+166], N[(0.5 * N[(N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.8000000000000003e166

   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 79.2%

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

      1. neg-mul-1 79.2%

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

      2. unsub-neg 79.2%

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

   5. Simplified 79.2%

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

   6. Taylor expanded in im around 0 70.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 70.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   8. Simplified 70.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   9. Taylor expanded in re around 0 46.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) - im\right)} \]

   10. Taylor expanded in im around 0 46.9%

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

   if 5.8000000000000003e166 < re

   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%

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

   4. Taylor expanded in re around 0 25.4%

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

      1. unpow2 25.4%

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

   6. Applied egg-rr 25.4%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 31.8% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 16000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 16000.0) 1.0 (+ 1.0 (* -0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 16000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 16000.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((-0.5d0) * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 16000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 16000.0:
		tmp = 1.0
	else:
		tmp = 1.0 + (-0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 16000.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 16000.0)
		tmp = 1.0;
	else
		tmp = 1.0 + (-0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 16000.0], 1.0, N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 16000

   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 72.0%

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

   4. Taylor expanded in re around 0 33.8%

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

   if 16000 < 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%

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

   4. Taylor expanded in re around 0 15.7%

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

      1. unpow2 15.7%

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

   6. Applied egg-rr 15.7%

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

Alternative 14: 28.8% 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 54.2%

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

4. Taylor expanded in re around 0 25.7%

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

Alternative 15: 9.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.75)

C

double code(double re, double im) {
	return 0.75;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.75d0
end function

Java

public static double code(double re, double im) {
	return 0.75;
}

Python

def code(re, im):
	return 0.75

Julia

function code(re, im)
	return 0.75
end

MATLAB

function tmp = code(re, im)
	tmp = 0.75;
end

Wolfram

code[re_, im_] := 0.75

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 re around 0 60.5%

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

5. Applied egg-rr 8.9%

   \[\leadsto 0.5 \cdot \color{blue}{1.5} \]
6. Step-by-step derivation

   1. metadata-eval 8.9%

      \[\leadsto \color{blue}{0.75} \]

7. Applied egg-rr 8.9%

   \[\leadsto \color{blue}{0.75} \]
8. Add Preprocessing

Alternative 16: 7.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.125)

C

double code(double re, double im) {
	return 0.125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.125d0
end function

Java

public static double code(double re, double im) {
	return 0.125;
}

Python

def code(re, im):
	return 0.125

Julia

function code(re, im)
	return 0.125
end

MATLAB

function tmp = code(re, im)
	tmp = 0.125;
end

Wolfram

code[re_, im_] := 0.125

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 re around 0 60.5%

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

5. Applied egg-rr 7.5%

   \[\leadsto 0.5 \cdot \color{blue}{0.25} \]
6. Step-by-step derivation

   1. metadata-eval 7.5%

      \[\leadsto \color{blue}{0.125} \]

7. Applied egg-rr 7.5%

   \[\leadsto \color{blue}{0.125} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))