math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 13

Speedup: 1.0×

• 50.1% 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: 70.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+69}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8000.0)
   (cos re)
   (if (<= im 6.8e+69)
     (+ 0.25 (pow re -2.0))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8000.0) {
		tmp = cos(re);
	} else if (im <= 6.8e+69) {
		tmp = 0.25 + pow(re, -2.0);
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 8000.0d0) then
        tmp = cos(re)
    else if (im <= 6.8d+69) then
        tmp = 0.25d0 + (re ** (-2.0d0))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8000.0) {
		tmp = Math.cos(re);
	} else if (im <= 6.8e+69) {
		tmp = 0.25 + Math.pow(re, -2.0);
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8000.0:
		tmp = math.cos(re)
	elif im <= 6.8e+69:
		tmp = 0.25 + math.pow(re, -2.0)
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8000.0)
		tmp = cos(re);
	elseif (im <= 6.8e+69)
		tmp = Float64(0.25 + (re ^ -2.0));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8000.0)
		tmp = cos(re);
	elseif (im <= 6.8e+69)
		tmp = 0.25 + (re ^ -2.0);
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 6.8e+69], N[(0.25 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8e3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 68.8%

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

   if 8e3 < im < 6.79999999999999973e69

   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%

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

   5. Taylor expanded in re around 0 3.1%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 3.1%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 3.1%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   9. Applied egg-rr 19.0%

      \[\leadsto 0.25 + \color{blue}{{re}^{-2}} \]

   if 6.79999999999999973e69 < 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 97.9%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 97.9%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 97.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 97.9%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 97.9%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 97.9%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   6. Taylor expanded in im around inf 97.9%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+69}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0019:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0019)
   (cos re)
   (if (<= im 1.16e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0019) {
		tmp = cos(re);
	} else if (im <= 1.16e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0019d0) then
        tmp = cos(re)
    else if (im <= 1.16d+77) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0019) {
		tmp = Math.cos(re);
	} else if (im <= 1.16e+77) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0019:
		tmp = math.cos(re)
	elif im <= 1.16e+77:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0019)
		tmp = cos(re);
	elseif (im <= 1.16e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0019)
		tmp = cos(re);
	elseif (im <= 1.16e+77)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0019], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.16e+77], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0019

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 69.4%

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

   if 0.0019 < im < 1.1600000000000001e77

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 73.2%

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

   if 1.1600000000000001e77 < 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%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0019:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.035:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.035)
   (* (* 0.5 (cos re)) (fma im im 2.0))
   (if (<= im 1.16e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.035) {
		tmp = (0.5 * cos(re)) * fma(im, im, 2.0);
	} else if (im <= 1.16e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.035)
		tmp = Float64(Float64(0.5 * cos(re)) * fma(im, im, 2.0));
	elseif (im <= 1.16e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.035], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.16e+77], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.035000000000000003

   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 83.7%

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

      1. +-commutative 83.7%

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

      2. unpow2 83.7%

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

      3. fma-define 83.7%

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

   5. Simplified 83.7%

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

   if 0.035000000000000003 < im < 1.1600000000000001e77

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 73.2%

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

   if 1.1600000000000001e77 < 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%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.035:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+95}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8000.0)
   (cos re)
   (if (<= im 1.8e+95)
     (+ 0.25 (pow re -2.0))
     (* (fma im im 2.0) (+ 0.5 (* (* re re) -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8000.0) {
		tmp = cos(re);
	} else if (im <= 1.8e+95) {
		tmp = 0.25 + pow(re, -2.0);
	} else {
		tmp = fma(im, im, 2.0) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8000.0)
		tmp = cos(re);
	elseif (im <= 1.8e+95)
		tmp = Float64(0.25 + (re ^ -2.0));
	else
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.8e+95], N[(0.25 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8e3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 68.8%

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

   if 8e3 < im < 1.79999999999999989e95

   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%

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

   5. Taylor expanded in re around 0 3.1%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 3.1%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 3.1%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   9. Applied egg-rr 16.0%

      \[\leadsto 0.25 + \color{blue}{{re}^{-2}} \]

   if 1.79999999999999989e95 < 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 73.0%

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

      1. +-commutative 73.0%

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

      2. unpow2 73.0%

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

      3. fma-define 73.0%

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

   5. Simplified 73.0%

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

   6. Taylor expanded in re around 0 13.3%

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

      1. associate-*r* 13.3%

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

      2. distribute-rgt-out 64.5%

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

      3. +-commutative 64.5%

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

      4. unpow2 64.5%

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

      5. fma-undefine 64.5%

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

      6. *-commutative 64.5%

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

   8. Simplified 64.5%

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

   10. Applied egg-rr 64.5%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+95}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+95}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 + re \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8000.0)
   (cos re)
   (if (<= im 2e+95)
     (+ 0.25 (pow re -2.0))
     (* (fma im im 2.0) (+ 0.5 (* re -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8000.0) {
		tmp = cos(re);
	} else if (im <= 2e+95) {
		tmp = 0.25 + pow(re, -2.0);
	} else {
		tmp = fma(im, im, 2.0) * (0.5 + (re * -0.25));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8000.0)
		tmp = cos(re);
	elseif (im <= 2e+95)
		tmp = Float64(0.25 + (re ^ -2.0));
	else
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 + Float64(re * -0.25)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2e+95], N[(0.25 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 + N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8e3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 68.8%

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

   if 8e3 < im < 2.00000000000000004e95

   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%

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

   5. Taylor expanded in re around 0 3.1%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 3.1%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 3.1%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   9. Applied egg-rr 16.0%

      \[\leadsto 0.25 + \color{blue}{{re}^{-2}} \]

   if 2.00000000000000004e95 < 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 73.0%

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

      1. +-commutative 73.0%

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

      2. unpow2 73.0%

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

      3. fma-define 73.0%

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

   5. Simplified 73.0%

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

   6. Taylor expanded in re around 0 13.3%

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

      1. associate-*r* 13.3%

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

      2. distribute-rgt-out 64.5%

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

      3. +-commutative 64.5%

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

      4. unpow2 64.5%

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

      5. fma-undefine 64.5%

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

      6. *-commutative 64.5%

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

   8. Simplified 64.5%

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

   10. Applied egg-rr 76.4%

       \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(\color{blue}{\log \left(e^{re}\right)} \cdot -0.25 + 0.5\right) \]
   11. Step-by-step derivation

       1. rem-log-exp 74.2%

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

   12. Simplified 74.2%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+95}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 + re \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8000.0)
   (cos re)
   (if (<= im 1.35e+154) (+ 0.25 (pow re -2.0)) (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8000.0) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.25 + pow(re, -2.0);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8000.0)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(0.25 + (re ^ -2.0));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.25 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8e3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 68.8%

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

   if 8e3 < im < 1.35000000000000003e154

   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 2.1%

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

   5. Taylor expanded in re around 0 6.2%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 6.2%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 6.2%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   9. Applied egg-rr 17.0%

      \[\leadsto 0.25 + \color{blue}{{re}^{-2}} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

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

   6. Taylor expanded in re around 0 72.4%

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

      1. +-commutative 72.4%

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

      2. unpow2 72.4%

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

      3. fma-undefine 72.4%

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

   8. Simplified 72.4%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8000.0) (cos re) (+ 0.25 (pow re -2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8000.0) {
		tmp = cos(re);
	} else {
		tmp = 0.25 + pow(re, -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 <= 8000.0d0) then
        tmp = cos(re)
    else
        tmp = 0.25d0 + (re ** (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8000.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.25 + Math.pow(re, -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8000.0:
		tmp = math.cos(re)
	else:
		tmp = 0.25 + math.pow(re, -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8000.0)
		tmp = cos(re);
	else
		tmp = Float64(0.25 + (re ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8000.0)
		tmp = cos(re);
	else
		tmp = 0.25 + (re ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8000.0], N[Cos[re], $MachinePrecision], N[(0.25 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8e3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 68.8%

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

   if 8e3 < 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 2.3%

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

   5. Taylor expanded in re around 0 7.9%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 7.9%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 7.9%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   9. Applied egg-rr 18.5%

      \[\leadsto 0.25 + \color{blue}{{re}^{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 320000000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 320000000.0) (cos re) (+ 0.25 (* 0.25 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 320000000.0) {
		tmp = cos(re);
	} else {
		tmp = 0.25 + (0.25 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 320000000.0d0) then
        tmp = cos(re)
    else
        tmp = 0.25d0 + (0.25d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 320000000.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.25 + (0.25 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 320000000.0:
		tmp = math.cos(re)
	else:
		tmp = 0.25 + (0.25 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 320000000.0)
		tmp = cos(re);
	else
		tmp = Float64(0.25 + Float64(0.25 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 320000000.0)
		tmp = cos(re);
	else
		tmp = 0.25 + (0.25 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 320000000.0], N[Cos[re], $MachinePrecision], N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.2e8

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 68.5%

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

   if 3.2e8 < 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 2.3%

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

   5. Taylor expanded in re around 0 8.0%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 8.0%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 8.0%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   9. Applied egg-rr 8.0%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 320000000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.4% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 900000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 900000000.0) 1.0 (+ 0.25 (* 0.25 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 900000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.25 + (0.25 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 900000000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.25d0 + (0.25d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 900000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.25 + (0.25 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 900000000.0:
		tmp = 1.0
	else:
		tmp = 0.25 + (0.25 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 900000000.0)
		tmp = 1.0;
	else
		tmp = Float64(0.25 + Float64(0.25 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 900000000.0)
		tmp = 1.0;
	else
		tmp = 0.25 + (0.25 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 900000000.0], 1.0, N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9e8

   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 39.8%

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

      1. +-inverses 39.8%

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

      2. +-rgt-identity 39.8%

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

      3. *-inverses 39.8%

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

   6. Simplified 39.8%

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

   if 9e8 < 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 2.3%

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

   5. Taylor expanded in re around 0 8.0%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 8.0%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 8.0%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   9. Applied egg-rr 8.0%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 900000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 2.4%

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

   1. pow-base-1 2.4%

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

   2. metadata-eval 2.4%

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

6. Simplified 2.4%

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

7. Final simplification 2.4%

   \[\leadsto 0 \]
8. Add Preprocessing

Alternative 12: 8.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

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 8.5%

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

5. Taylor expanded in re around 0 8.7%

   \[\leadsto \color{blue}{0.25} \]

6. Final simplification 8.7%

   \[\leadsto 0.25 \]
7. Add Preprocessing

Alternative 13: 28.2% 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 31.7%

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

   1. +-inverses 31.7%

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

   2. +-rgt-identity 31.7%

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

   3. *-inverses 31.7%

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

6. Simplified 31.7%

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

7. Final simplification 31.7%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))