math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 8

Speedup: 1.0×

• 51.0% 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: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 360:\\ \;\;\;\;\cos re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \left(0.5 \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 360.0)
   (* (cos re) (+ 1.0 (* 0.5 (pow im 2.0))))
   (* im (log1p (expm1 (* (cos re) (* 0.5 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 360.0) {
		tmp = cos(re) * (1.0 + (0.5 * pow(im, 2.0)));
	} else {
		tmp = im * log1p(expm1((cos(re) * (0.5 * im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 360.0) {
		tmp = Math.cos(re) * (1.0 + (0.5 * Math.pow(im, 2.0)));
	} else {
		tmp = im * Math.log1p(Math.expm1((Math.cos(re) * (0.5 * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 360.0:
		tmp = math.cos(re) * (1.0 + (0.5 * math.pow(im, 2.0)))
	else:
		tmp = im * math.log1p(math.expm1((math.cos(re) * (0.5 * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 360.0)
		tmp = Float64(cos(re) * Float64(1.0 + Float64(0.5 * (im ^ 2.0))));
	else
		tmp = Float64(im * log1p(expm1(Float64(cos(re) * Float64(0.5 * im)))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 360.0], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[Log[1 + N[(Exp[N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 360

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.3%

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

      1. associate-*r* 86.3%

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

      2. distribute-rgt1-in 86.3%

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

      3. *-commutative 86.3%

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

      4. fma-define 86.3%

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

   5. Simplified 86.3%

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

   6. Taylor expanded in re around inf 86.3%

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

   if 360 < 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 51.2%

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

      1. associate-*r* 51.2%

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

      2. distribute-rgt1-in 51.2%

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

      3. *-commutative 51.2%

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

      4. fma-define 51.2%

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

   5. Simplified 51.2%

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

   6. Taylor expanded in im around inf 51.2%

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

      1. associate-*r* 51.2%

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

      2. *-commutative 51.2%

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

      3. associate-*r* 51.2%

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

   8. Simplified 51.2%

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

      1. *-commutative 51.2%

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

      2. unpow2 51.2%

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

      3. associate-*r* 51.2%

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

      4. *-commutative 51.2%

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

   10. Applied egg-rr 51.2%

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

       1. log1p-expm1-u 100.0%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos re \cdot 0.5\right) \cdot im\right)\right)} \cdot im \]

       2. associate-*l* 100.0%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\cos re \cdot \left(0.5 \cdot im\right)}\right)\right) \cdot im \]

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \left(0.5 \cdot im\right)\right)\right)} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 360:\\ \;\;\;\;\cos re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \left(0.5 \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 250.0)
   (* (cos re) (+ 1.0 (* 0.5 (pow im 2.0))))
   (if (<= im 2.2e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* im (* (* 0.5 (cos re)) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = cos(re) * (1.0 + (0.5 * pow(im, 2.0)));
	} else if (im <= 2.2e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = im * ((0.5 * cos(re)) * 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 <= 250.0d0) then
        tmp = cos(re) * (1.0d0 + (0.5d0 * (im ** 2.0d0)))
    else if (im <= 2.2d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = im * ((0.5d0 * cos(re)) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = Math.cos(re) * (1.0 + (0.5 * Math.pow(im, 2.0)));
	} else if (im <= 2.2e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = im * ((0.5 * Math.cos(re)) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 250.0:
		tmp = math.cos(re) * (1.0 + (0.5 * math.pow(im, 2.0)))
	elif im <= 2.2e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = im * ((0.5 * math.cos(re)) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 250.0)
		tmp = Float64(cos(re) * Float64(1.0 + Float64(0.5 * (im ^ 2.0))));
	elseif (im <= 2.2e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(im * Float64(Float64(0.5 * cos(re)) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 250.0)
		tmp = cos(re) * (1.0 + (0.5 * (im ^ 2.0)));
	elseif (im <= 2.2e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = im * ((0.5 * cos(re)) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 250.0], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.2e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 250

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.3%

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

      1. associate-*r* 86.3%

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

      2. distribute-rgt1-in 86.3%

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

      3. *-commutative 86.3%

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

      4. fma-define 86.3%

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

   5. Simplified 86.3%

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

   6. Taylor expanded in re around inf 86.3%

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

   if 250 < im < 2.2000000000000001e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 81.3%

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

   if 2.2000000000000001e154 < 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%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt1-in 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 250.0)
   (cos re)
   (if (<= im 2.1e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* im (* (* 0.5 (cos re)) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = cos(re);
	} else if (im <= 2.1e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = im * ((0.5 * cos(re)) * 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 <= 250.0d0) then
        tmp = cos(re)
    else if (im <= 2.1d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = im * ((0.5d0 * cos(re)) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = Math.cos(re);
	} else if (im <= 2.1e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = im * ((0.5 * Math.cos(re)) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 250.0:
		tmp = math.cos(re)
	elif im <= 2.1e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = im * ((0.5 * math.cos(re)) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 250.0)
		tmp = cos(re);
	elseif (im <= 2.1e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(im * Float64(Float64(0.5 * cos(re)) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 250.0)
		tmp = cos(re);
	elseif (im <= 2.1e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = im * ((0.5 * cos(re)) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 250.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.1e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 250

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 67.1%

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

   if 250 < im < 2.09999999999999994e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 81.3%

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

   if 2.09999999999999994e154 < 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%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt1-in 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+153}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im + -0.25 \cdot \left(im \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.2e+14)
   (cos re)
   (if (<= im 2.8e+153)
     (* im (+ (* 0.5 im) (* -0.25 (* im (pow re 2.0)))))
     (* im (* (* 0.5 (cos re)) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.2e+14) {
		tmp = cos(re);
	} else if (im <= 2.8e+153) {
		tmp = im * ((0.5 * im) + (-0.25 * (im * pow(re, 2.0))));
	} else {
		tmp = im * ((0.5 * cos(re)) * 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 <= 2.2d+14) then
        tmp = cos(re)
    else if (im <= 2.8d+153) then
        tmp = im * ((0.5d0 * im) + ((-0.25d0) * (im * (re ** 2.0d0))))
    else
        tmp = im * ((0.5d0 * cos(re)) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.2e+14) {
		tmp = Math.cos(re);
	} else if (im <= 2.8e+153) {
		tmp = im * ((0.5 * im) + (-0.25 * (im * Math.pow(re, 2.0))));
	} else {
		tmp = im * ((0.5 * Math.cos(re)) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.2e+14:
		tmp = math.cos(re)
	elif im <= 2.8e+153:
		tmp = im * ((0.5 * im) + (-0.25 * (im * math.pow(re, 2.0))))
	else:
		tmp = im * ((0.5 * math.cos(re)) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.2e+14)
		tmp = cos(re);
	elseif (im <= 2.8e+153)
		tmp = Float64(im * Float64(Float64(0.5 * im) + Float64(-0.25 * Float64(im * (re ^ 2.0)))));
	else
		tmp = Float64(im * Float64(Float64(0.5 * cos(re)) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.2e+14)
		tmp = cos(re);
	elseif (im <= 2.8e+153)
		tmp = im * ((0.5 * im) + (-0.25 * (im * (re ^ 2.0))));
	else
		tmp = im * ((0.5 * cos(re)) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.2e+14], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.8e+153], N[(im * N[(N[(0.5 * im), $MachinePrecision] + N[(-0.25 * N[(im * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.2e14

   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%

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

   if 2.2e14 < im < 2.79999999999999985e153

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 5.2%

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

      1. associate-*r* 5.2%

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

      2. distribute-rgt1-in 5.2%

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

      3. *-commutative 5.2%

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

      4. fma-define 5.2%

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

   5. Simplified 5.2%

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

   6. Taylor expanded in im around inf 5.2%

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

      1. associate-*r* 5.2%

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

      2. *-commutative 5.2%

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

      3. associate-*r* 5.2%

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

   8. Simplified 5.2%

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

      1. *-commutative 5.2%

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

      2. unpow2 5.2%

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

      3. associate-*r* 5.2%

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

      4. *-commutative 5.2%

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

   10. Applied egg-rr 5.2%

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

   11. Taylor expanded in re around 0 21.0%

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

   if 2.79999999999999985e153 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 97.2%

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

      1. associate-*r* 97.2%

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

      2. distribute-rgt1-in 97.2%

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

      3. *-commutative 97.2%

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

      4. fma-define 97.2%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in im around inf 97.2%

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

      1. associate-*r* 97.2%

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

      2. *-commutative 97.2%

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

      3. associate-*r* 97.2%

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

   8. Simplified 97.2%

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

      1. *-commutative 97.2%

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

      2. unpow2 97.2%

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

      3. associate-*r* 97.2%

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

      4. *-commutative 97.2%

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

   10. Applied egg-rr 97.2%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+153}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im + -0.25 \cdot \left(im \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.4) (cos re) (* im (* (* 0.5 (cos re)) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = cos(re);
	} else {
		tmp = im * ((0.5 * cos(re)) * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.4d0) then
        tmp = cos(re)
    else
        tmp = im * ((0.5d0 * cos(re)) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = Math.cos(re);
	} else {
		tmp = im * ((0.5 * Math.cos(re)) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.4:
		tmp = math.cos(re)
	else:
		tmp = im * ((0.5 * math.cos(re)) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.4)
		tmp = cos(re);
	else
		tmp = Float64(im * Float64(Float64(0.5 * cos(re)) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.4)
		tmp = cos(re);
	else
		tmp = im * ((0.5 * cos(re)) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.4], N[Cos[re], $MachinePrecision], N[(im * N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * im), $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 67.5%

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

   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

   3. Taylor expanded in im around 0 50.4%

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

      1. associate-*r* 50.4%

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

      2. distribute-rgt1-in 50.4%

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

      3. *-commutative 50.4%

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

      4. fma-define 50.4%

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

   5. Simplified 50.4%

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

   6. Taylor expanded in im around inf 50.4%

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

      1. associate-*r* 50.4%

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

      2. *-commutative 50.4%

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

      3. associate-*r* 50.4%

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

   8. Simplified 50.4%

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

      1. *-commutative 50.4%

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

      2. unpow2 50.4%

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

      3. associate-*r* 50.4%

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

      4. *-commutative 50.4%

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

   10. Applied egg-rr 50.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{+38}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.2e+38) (cos re) (* im (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.2e+38) {
		tmp = cos(re);
	} else {
		tmp = im * (0.5 * 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 <= 2.2d+38) then
        tmp = cos(re)
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.2e+38) {
		tmp = Math.cos(re);
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.2e+38:
		tmp = math.cos(re)
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.2e+38)
		tmp = cos(re);
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.2e+38)
		tmp = cos(re);
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.2e+38], N[Cos[re], $MachinePrecision], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.20000000000000006e38

   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 64.3%

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

   if 2.20000000000000006e38 < 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 59.3%

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

      1. associate-*r* 59.3%

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

      2. distribute-rgt1-in 59.3%

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

      3. *-commutative 59.3%

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

      4. fma-define 59.3%

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

   5. Simplified 59.3%

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

   6. Taylor expanded in im around inf 59.3%

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

      1. associate-*r* 59.3%

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

      2. *-commutative 59.3%

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

      3. associate-*r* 59.3%

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

   8. Simplified 59.3%

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

      1. *-commutative 59.3%

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

      2. unpow2 59.3%

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

      3. associate-*r* 59.3%

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

      4. *-commutative 59.3%

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

   10. Applied egg-rr 59.3%

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

   11. Taylor expanded in re around 0 41.9%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{+38}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 22.6% accurate, 61.6× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* im (* 0.5 im)))

C

double code(double re, double im) {
	return im * (0.5 * im);
}

Fortran

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

Java

public static double code(double re, double im) {
	return im * (0.5 * im);
}

Python

def code(re, im):
	return im * (0.5 * im)

Julia

function code(re, im)
	return Float64(im * Float64(0.5 * im))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (0.5 * im);
end

Wolfram

code[re_, im_] := N[(im * N[(0.5 * 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 77.8%

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

   1. associate-*r* 77.8%

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

   2. distribute-rgt1-in 77.8%

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

   3. *-commutative 77.8%

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

   4. fma-define 77.8%

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

5. Simplified 77.8%

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

6. Taylor expanded in im around inf 29.5%

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

   1. associate-*r* 29.5%

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

   2. *-commutative 29.5%

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

   3. associate-*r* 29.5%

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

8. Simplified 29.5%

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

   1. *-commutative 29.5%

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

   2. unpow2 29.5%

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

   3. associate-*r* 29.5%

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

   4. *-commutative 29.5%

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

10. Applied egg-rr 29.5%

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

11. Taylor expanded in re around 0 20.5%

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

12. Final simplification 20.5%

    \[\leadsto im \cdot \left(0.5 \cdot im\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))