math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 13

Speedup: 1.0×

• 49.9% 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: 68.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 485000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+48}:\\ \;\;\;\;0.25 + 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 485000.0)
   (cos re)
   (if (<= im 3.9e+48)
     (+ 0.25 (* 0.25 (log1p (expm1 re))))
     (if (<= im 1.35e+154)
       (* 0.5 (+ 2.0 (* 0.08333333333333333 (pow im 4.0))))
       (* (cos re) (* 0.5 (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 485000.0) {
		tmp = cos(re);
	} else if (im <= 3.9e+48) {
		tmp = 0.25 + (0.25 * log1p(expm1(re)));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (2.0 + (0.08333333333333333 * pow(im, 4.0)));
	} else {
		tmp = cos(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 485000.0) {
		tmp = Math.cos(re);
	} else if (im <= 3.9e+48) {
		tmp = 0.25 + (0.25 * Math.log1p(Math.expm1(re)));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (2.0 + (0.08333333333333333 * Math.pow(im, 4.0)));
	} else {
		tmp = Math.cos(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 485000.0:
		tmp = math.cos(re)
	elif im <= 3.9e+48:
		tmp = 0.25 + (0.25 * math.log1p(math.expm1(re)))
	elif im <= 1.35e+154:
		tmp = 0.5 * (2.0 + (0.08333333333333333 * math.pow(im, 4.0)))
	else:
		tmp = math.cos(re) * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 485000.0)
		tmp = cos(re);
	elseif (im <= 3.9e+48)
		tmp = Float64(0.25 + Float64(0.25 * log1p(expm1(re))));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(2.0 + Float64(0.08333333333333333 * (im ^ 4.0))));
	else
		tmp = Float64(cos(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 485000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.9e+48], N[(0.25 + N[(0.25 * N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(2.0 + N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 485000

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 66.7%

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

   if 485000 < im < 3.9000000000000001e48

   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 1.7%

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

      1. *-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in re around 0 15.2%

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

      1. *-commutative 15.2%

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

   9. Simplified 15.2%

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

   11. Applied egg-rr 29.5%

       \[\leadsto 0.25 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)} \cdot 0.25 \]

   if 3.9000000000000001e48 < 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

   3. Taylor expanded in re around 0 85.7%

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

      1. *-commutative 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in im around 0 79.0%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 79.0%

         \[\leadsto \left(2 + {im}^{2} \cdot \left(1 + \color{blue}{{im}^{2} \cdot 0.08333333333333333}\right)\right) \cdot 0.5 \]

   8. Simplified 79.0%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \cdot 0.5 \]

   9. Taylor expanded in im around inf 79.0%

      \[\leadsto \left(2 + \color{blue}{0.08333333333333333 \cdot {im}^{4}}\right) \cdot 0.5 \]

   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 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%

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

   8. Simplified 100.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 485000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+48}:\\ \;\;\;\;0.25 + 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.9e-5)
   (cos re)
   (if (<= im 1.35e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.9e-5) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (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 <= 1.9d-5) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.9e-5) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.9e-5:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.9e-5)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.9e-5)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.9e-5], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.9000000000000001e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 67.5%

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

   if 1.9000000000000001e-5 < im < 1.35000000000000003e154

   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 69.0%

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

      1. *-commutative 69.0%

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

   5. Simplified 69.0%

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

   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 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%

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

   8. Simplified 100.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.8:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.8)
   (* (* 0.5 (cos re)) (fma im im 2.0))
   (if (<= im 1.35e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.8) {
		tmp = (0.5 * cos(re)) * fma(im, im, 2.0);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.8)
		tmp = Float64(Float64(0.5 * cos(re)) * fma(im, im, 2.0));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.8], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.80000000000000004

   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 81.9%

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

      1. +-commutative 81.9%

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

      2. unpow2 81.9%

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

      3. fma-define 81.9%

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

   5. Simplified 81.9%

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

   if 1.80000000000000004 < 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

   3. Taylor expanded in re around 0 72.7%

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

      1. *-commutative 72.7%

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

   5. Simplified 72.7%

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

   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 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%

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

   8. Simplified 100.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.8:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 485000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+48}:\\ \;\;\;\;0.25 + 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.08333333333333333 \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 485000.0)
   (cos re)
   (if (<= im 3.9e+48)
     (+ 0.25 (* 0.25 (log1p (expm1 re))))
     (* 0.5 (+ 2.0 (* 0.08333333333333333 (pow im 4.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 485000.0) {
		tmp = cos(re);
	} else if (im <= 3.9e+48) {
		tmp = 0.25 + (0.25 * log1p(expm1(re)));
	} else {
		tmp = 0.5 * (2.0 + (0.08333333333333333 * pow(im, 4.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 485000.0) {
		tmp = Math.cos(re);
	} else if (im <= 3.9e+48) {
		tmp = 0.25 + (0.25 * Math.log1p(Math.expm1(re)));
	} else {
		tmp = 0.5 * (2.0 + (0.08333333333333333 * Math.pow(im, 4.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 485000.0:
		tmp = math.cos(re)
	elif im <= 3.9e+48:
		tmp = 0.25 + (0.25 * math.log1p(math.expm1(re)))
	else:
		tmp = 0.5 * (2.0 + (0.08333333333333333 * math.pow(im, 4.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 485000.0)
		tmp = cos(re);
	elseif (im <= 3.9e+48)
		tmp = Float64(0.25 + Float64(0.25 * log1p(expm1(re))));
	else
		tmp = Float64(0.5 * Float64(2.0 + Float64(0.08333333333333333 * (im ^ 4.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 485000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.9e+48], N[(0.25 + N[(0.25 * N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 + N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 485000

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 66.7%

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

   if 485000 < im < 3.9000000000000001e48

   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 1.7%

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

      1. *-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in re around 0 15.2%

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

      1. *-commutative 15.2%

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

   9. Simplified 15.2%

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

   11. Applied egg-rr 29.5%

       \[\leadsto 0.25 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)} \cdot 0.25 \]

   if 3.9000000000000001e48 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 75.0%

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

      1. *-commutative 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in im around 0 73.2%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 73.2%

         \[\leadsto \left(2 + {im}^{2} \cdot \left(1 + \color{blue}{{im}^{2} \cdot 0.08333333333333333}\right)\right) \cdot 0.5 \]

   8. Simplified 73.2%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \cdot 0.5 \]

   9. Taylor expanded in im around inf 73.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 485000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+48}:\\ \;\;\;\;0.25 + 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.08333333333333333 \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.22 \cdot 10^{+121}:\\ \;\;\;\;1 + 0.5 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.6e+26)
   (cos re)
   (if (<= im 1.22e+121)
     (+ 1.0 (* 0.5 (pow re 2.0)))
     (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.6e+26) {
		tmp = cos(re);
	} else if (im <= 1.22e+121) {
		tmp = 1.0 + (0.5 * 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 <= 4.6e+26)
		tmp = cos(re);
	elseif (im <= 1.22e+121)
		tmp = Float64(1.0 + Float64(0.5 * (re ^ 2.0)));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.6e+26], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.22e+121], N[(1.0 + N[(0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.6000000000000001e26

   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 65.4%

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

   if 4.6000000000000001e26 < im < 1.22000000000000011e121

   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 4.7%

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

      1. +-commutative 4.7%

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

      2. unpow2 4.7%

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

      3. fma-define 4.7%

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

   5. Simplified 4.7%

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

   6. Taylor expanded in im around inf 4.7%

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

      1. associate-*r* 4.7%

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

   8. Simplified 4.7%

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

   10. Applied egg-rr 3.1%

       \[\leadsto \color{blue}{{\cos re}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 3.1%

          \[\leadsto \color{blue}{\frac{1}{\cos re}} \]

   12. Simplified 3.1%

       \[\leadsto \color{blue}{\frac{1}{\cos re}} \]

   13. Taylor expanded in re around 0 26.7%

       \[\leadsto \color{blue}{1 + 0.5 \cdot {re}^{2}} \]

   if 1.22000000000000011e121 < 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 89.3%

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

      1. +-commutative 89.3%

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

      2. unpow2 89.3%

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

      3. fma-define 89.3%

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

   5. Simplified 89.3%

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

   6. Taylor expanded in re around 0 63.7%

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

      1. +-commutative 63.7%

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

      2. unpow2 63.7%

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

      3. fma-undefine 63.7%

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

   8. Simplified 63.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.22 \cdot 10^{+121}:\\ \;\;\;\;1 + 0.5 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.36 \cdot 10^{+121}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.8e+26)
   (cos re)
   (if (<= im 1.36e+121) (+ 0.25 (* 0.25 (* re re))) (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.8e+26) {
		tmp = cos(re);
	} else if (im <= 1.36e+121) {
		tmp = 0.25 + (0.25 * (re * re));
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.8e+26)
		tmp = cos(re);
	elseif (im <= 1.36e+121)
		tmp = Float64(0.25 + Float64(0.25 * Float64(re * re)));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.8e+26], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.36e+121], N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.8000000000000005e26

   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 65.4%

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

   if 6.8000000000000005e26 < im < 1.36e121

   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 2.5%

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

      1. *-commutative 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in re around 0 26.7%

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

      1. *-commutative 26.7%

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

   9. Simplified 26.7%

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

   11. Applied egg-rr 26.7%

       \[\leadsto 0.25 + \color{blue}{\left(re \cdot re\right)} \cdot 0.25 \]

   if 1.36e121 < 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 89.3%

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

      1. +-commutative 89.3%

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

      2. unpow2 89.3%

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

      3. fma-define 89.3%

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

   5. Simplified 89.3%

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

   6. Taylor expanded in re around 0 63.7%

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

      1. +-commutative 63.7%

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

      2. unpow2 63.7%

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

      3. fma-undefine 63.7%

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

   8. Simplified 63.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.36 \cdot 10^{+121}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.36 \cdot 10^{+121}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.1e+26)
   (cos re)
   (if (<= im 1.36e+121) (+ 0.25 (* 0.25 (* re re))) (* 0.5 (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.1e+26) {
		tmp = cos(re);
	} else if (im <= 1.36e+121) {
		tmp = 0.25 + (0.25 * (re * 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 <= 4.1d+26) then
        tmp = cos(re)
    else if (im <= 1.36d+121) then
        tmp = 0.25d0 + (0.25d0 * (re * 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 <= 4.1e+26) {
		tmp = Math.cos(re);
	} else if (im <= 1.36e+121) {
		tmp = 0.25 + (0.25 * (re * re));
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.1e+26:
		tmp = math.cos(re)
	elif im <= 1.36e+121:
		tmp = 0.25 + (0.25 * (re * re))
	else:
		tmp = 0.5 * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.1e+26)
		tmp = cos(re);
	elseif (im <= 1.36e+121)
		tmp = Float64(0.25 + Float64(0.25 * Float64(re * 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 <= 4.1e+26)
		tmp = cos(re);
	elseif (im <= 1.36e+121)
		tmp = 0.25 + (0.25 * (re * re));
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.1e+26], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.36e+121], N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.09999999999999983e26

   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 65.4%

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

   if 4.09999999999999983e26 < im < 1.36e121

   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 2.5%

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

      1. *-commutative 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in re around 0 26.7%

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

      1. *-commutative 26.7%

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

   9. Simplified 26.7%

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

   11. Applied egg-rr 26.7%

       \[\leadsto 0.25 + \color{blue}{\left(re \cdot re\right)} \cdot 0.25 \]

   if 1.36e121 < 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 89.3%

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

      1. +-commutative 89.3%

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

      2. unpow2 89.3%

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

      3. fma-define 89.3%

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

   5. Simplified 89.3%

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

   6. Taylor expanded in re around 0 63.7%

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

      1. +-commutative 63.7%

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

      2. unpow2 63.7%

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

      3. fma-undefine 63.7%

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

   8. Simplified 63.7%

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

   9. Taylor expanded in im around inf 63.7%

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

       1. *-commutative 63.7%

          \[\leadsto \color{blue}{{im}^{2} \cdot 0.5} \]

   11. Simplified 63.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.36 \cdot 10^{+121}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{+29}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.08333333333333333 \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.5e+29)
   (cos re)
   (* 0.5 (+ 2.0 (* 0.08333333333333333 (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.5e+29) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (2.0 + (0.08333333333333333 * 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 <= 9.5d+29) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (2.0d0 + (0.08333333333333333d0 * (im ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.5e+29) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (2.0 + (0.08333333333333333 * Math.pow(im, 4.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.5e+29:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (2.0 + (0.08333333333333333 * math.pow(im, 4.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.5e+29)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(2.0 + Float64(0.08333333333333333 * (im ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.5e+29)
		tmp = cos(re);
	else
		tmp = 0.5 * (2.0 + (0.08333333333333333 * (im ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.5e+29], N[Cos[re], $MachinePrecision], N[(0.5 * N[(2.0 + N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.5000000000000003e29

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 64.8%

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

   if 9.5000000000000003e29 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 75.5%

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

      1. *-commutative 75.5%

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

   5. Simplified 75.5%

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

   6. Taylor expanded in im around 0 71.9%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 71.9%

         \[\leadsto \left(2 + {im}^{2} \cdot \left(1 + \color{blue}{{im}^{2} \cdot 0.08333333333333333}\right)\right) \cdot 0.5 \]

   8. Simplified 71.9%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \cdot 0.5 \]

   9. Taylor expanded in im around inf 71.9%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{+29}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.08333333333333333 \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.2e+26) (cos re) (+ 0.25 (* 0.25 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.2e+26) {
		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 <= 4.2d+26) 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 <= 4.2e+26) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.25 + (0.25 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.2e+26:
		tmp = math.cos(re)
	else:
		tmp = 0.25 + (0.25 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.2e+26)
		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 <= 4.2e+26)
		tmp = cos(re);
	else
		tmp = 0.25 + (0.25 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.2e+26], 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 < 4.2000000000000002e26

   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 65.4%

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

   if 4.2000000000000002e26 < 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 2.5%

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

      1. *-commutative 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in re around 0 14.9%

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

      1. *-commutative 14.9%

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

   9. Simplified 14.9%

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

   11. Applied egg-rr 14.9%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 31.2% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 485000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 485000.0) 1.0 (+ 0.25 (* 0.25 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 485000.0) {
		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 <= 485000.0d0) 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 <= 485000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.25 + (0.25 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 485000.0:
		tmp = 1.0
	else:
		tmp = 0.25 + (0.25 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 485000.0)
		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 <= 485000.0)
		tmp = 1.0;
	else
		tmp = 0.25 + (0.25 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 485000.0], 1.0, N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 485000

   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 39.1%

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

      1. +-inverses 39.1%

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

      2. +-rgt-identity 39.1%

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

      3. *-inverses 39.1%

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

   6. Simplified 39.1%

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

   if 485000 < 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 2.4%

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

      1. *-commutative 2.4%

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

   6. Simplified 2.4%

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

   7. Taylor expanded in re around 0 13.9%

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

      1. *-commutative 13.9%

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

   9. Simplified 13.9%

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

   11. Applied egg-rr 13.9%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 485000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \]
5. 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 7.9%

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

   1. *-commutative 7.9%

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

6. Simplified 7.9%

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

7. Taylor expanded in re around 0 7.9%

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

8. Final simplification 7.9%

   \[\leadsto 0.25 \]
9. Add Preprocessing

Alternative 13: 28.1% 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 30.6%

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

   1. +-inverses 30.6%

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

   2. +-rgt-identity 30.6%

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

   3. *-inverses 30.6%

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

6. Simplified 30.6%

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

7. Final simplification 30.6%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))