math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.1s

Alternatives: 15

Speedup: 1.0×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 73.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7e-9)
   (cos re)
   (if (<= im 1.05e+103)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (*
      0.5
      (*
       (cos re)
       (-
        (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
        im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = cos(re);
	} else if (im <= 1.05e+103) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (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 <= 3.7d-9) then
        tmp = cos(re)
    else if (im <= 1.05d+103) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = 0.5d0 * (cos(re) * ((2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = Math.cos(re);
	} else if (im <= 1.05e+103) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.5 * (Math.cos(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7e-9:
		tmp = math.cos(re)
	elif im <= 1.05e+103:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.5 * (math.cos(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.05e+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(0.5 + Float64(im * 0.16666666666666666)))))) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.05e+103)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7e-9], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.05e+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[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.7e-9

   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 63.1%

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

   if 3.7e-9 < im < 1.0500000000000001e103

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 75.0%

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

   if 1.0500000000000001e103 < 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%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% 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 73.1%

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

   1. neg-mul-1 73.1%

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

   2. unsub-neg 73.1%

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

5. Simplified 73.1%

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

6. Final simplification 73.1%

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

Alternative 4: 72.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7e-9)
   (cos re)
   (if (<= im 1.05e+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 (+ 0.5 (* im 0.16666666666666666))))))
        im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = cos(re);
	} else if (im <= 1.05e+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 * (0.5 + (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 <= 3.7d-9) then
        tmp = cos(re)
    else if (im <= 1.05d+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 * (0.5d0 + (im * 0.16666666666666666d0)))))) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = Math.cos(re);
	} else if (im <= 1.05e+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 * (0.5 + (im * 0.16666666666666666)))))) - im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7e-9:
		tmp = math.cos(re)
	elif im <= 1.05e+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 * (0.5 + (im * 0.16666666666666666)))))) - im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.05e+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(0.5 + Float64(im * 0.16666666666666666)))))) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.05e+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 * (0.5 + (im * 0.16666666666666666)))))) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7e-9], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.05e+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[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.7e-9

   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 63.1%

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

   if 3.7e-9 < im < 1.0500000000000001e103

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 75.0%

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

   4. Taylor expanded in im around 0 75.0%

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

   if 1.0500000000000001e103 < 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%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 235 \lor \neg \left(im \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 235.0) (not (<= im 1.9e+154)))
   (* (* 0.5 (cos re)) (+ (- 1.0 im) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
   (* 0.5 (- (+ (exp im) 1.0) im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 235.0) || !(im <= 1.9e+154)) {
		tmp = (0.5 * cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else {
		tmp = 0.5 * ((exp(im) + 1.0) - 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 <= 235.0d0) .or. (.not. (im <= 1.9d+154))) then
        tmp = (0.5d0 * cos(re)) * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else
        tmp = 0.5d0 * ((exp(im) + 1.0d0) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 235.0) || !(im <= 1.9e+154)) {
		tmp = (0.5 * Math.cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else {
		tmp = 0.5 * ((Math.exp(im) + 1.0) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 235.0) or not (im <= 1.9e+154):
		tmp = (0.5 * math.cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))))
	else:
		tmp = 0.5 * ((math.exp(im) + 1.0) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 235.0) || !(im <= 1.9e+154))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	else
		tmp = Float64(0.5 * Float64(Float64(exp(im) + 1.0) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 235.0) || ~((im <= 1.9e+154)))
		tmp = (0.5 * cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	else
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 235.0], N[Not[LessEqual[im, 1.9e+154]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 235 or 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 68.7%

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

      1. neg-mul-1 68.7%

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

      2. unsub-neg 68.7%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in im around 0 81.6%

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

   if 235 < 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 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   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 re around 0 72.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235 \lor \neg \left(im \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+75}:\\ \;\;\;\;8 \cdot {re}^{2} - 8\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 700.0)
   (cos re)
   (if (<= im 5.4e+75)
     (- (* 8.0 (pow re 2.0)) 8.0)
     (*
      0.5
      (-
       (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
       im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 700.0) {
		tmp = cos(re);
	} else if (im <= 5.4e+75) {
		tmp = (8.0 * pow(re, 2.0)) - 8.0;
	} else {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (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 <= 700.0d0) then
        tmp = cos(re)
    else if (im <= 5.4d+75) then
        tmp = (8.0d0 * (re ** 2.0d0)) - 8.0d0
    else
        tmp = 0.5d0 * ((2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 700.0) {
		tmp = Math.cos(re);
	} else if (im <= 5.4e+75) {
		tmp = (8.0 * Math.pow(re, 2.0)) - 8.0;
	} else {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 700.0:
		tmp = math.cos(re)
	elif im <= 5.4e+75:
		tmp = (8.0 * math.pow(re, 2.0)) - 8.0
	else:
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 700.0)
		tmp = cos(re);
	elseif (im <= 5.4e+75)
		tmp = Float64(Float64(8.0 * (re ^ 2.0)) - 8.0);
	else
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 700.0)
		tmp = cos(re);
	elseif (im <= 5.4e+75)
		tmp = (8.0 * (re ^ 2.0)) - 8.0;
	else
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 700.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5.4e+75], N[(N[(8.0 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if im < 700

   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 62.9%

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

   if 700 < im < 5.39999999999999996e75

   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 3.1%

      \[\leadsto \color{blue}{\cos re + \cos re} \]
   5. Step-by-step derivation

      1. count-2 3.1%

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

   6. Simplified 3.1%

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

   8. Applied egg-rr 1.3%

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

      1. *-commutative 1.3%

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

   10. Simplified 1.3%

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

   11. Taylor expanded in re around 0 22.9%

       \[\leadsto \color{blue}{8 \cdot {re}^{2} - 8} \]

   if 5.39999999999999996e75 < 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 84.2%

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

      1. *-commutative 84.2%

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

   8. Simplified 84.2%

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

   9. Taylor expanded in re around 0 57.1%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+75}:\\ \;\;\;\;8 \cdot {re}^{2} - 8\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 235.0) (cos re) (* 0.5 (- (+ (exp im) 1.0) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 235.0) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * ((exp(im) + 1.0) - 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 <= 235.0d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * ((exp(im) + 1.0d0) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 235.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * ((Math.exp(im) + 1.0) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 235.0:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * ((math.exp(im) + 1.0) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 235.0)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(Float64(exp(im) + 1.0) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 235.0)
		tmp = cos(re);
	else
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 235.0], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 235

   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 62.9%

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

   if 235 < 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 re around 0 71.9%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.5 \cdot 10^{+25}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.5e+25)
   (cos re)
   (*
    0.5
    (- (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.5e+25) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (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 <= 4.5d+25) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * ((2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.5e+25) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.5e+25:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.5e+25)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.5e+25)
		tmp = cos(re);
	else
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.5e+25], N[Cos[re], $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if im < 4.5000000000000003e25

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 61.0%

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

   if 4.5000000000000003e25 < 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 60.9%

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

      1. *-commutative 60.9%

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

   8. Simplified 60.9%

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

   9. Taylor expanded in re around 0 41.4%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.5 \cdot 10^{+25}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.5% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  0.5
  (- (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))) im)))

C

double code(double re, double im) {
	return 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * ((2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) - im)
end function

Java

public static double code(double re, double im) {
	return 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
}

Python

def code(re, im):
	return 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im)

Julia

function code(re, im)
	return Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
end

Wolfram

code[re_, im_] := 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]

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 73.1%

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

   1. neg-mul-1 73.1%

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

   2. unsub-neg 73.1%

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

5. Simplified 73.1%

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

6. Taylor expanded in im around 0 60.2%

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

   1. *-commutative 60.2%

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

8. Simplified 60.2%

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

9. Taylor expanded in re around 0 38.3%

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

10. Final simplification 38.3%

    \[\leadsto 0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right) \]
11. Add Preprocessing

Alternative 10: 47.0% accurate, 23.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (- (+ 2.0 (* im (+ 1.0 (* 0.5 im)))) im)))

C

double code(double re, double im) {
	return 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * ((2.0d0 + (im * (1.0d0 + (0.5d0 * im)))) - im)
end function

Java

public static double code(double re, double im) {
	return 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im);
}

Python

def code(re, im):
	return 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im)

Julia

function code(re, im)
	return Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $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 73.1%

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

   1. neg-mul-1 73.1%

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

   2. unsub-neg 73.1%

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

5. Simplified 73.1%

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

6. Taylor expanded in re around 0 44.7%

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

7. Taylor expanded in im around 0 43.9%

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

   1. +-commutative 43.9%

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

   2. *-commutative 43.9%

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

9. Simplified 43.9%

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

10. Final simplification 43.9%

    \[\leadsto 0.5 \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right) \]
11. Add Preprocessing

Alternative 11: 37.9% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 235.0) 1.0 (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 235.0) {
		tmp = 1.0;
	} else {
		tmp = 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 <= 235.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 235.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 235.0:
		tmp = 1.0
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 235.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 235.0)
		tmp = 1.0;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 235.0], 1.0, N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 235

   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 64.6%

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

   4. Taylor expanded in im around 0 47.9%

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

      1. +-commutative 47.9%

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

      2. unpow2 47.9%

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

      3. fma-define 47.9%

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

   6. Simplified 47.9%

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

   7. Taylor expanded in im around 0 35.1%

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

   if 235 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 71.9%

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

   4. Taylor expanded in im around 0 33.1%

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

      1. +-commutative 33.1%

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

      2. unpow2 33.1%

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

      3. fma-define 33.1%

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

   6. Simplified 33.1%

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

   7. Taylor expanded in im around inf 33.1%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 33.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]

   9. Applied egg-rr 33.1%

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

Alternative 12: 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

3. Taylor expanded in re around 0 66.4%

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

4. Taylor expanded in im around 0 44.2%

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

   1. +-commutative 44.2%

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

   2. unpow2 44.2%

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

   3. fma-define 44.2%

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

6. Simplified 44.2%

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

7. Taylor expanded in im around 0 26.9%

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

Alternative 13: 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 66.4%

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

5. Applied egg-rr 8.6%

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

   1. metadata-eval 8.6%

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

7. Applied egg-rr 8.6%

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

Alternative 14: 3.9% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

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

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 66.4%

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

5. Applied egg-rr 3.8%

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

   1. metadata-eval 3.8%

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

7. Applied egg-rr 3.8%

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

Alternative 15: 3.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -8 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -8.0)

C

double code(double re, double im) {
	return -8.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -8.0d0
end function

Java

public static double code(double re, double im) {
	return -8.0;
}

Python

def code(re, im):
	return -8.0

Julia

function code(re, im)
	return -8.0
end

MATLAB

function tmp = code(re, im)
	tmp = -8.0;
end

Wolfram

code[re_, im_] := -8.0

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 10.5%

   \[\leadsto \color{blue}{\cos re + \cos re} \]
5. Step-by-step derivation

   1. count-2 10.5%

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

6. Simplified 10.5%

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

8. Applied egg-rr 3.3%

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

   1. *-commutative 3.3%

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

10. Simplified 3.3%

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

11. Taylor expanded in re around 0 3.1%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))