math.cos on complex, real part

Percentage Accurate: 100.0% → 99.7%

Time: 7.1s

Alternatives: 15

Speedup: 1.0×

• 49.3% 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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;e^{-im\_m} + e^{im\_m} \leq 4:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{im\_m} + 3\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= (+ (exp (- im_m)) (exp im_m)) 4.0)
     (* t_0 (fma im_m im_m 2.0))
     (* t_0 (+ (exp im_m) 3.0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if ((exp(-im_m) + exp(im_m)) <= 4.0) {
		tmp = t_0 * fma(im_m, im_m, 2.0);
	} else {
		tmp = t_0 * (exp(im_m) + 3.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) + exp(im_m)) <= 4.0)
		tmp = Float64(t_0 * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(t_0 * Float64(exp(im_m) + 3.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], 4.0], N[(t$95$0 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Exp[im$95$m], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   5. Simplified 100.0%

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

   if 4 < (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

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

   4. Applied egg-rr 53.0%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;e^{-im\_m} + e^{im\_m} \leq 4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im\_m} + 3\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ (exp (- im_m)) (exp im_m)) 4.0)
   (cos re)
   (* (* 0.5 (cos re)) (+ (exp im_m) 3.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((exp(-im_m) + exp(im_m)) <= 4.0) {
		tmp = cos(re);
	} else {
		tmp = (0.5 * cos(re)) * (exp(im_m) + 3.0);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(-im_m) + exp(im_m)) <= 4.0d0) then
        tmp = cos(re)
    else
        tmp = (0.5d0 * cos(re)) * (exp(im_m) + 3.0d0)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if ((Math.exp(-im_m) + Math.exp(im_m)) <= 4.0) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp(im_m) + 3.0);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if (math.exp(-im_m) + math.exp(im_m)) <= 4.0:
		tmp = math.cos(re)
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp(im_m) + 3.0)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) + exp(im_m)) <= 4.0)
		tmp = cos(re);
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(im_m) + 3.0));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if ((exp(-im_m) + exp(im_m)) <= 4.0)
		tmp = cos(re);
	else
		tmp = (0.5 * cos(re)) * (exp(im_m) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], 4.0], N[Cos[re], $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 4

   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%

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

   4. Taylor expanded in re around inf 99.3%

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

   if 4 < (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

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

   4. Applied egg-rr 53.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im_m)) (exp im_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $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 4: 95.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im\_m \leq 2.4 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} + e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im\_m \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 4e-6)
   (cos re)
   (if (<= im_m 2.4e+101)
     (* 0.5 (+ (exp (- im_m)) (exp im_m)))
     (*
      (* 0.5 (cos re))
      (+
       4.0
       (* im_m (+ 1.0 (* im_m (+ 0.5 (* im_m 0.16666666666666666))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 4e-6) {
		tmp = cos(re);
	} else if (im_m <= 2.4e+101) {
		tmp = 0.5 * (exp(-im_m) + exp(im_m));
	} else {
		tmp = (0.5 * cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4d-6) then
        tmp = cos(re)
    else if (im_m <= 2.4d+101) then
        tmp = 0.5d0 * (exp(-im_m) + exp(im_m))
    else
        tmp = (0.5d0 * cos(re)) * (4.0d0 + (im_m * (1.0d0 + (im_m * (0.5d0 + (im_m * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 4e-6) {
		tmp = Math.cos(re);
	} else if (im_m <= 2.4e+101) {
		tmp = 0.5 * (Math.exp(-im_m) + Math.exp(im_m));
	} else {
		tmp = (0.5 * Math.cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 4e-6:
		tmp = math.cos(re)
	elif im_m <= 2.4e+101:
		tmp = 0.5 * (math.exp(-im_m) + math.exp(im_m))
	else:
		tmp = (0.5 * math.cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 4e-6)
		tmp = cos(re);
	elseif (im_m <= 2.4e+101)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) + exp(im_m)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(4.0 + Float64(im_m * Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 4e-6)
		tmp = cos(re);
	elseif (im_m <= 2.4e+101)
		tmp = 0.5 * (exp(-im_m) + exp(im_m));
	else
		tmp = (0.5 * cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 4e-6], N[Cos[re], $MachinePrecision], If[LessEqual[im$95$m, 2.4e+101], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im$95$m * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.99999999999999982e-6

   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 66.6%

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

   4. Taylor expanded in re around inf 66.6%

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

   if 3.99999999999999982e-6 < im < 2.39999999999999988e101

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 76.5%

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

   if 2.39999999999999988e101 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in im around 0 98.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   7. Simplified 98.3%

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

Alternative 5: 95.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 2.8:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im\_m \leq 2.4 \cdot 10^{+101}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im\_m \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 2.8)
   (cos re)
   (if (<= im_m 2.4e+101)
     (+ 1.5 (* 0.5 (exp im_m)))
     (*
      (* 0.5 (cos re))
      (+
       4.0
       (* im_m (+ 1.0 (* im_m (+ 0.5 (* im_m 0.16666666666666666))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 2.8) {
		tmp = cos(re);
	} else if (im_m <= 2.4e+101) {
		tmp = 1.5 + (0.5 * exp(im_m));
	} else {
		tmp = (0.5 * cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.8d0) then
        tmp = cos(re)
    else if (im_m <= 2.4d+101) then
        tmp = 1.5d0 + (0.5d0 * exp(im_m))
    else
        tmp = (0.5d0 * cos(re)) * (4.0d0 + (im_m * (1.0d0 + (im_m * (0.5d0 + (im_m * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 2.8) {
		tmp = Math.cos(re);
	} else if (im_m <= 2.4e+101) {
		tmp = 1.5 + (0.5 * Math.exp(im_m));
	} else {
		tmp = (0.5 * Math.cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 2.8:
		tmp = math.cos(re)
	elif im_m <= 2.4e+101:
		tmp = 1.5 + (0.5 * math.exp(im_m))
	else:
		tmp = (0.5 * math.cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 2.8)
		tmp = cos(re);
	elseif (im_m <= 2.4e+101)
		tmp = Float64(1.5 + Float64(0.5 * exp(im_m)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(4.0 + Float64(im_m * Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 2.8)
		tmp = cos(re);
	elseif (im_m <= 2.4e+101)
		tmp = 1.5 + (0.5 * exp(im_m));
	else
		tmp = (0.5 * cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 2.8], N[Cos[re], $MachinePrecision], If[LessEqual[im$95$m, 2.4e+101], N[(1.5 + N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im$95$m * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.7999999999999998

   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 66.5%

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

   4. Taylor expanded in re around inf 66.5%

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

   if 2.7999999999999998 < im < 2.39999999999999988e101

   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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 75.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 75.0%

         \[\leadsto \color{blue}{0.5 \cdot 3 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 75.0%

         \[\leadsto \color{blue}{1.5} + 0.5 \cdot e^{im} \]

   7. Simplified 75.0%

      \[\leadsto \color{blue}{1.5 + 0.5 \cdot e^{im}} \]

   if 2.39999999999999988e101 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in im around 0 98.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   7. Simplified 98.3%

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

Alternative 6: 93.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 3:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im\_m \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im\_m \cdot \left(1 + 0.5 \cdot im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 3.0)
   (cos re)
   (if (<= im_m 1.9e+154)
     (+ 1.5 (* 0.5 (exp im_m)))
     (* (* 0.5 (cos re)) (+ 4.0 (* im_m (+ 1.0 (* 0.5 im_m))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3.0) {
		tmp = cos(re);
	} else if (im_m <= 1.9e+154) {
		tmp = 1.5 + (0.5 * exp(im_m));
	} else {
		tmp = (0.5 * cos(re)) * (4.0 + (im_m * (1.0 + (0.5 * im_m))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.0d0) then
        tmp = cos(re)
    else if (im_m <= 1.9d+154) then
        tmp = 1.5d0 + (0.5d0 * exp(im_m))
    else
        tmp = (0.5d0 * cos(re)) * (4.0d0 + (im_m * (1.0d0 + (0.5d0 * im_m))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3.0) {
		tmp = Math.cos(re);
	} else if (im_m <= 1.9e+154) {
		tmp = 1.5 + (0.5 * Math.exp(im_m));
	} else {
		tmp = (0.5 * Math.cos(re)) * (4.0 + (im_m * (1.0 + (0.5 * im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 3.0:
		tmp = math.cos(re)
	elif im_m <= 1.9e+154:
		tmp = 1.5 + (0.5 * math.exp(im_m))
	else:
		tmp = (0.5 * math.cos(re)) * (4.0 + (im_m * (1.0 + (0.5 * im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 3.0)
		tmp = cos(re);
	elseif (im_m <= 1.9e+154)
		tmp = Float64(1.5 + Float64(0.5 * exp(im_m)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(4.0 + Float64(im_m * Float64(1.0 + Float64(0.5 * im_m)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 3.0)
		tmp = cos(re);
	elseif (im_m <= 1.9e+154)
		tmp = 1.5 + (0.5 * exp(im_m));
	else
		tmp = (0.5 * cos(re)) * (4.0 + (im_m * (1.0 + (0.5 * im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 3.0], N[Cos[re], $MachinePrecision], If[LessEqual[im$95$m, 1.9e+154], N[(1.5 + N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im$95$m * N[(1.0 + N[(0.5 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3

   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 66.5%

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

   4. Taylor expanded in re around inf 66.5%

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

   if 3 < 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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 74.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 74.2%

         \[\leadsto \color{blue}{0.5 \cdot 3 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 74.2%

         \[\leadsto \color{blue}{1.5} + 0.5 \cdot e^{im} \]

   7. Simplified 74.2%

      \[\leadsto \color{blue}{1.5 + 0.5 \cdot e^{im}} \]

   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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + \color{blue}{im \cdot 0.5}\right)\right) \]

   7. Simplified 100.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 3100000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im\_m \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;2 - {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 + im\_m \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 3100000.0)
   (cos re)
   (if (<= im_m 1.05e+103)
     (- 2.0 (pow re 2.0))
     (+ 2.0 (* im_m (+ 0.5 (* im_m (* im_m 0.08333333333333333))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3100000.0) {
		tmp = cos(re);
	} else if (im_m <= 1.05e+103) {
		tmp = 2.0 - pow(re, 2.0);
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (im_m * 0.08333333333333333))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3100000.0d0) then
        tmp = cos(re)
    else if (im_m <= 1.05d+103) then
        tmp = 2.0d0 - (re ** 2.0d0)
    else
        tmp = 2.0d0 + (im_m * (0.5d0 + (im_m * (im_m * 0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3100000.0) {
		tmp = Math.cos(re);
	} else if (im_m <= 1.05e+103) {
		tmp = 2.0 - Math.pow(re, 2.0);
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (im_m * 0.08333333333333333))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 3100000.0:
		tmp = math.cos(re)
	elif im_m <= 1.05e+103:
		tmp = 2.0 - math.pow(re, 2.0)
	else:
		tmp = 2.0 + (im_m * (0.5 + (im_m * (im_m * 0.08333333333333333))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 3100000.0)
		tmp = cos(re);
	elseif (im_m <= 1.05e+103)
		tmp = Float64(2.0 - (re ^ 2.0));
	else
		tmp = Float64(2.0 + Float64(im_m * Float64(0.5 + Float64(im_m * Float64(im_m * 0.08333333333333333)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 3100000.0)
		tmp = cos(re);
	elseif (im_m <= 1.05e+103)
		tmp = 2.0 - (re ^ 2.0);
	else
		tmp = 2.0 + (im_m * (0.5 + (im_m * (im_m * 0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 3100000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im$95$m, 1.05e+103], N[(2.0 - N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * N[(im$95$m * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.1e6

   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 66.2%

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

   4. Taylor expanded in re around inf 66.2%

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

   if 3.1e6 < 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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in im around 0 3.1%

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

   6. Taylor expanded in re around 0 26.4%

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

      1. mul-1-neg 26.4%

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

      2. unsub-neg 26.4%

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

   8. Simplified 26.4%

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

   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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 84.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 84.3%

         \[\leadsto \color{blue}{0.5 \cdot 3 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 84.3%

         \[\leadsto \color{blue}{1.5} + 0.5 \cdot e^{im} \]

   7. Simplified 84.3%

      \[\leadsto \color{blue}{1.5 + 0.5 \cdot e^{im}} \]

   8. Taylor expanded in im around 0 84.3%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]

   9. Taylor expanded in im around inf 84.3%

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

       1. *-commutative 84.3%

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

   11. Simplified 84.3%

       \[\leadsto 2 + im \cdot \left(0.5 + im \cdot \color{blue}{\left(im \cdot 0.08333333333333333\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 87.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.75:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im\_m}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.75) (cos re) (+ 1.5 (* 0.5 (exp im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.75) {
		tmp = cos(re);
	} else {
		tmp = 1.5 + (0.5 * exp(im_m));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.75d0) then
        tmp = cos(re)
    else
        tmp = 1.5d0 + (0.5d0 * exp(im_m))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.75) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.5 + (0.5 * Math.exp(im_m));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.75:
		tmp = math.cos(re)
	else:
		tmp = 1.5 + (0.5 * math.exp(im_m))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.75)
		tmp = cos(re);
	else
		tmp = Float64(1.5 + Float64(0.5 * exp(im_m)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.75)
		tmp = cos(re);
	else
		tmp = 1.5 + (0.5 * exp(im_m));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.75], N[Cos[re], $MachinePrecision], N[(1.5 + N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.75

   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 66.5%

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

   4. Taylor expanded in re around inf 66.5%

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

   if 1.75 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 80.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 80.9%

         \[\leadsto \color{blue}{0.5 \cdot 3 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 80.9%

         \[\leadsto \color{blue}{1.5} + 0.5 \cdot e^{im} \]

   7. Simplified 80.9%

      \[\leadsto \color{blue}{1.5 + 0.5 \cdot e^{im}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;2 + im\_m \cdot \left(0.5 + im\_m \cdot \left(0.25 + im\_m \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.5e+39)
   (cos re)
   (+ 2.0 (* im_m (+ 0.5 (* im_m (+ 0.25 (* im_m 0.08333333333333333))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.5e+39) {
		tmp = cos(re);
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.5d+39) then
        tmp = cos(re)
    else
        tmp = 2.0d0 + (im_m * (0.5d0 + (im_m * (0.25d0 + (im_m * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.5e+39) {
		tmp = Math.cos(re);
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.5e+39:
		tmp = math.cos(re)
	else:
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.5e+39)
		tmp = cos(re);
	else
		tmp = Float64(2.0 + Float64(im_m * Float64(0.5 + Float64(im_m * Float64(0.25 + Float64(im_m * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.5e+39)
		tmp = cos(re);
	else
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.5e+39], N[Cos[re], $MachinePrecision], N[(2.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * N[(0.25 + N[(im$95$m * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.5e39

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 64.6%

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

   4. Taylor expanded in re around inf 64.6%

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

   if 1.5e39 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 82.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 82.3%

         \[\leadsto \color{blue}{0.5 \cdot 3 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 82.3%

         \[\leadsto \color{blue}{1.5} + 0.5 \cdot e^{im} \]

   7. Simplified 82.3%

      \[\leadsto \color{blue}{1.5 + 0.5 \cdot e^{im}} \]

   8. Taylor expanded in im around 0 70.2%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.15:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im\_m \cdot \left(0.5 + im\_m \cdot \left(0.25 + im\_m \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.15)
   1.0
   (+ 2.0 (* im_m (+ 0.5 (* im_m (+ 0.25 (* im_m 0.08333333333333333))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.15) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.15d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 + (im_m * (0.5d0 + (im_m * (0.25d0 + (im_m * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.15) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.15:
		tmp = 1.0
	else:
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.15)
		tmp = 1.0;
	else
		tmp = Float64(2.0 + Float64(im_m * Float64(0.5 + Float64(im_m * Float64(0.25 + Float64(im_m * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.15)
		tmp = 1.0;
	else
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.15], 1.0, N[(2.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * N[(0.25 + N[(im$95$m * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.1499999999999999

   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 66.5%

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

   4. Taylor expanded in re around 0 36.7%

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

   if 1.1499999999999999 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 80.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 80.9%

         \[\leadsto \color{blue}{0.5 \cdot 3 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 80.9%

         \[\leadsto \color{blue}{1.5} + 0.5 \cdot e^{im} \]

   7. Simplified 80.9%

      \[\leadsto \color{blue}{1.5 + 0.5 \cdot e^{im}} \]

   8. Taylor expanded in im around 0 64.2%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.15:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.7% accurate, 19.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im\_m \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.2)
   1.0
   (+ 2.0 (* im_m (+ 0.5 (* im_m (* im_m 0.08333333333333333)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (im_m * 0.08333333333333333))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.2d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 + (im_m * (0.5d0 + (im_m * (im_m * 0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (im_m * 0.08333333333333333))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.2:
		tmp = 1.0
	else:
		tmp = 2.0 + (im_m * (0.5 + (im_m * (im_m * 0.08333333333333333))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.2)
		tmp = 1.0;
	else
		tmp = Float64(2.0 + Float64(im_m * Float64(0.5 + Float64(im_m * Float64(im_m * 0.08333333333333333)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.2)
		tmp = 1.0;
	else
		tmp = 2.0 + (im_m * (0.5 + (im_m * (im_m * 0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.2], 1.0, N[(2.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * N[(im$95$m * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.19999999999999996

   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 66.5%

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

   4. Taylor expanded in re around 0 36.7%

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

   if 1.19999999999999996 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 80.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 80.9%

         \[\leadsto \color{blue}{0.5 \cdot 3 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 80.9%

         \[\leadsto \color{blue}{1.5} + 0.5 \cdot e^{im} \]

   7. Simplified 80.9%

      \[\leadsto \color{blue}{1.5 + 0.5 \cdot e^{im}} \]

   8. Taylor expanded in im around 0 64.2%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]

   9. Taylor expanded in im around inf 64.2%

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

       1. *-commutative 64.2%

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

   11. Simplified 64.2%

       \[\leadsto 2 + im \cdot \left(0.5 + im \cdot \color{blue}{\left(im \cdot 0.08333333333333333\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 47.8% accurate, 22.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im\_m \cdot \left(0.5 + im\_m \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.4) 1.0 (+ 2.0 (* im_m (+ 0.5 (* im_m 0.25))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * 0.25)));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.4d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 + (im_m * (0.5d0 + (im_m * 0.25d0)))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * 0.25)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.4:
		tmp = 1.0
	else:
		tmp = 2.0 + (im_m * (0.5 + (im_m * 0.25)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.4)
		tmp = 1.0;
	else
		tmp = Float64(2.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.25))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.4)
		tmp = 1.0;
	else
		tmp = 2.0 + (im_m * (0.5 + (im_m * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.4], 1.0, N[(2.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3999999999999999

   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 66.5%

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

   4. Taylor expanded in re around 0 36.7%

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

   if 1.3999999999999999 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 80.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 80.9%

         \[\leadsto \color{blue}{0.5 \cdot 3 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 80.9%

         \[\leadsto \color{blue}{1.5} + 0.5 \cdot e^{im} \]

   7. Simplified 80.9%

      \[\leadsto \color{blue}{1.5 + 0.5 \cdot e^{im}} \]

   8. Taylor expanded in im around 0 49.2%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + 0.25 \cdot im\right)} \]
   9. Step-by-step derivation

      1. *-commutative 49.2%

         \[\leadsto 2 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.25}\right) \]

   10. Simplified 49.2%

       \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot 0.25\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 29.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 1 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 1.0)

C

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

Fortran

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

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 1.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 1.0

Julia

im_m = abs(im)
function code(re, im_m)
	return 1.0
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 49.7%

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

4. Taylor expanded in re around 0 27.7%

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

Alternative 14: 7.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.125 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 0.125)

C

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

Fortran

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

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.125;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.125

Julia

im_m = abs(im)
function code(re, im_m)
	return 0.125
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 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 67.2%

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

5. Applied egg-rr 7.3%

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

   1. metadata-eval 7.3%

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

7. Applied egg-rr 7.3%

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

Alternative 15: 3.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ -1 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 -1.0)

C

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

Fortran

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

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return -1.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return -1.0

Julia

im_m = abs(im)
function code(re, im_m)
	return -1.0
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := -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 67.2%

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

5. Applied egg-rr 4.1%

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

   1. metadata-eval 4.1%

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

7. Applied egg-rr 4.1%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))