math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 11

Speedup: 1.0×

• 50.2% 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. Final simplification 100.0%

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

Alternative 2: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1350000:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+66}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \sqrt{{im}^{4} \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1350000.0)
   (* (cos re) (+ (* 0.5 (pow im 2.0)) 1.0))
   (if (<= im 3e+66)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (sqrt (* (pow im 4.0) 0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1350000.0) {
		tmp = cos(re) * ((0.5 * pow(im, 2.0)) + 1.0);
	} else if (im <= 3e+66) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * sqrt((pow(im, 4.0) * 0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1350000.0d0) then
        tmp = cos(re) * ((0.5d0 * (im ** 2.0d0)) + 1.0d0)
    else if (im <= 3d+66) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * sqrt(((im ** 4.0d0) * 0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1350000.0) {
		tmp = Math.cos(re) * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	} else if (im <= 3e+66) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * Math.sqrt((Math.pow(im, 4.0) * 0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1350000.0:
		tmp = math.cos(re) * ((0.5 * math.pow(im, 2.0)) + 1.0)
	elif im <= 3e+66:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * math.sqrt((math.pow(im, 4.0) * 0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1350000.0)
		tmp = Float64(cos(re) * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	elseif (im <= 3e+66)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * sqrt(Float64((im ^ 4.0) * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1350000.0)
		tmp = cos(re) * ((0.5 * (im ^ 2.0)) + 1.0);
	elseif (im <= 3e+66)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * sqrt(((im ^ 4.0) * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1350000.0], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3e+66], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[Sqrt[N[(N[Power[im, 4.0], $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.35e6

   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 83.4%

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

      1. associate-*r* 83.4%

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

      2. distribute-rgt1-in 83.4%

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

   5. Simplified 83.4%

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

   if 1.35e6 < im < 3.00000000000000002e66

   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 93.3%

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

   if 3.00000000000000002e66 < 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 62.6%

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

      1. associate-*r* 62.6%

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

      2. distribute-rgt1-in 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in im around inf 62.6%

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

      1. associate-*r* 62.6%

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

      2. *-commutative 62.6%

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

   8. Simplified 62.6%

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

      1. add-sqr-sqrt 62.6%

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

      2. sqrt-unprod 94.3%

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

      3. *-commutative 94.3%

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

      4. *-commutative 94.3%

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

      5. swap-sqr 94.3%

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

      6. pow-prod-up 94.3%

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

      7. metadata-eval 94.3%

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

      8. metadata-eval 94.3%

         \[\leadsto \cos re \cdot \sqrt{{im}^{4} \cdot \color{blue}{0.25}} \]

   10. Applied egg-rr 94.3%

       \[\leadsto \cos re \cdot \color{blue}{\sqrt{{im}^{4} \cdot 0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1350000:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+66}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \sqrt{{im}^{4} \cdot 0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00175:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00175)
   (cos re)
   (if (<= im 1.9e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00175) {
		tmp = cos(re);
	} else if (im <= 1.9e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (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 <= 0.00175d0) then
        tmp = cos(re)
    else if (im <= 1.9d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * (im * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00175) {
		tmp = Math.cos(re);
	} else if (im <= 1.9e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * (im * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00175:
		tmp = math.cos(re)
	elif im <= 1.9e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * (im * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00175)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00175)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * (im * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00175], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00175000000000000004

   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 65.5%

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

   if 0.00175000000000000004 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 69.4%

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

   if 1.8999999999999999e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

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

   5. Simplified 100.0%

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

   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%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. swap-sqr 100.0%

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

      6. pow-prod-up 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

         \[\leadsto \cos re \cdot \sqrt{{im}^{4} \cdot \color{blue}{0.25}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \cos re \cdot \color{blue}{\sqrt{{im}^{4} \cdot 0.25}} \]
   11. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto \cos re \cdot \sqrt{\color{blue}{0.25 \cdot {im}^{4}}} \]

       2. sqrt-prod 100.0%

          \[\leadsto \cos re \cdot \color{blue}{\left(\sqrt{0.25} \cdot \sqrt{{im}^{4}}\right)} \]

       3. metadata-eval 100.0%

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

       4. sqrt-pow1 100.0%

          \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{{im}^{\left(\frac{4}{2}\right)}}\right) \]

       5. metadata-eval 100.0%

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

       6. unpow2 100.0%

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

       7. associate-*r* 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00175:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1350000:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1350000.0)
   (* (cos re) (+ (* 0.5 (pow im 2.0)) 1.0))
   (if (<= im 1.9e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1350000.0) {
		tmp = cos(re) * ((0.5 * pow(im, 2.0)) + 1.0);
	} else if (im <= 1.9e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (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 <= 1350000.0d0) then
        tmp = cos(re) * ((0.5d0 * (im ** 2.0d0)) + 1.0d0)
    else if (im <= 1.9d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * (im * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1350000.0) {
		tmp = Math.cos(re) * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	} else if (im <= 1.9e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * (im * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1350000.0:
		tmp = math.cos(re) * ((0.5 * math.pow(im, 2.0)) + 1.0)
	elif im <= 1.9e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * (im * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1350000.0)
		tmp = Float64(cos(re) * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	elseif (im <= 1.9e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1350000.0)
		tmp = cos(re) * ((0.5 * (im ^ 2.0)) + 1.0);
	elseif (im <= 1.9e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * (im * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1350000.0], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.35e6

   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 83.4%

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

      1. associate-*r* 83.4%

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

      2. distribute-rgt1-in 83.4%

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

   5. Simplified 83.4%

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

   if 1.35e6 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 71.4%

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

   if 1.8999999999999999e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

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

   5. Simplified 100.0%

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

   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%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. swap-sqr 100.0%

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

      6. pow-prod-up 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

         \[\leadsto \cos re \cdot \sqrt{{im}^{4} \cdot \color{blue}{0.25}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \cos re \cdot \color{blue}{\sqrt{{im}^{4} \cdot 0.25}} \]
   11. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto \cos re \cdot \sqrt{\color{blue}{0.25 \cdot {im}^{4}}} \]

       2. sqrt-prod 100.0%

          \[\leadsto \cos re \cdot \color{blue}{\left(\sqrt{0.25} \cdot \sqrt{{im}^{4}}\right)} \]

       3. metadata-eval 100.0%

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

       4. sqrt-pow1 100.0%

          \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{{im}^{\left(\frac{4}{2}\right)}}\right) \]

       5. metadata-eval 100.0%

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

       6. unpow2 100.0%

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

       7. associate-*r* 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1350000:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2 \cdot 10^{+39}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{im}^{6} \cdot 0.125}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.2e+39)
   (cos re)
   (if (<= im 1.9e+154)
     (cbrt (* (pow im 6.0) 0.125))
     (* (cos re) (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.2e+39) {
		tmp = cos(re);
	} else if (im <= 1.9e+154) {
		tmp = cbrt((pow(im, 6.0) * 0.125));
	} else {
		tmp = cos(re) * (im * (0.5 * im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.2e+39) {
		tmp = Math.cos(re);
	} else if (im <= 1.9e+154) {
		tmp = Math.cbrt((Math.pow(im, 6.0) * 0.125));
	} else {
		tmp = Math.cos(re) * (im * (0.5 * im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.2e+39)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = cbrt(Float64((im ^ 6.0) * 0.125));
	else
		tmp = Float64(cos(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.2e+39], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.9e+154], N[Power[N[(N[Power[im, 6.0], $MachinePrecision] * 0.125), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.2000000000000005e39

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 62.1%

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

   if 6.2000000000000005e39 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 6.0%

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

      1. associate-*r* 6.0%

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

      2. distribute-rgt1-in 6.0%

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

   5. Simplified 6.0%

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

   6. Taylor expanded in im around inf 6.0%

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

      1. associate-*r* 6.0%

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

      2. *-commutative 6.0%

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

   8. Simplified 6.0%

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

   9. Taylor expanded in re around 0 4.1%

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

       1. add-cbrt-cube 56.3%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.5 \cdot {im}^{2}\right) \cdot \left(0.5 \cdot {im}^{2}\right)\right) \cdot \left(0.5 \cdot {im}^{2}\right)}} \]

       2. pow1/3 56.3%

          \[\leadsto \color{blue}{{\left(\left(\left(0.5 \cdot {im}^{2}\right) \cdot \left(0.5 \cdot {im}^{2}\right)\right) \cdot \left(0.5 \cdot {im}^{2}\right)\right)}^{0.3333333333333333}} \]

       3. pow3 56.3%

          \[\leadsto {\color{blue}{\left({\left(0.5 \cdot {im}^{2}\right)}^{3}\right)}}^{0.3333333333333333} \]

       4. *-commutative 56.3%

          \[\leadsto {\left({\color{blue}{\left({im}^{2} \cdot 0.5\right)}}^{3}\right)}^{0.3333333333333333} \]

       5. unpow-prod-down 56.3%

          \[\leadsto {\color{blue}{\left({\left({im}^{2}\right)}^{3} \cdot {0.5}^{3}\right)}}^{0.3333333333333333} \]

       6. pow-pow 56.3%

          \[\leadsto {\left(\color{blue}{{im}^{\left(2 \cdot 3\right)}} \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

       7. metadata-eval 56.3%

          \[\leadsto {\left({im}^{\color{blue}{6}} \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

       8. metadata-eval 56.3%

          \[\leadsto {\left({im}^{6} \cdot \color{blue}{0.125}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 56.3%

       \[\leadsto \color{blue}{{\left({im}^{6} \cdot 0.125\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 56.3%

          \[\leadsto \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]

   13. Simplified 56.3%

       \[\leadsto \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]

   if 1.8999999999999999e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

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

   5. Simplified 100.0%

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

   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%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. swap-sqr 100.0%

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

      6. pow-prod-up 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

         \[\leadsto \cos re \cdot \sqrt{{im}^{4} \cdot \color{blue}{0.25}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \cos re \cdot \color{blue}{\sqrt{{im}^{4} \cdot 0.25}} \]
   11. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto \cos re \cdot \sqrt{\color{blue}{0.25 \cdot {im}^{4}}} \]

       2. sqrt-prod 100.0%

          \[\leadsto \cos re \cdot \color{blue}{\left(\sqrt{0.25} \cdot \sqrt{{im}^{4}}\right)} \]

       3. metadata-eval 100.0%

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

       4. sqrt-pow1 100.0%

          \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{{im}^{\left(\frac{4}{2}\right)}}\right) \]

       5. metadata-eval 100.0%

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

       6. unpow2 100.0%

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

       7. associate-*r* 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2 \cdot 10^{+39}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{im}^{6} \cdot 0.125}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.45:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.45) (cos re) (* (cos re) (* im (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.45) {
		tmp = cos(re);
	} else {
		tmp = cos(re) * (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 <= 1.45d0) then
        tmp = cos(re)
    else
        tmp = cos(re) * (im * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.45) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.cos(re) * (im * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.45:
		tmp = math.cos(re)
	else:
		tmp = math.cos(re) * (im * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.45)
		tmp = cos(re);
	else
		tmp = Float64(cos(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.45)
		tmp = cos(re);
	else
		tmp = cos(re) * (im * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.45], N[Cos[re], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.44999999999999996

   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 65.5%

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

   if 1.44999999999999996 < 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 48.3%

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

      1. associate-*r* 48.3%

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

      2. distribute-rgt1-in 48.3%

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

   5. Simplified 48.3%

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

   6. Taylor expanded in im around inf 48.3%

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

      1. associate-*r* 48.3%

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

      2. *-commutative 48.3%

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

   8. Simplified 48.3%

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

      1. add-sqr-sqrt 48.3%

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

      2. sqrt-unprod 72.3%

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

      3. *-commutative 72.3%

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

      4. *-commutative 72.3%

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

      5. swap-sqr 72.3%

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

      6. pow-prod-up 72.3%

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

      7. metadata-eval 72.3%

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

      8. metadata-eval 72.3%

         \[\leadsto \cos re \cdot \sqrt{{im}^{4} \cdot \color{blue}{0.25}} \]

   10. Applied egg-rr 72.3%

       \[\leadsto \cos re \cdot \color{blue}{\sqrt{{im}^{4} \cdot 0.25}} \]
   11. Step-by-step derivation

       1. *-commutative 72.3%

          \[\leadsto \cos re \cdot \sqrt{\color{blue}{0.25 \cdot {im}^{4}}} \]

       2. sqrt-prod 72.3%

          \[\leadsto \cos re \cdot \color{blue}{\left(\sqrt{0.25} \cdot \sqrt{{im}^{4}}\right)} \]

       3. metadata-eval 72.3%

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

       4. sqrt-pow1 48.3%

          \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{{im}^{\left(\frac{4}{2}\right)}}\right) \]

       5. metadata-eval 48.3%

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

       6. unpow2 48.3%

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

       7. associate-*r* 48.3%

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

   12. Applied egg-rr 48.3%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.45:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.03:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.03) (cos re) (+ (* 0.5 (pow im 2.0)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.03) {
		tmp = cos(re);
	} else {
		tmp = (0.5 * pow(im, 2.0)) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.03d0) then
        tmp = cos(re)
    else
        tmp = (0.5d0 * (im ** 2.0d0)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.03) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * Math.pow(im, 2.0)) + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.03:
		tmp = math.cos(re)
	else:
		tmp = (0.5 * math.pow(im, 2.0)) + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.03)
		tmp = cos(re);
	else
		tmp = Float64(Float64(0.5 * (im ^ 2.0)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.03)
		tmp = cos(re);
	else
		tmp = (0.5 * (im ^ 2.0)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.03], N[Cos[re], $MachinePrecision], N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.029999999999999999

   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 65.5%

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

   if 0.029999999999999999 < 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 48.3%

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

      1. associate-*r* 48.3%

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

      2. distribute-rgt1-in 48.3%

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

   5. Simplified 48.3%

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

   6. Taylor expanded in re around 0 30.8%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {im}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.03:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.25 \cdot 10^{+92}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.25e+92) (cos re) (* 0.5 (pow im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.25e+92) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * pow(im, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.25d+92) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.25e+92) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.25e+92:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.25e+92)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.25e+92)
		tmp = cos(re);
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.25e+92], N[Cos[re], $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.25000000000000005e92

   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 59.0%

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

   if 1.25000000000000005e92 < 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 70.5%

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

      1. associate-*r* 70.5%

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

      2. distribute-rgt1-in 70.5%

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

   5. Simplified 70.5%

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

   6. Taylor expanded in im around inf 70.5%

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

      1. associate-*r* 70.5%

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

      2. *-commutative 70.5%

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

   8. Simplified 70.5%

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

   9. Taylor expanded in re around 0 44.9%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.25 \cdot 10^{+92}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \cos re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (cos re))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return Math.cos(re);
}

Python

def code(re, im):
	return math.cos(re)

Julia

function code(re, im)
	return cos(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Cos[re], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 49.4%

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

4. Final simplification 49.4%

   \[\leadsto \cos re \]
5. Add Preprocessing

Alternative 10: 2.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 2.3%

   \[\leadsto \color{blue}{\log \left({1}^{\cos re}\right)} \]
5. Step-by-step derivation

   1. pow-base-1 2.3%

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

   2. metadata-eval 2.3%

      \[\leadsto \color{blue}{0} \]

6. Simplified 2.3%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 2.3%

   \[\leadsto 0 \]
8. Add Preprocessing

Alternative 11: 28.0% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 28.0%

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

   1. +-inverses 28.0%

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

   2. +-rgt-identity 28.0%

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

   3. *-inverses 28.0%

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

6. Simplified 28.0%

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

7. Final simplification 28.0%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))