math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 14

Speedup: 1.0×

• 48.8% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (cos re) (fma 0.5 (exp im) (/ 0.5 (exp im)))))

C

double code(double re, double im) {
	return cos(re) * fma(0.5, exp(im), (0.5 / exp(im)));
}

Julia

function code(re, im)
	return Float64(cos(re) * fma(0.5, exp(im), Float64(0.5 / exp(im))))
end

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (* (cos re) 0.5) (+ (exp im) (exp (- im)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $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. Final simplification 100.0%

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

Alternative 3: 88.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\cos re + \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.35)
   (+ (cos re) (* (cos re) (* 0.5 (* im im))))
   (* (cos re) (+ 0.001953125 (* 0.5 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		tmp = cos(re) + (cos(re) * (0.5 * (im * im)));
	} else {
		tmp = cos(re) * (0.001953125 + (0.5 * exp(im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.35d0) then
        tmp = cos(re) + (cos(re) * (0.5d0 * (im * im)))
    else
        tmp = cos(re) * (0.001953125d0 + (0.5d0 * exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		tmp = Math.cos(re) + (Math.cos(re) * (0.5 * (im * im)));
	} else {
		tmp = Math.cos(re) * (0.001953125 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.35:
		tmp = math.cos(re) + (math.cos(re) * (0.5 * (im * im)))
	else:
		tmp = math.cos(re) * (0.001953125 + (0.5 * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.35)
		tmp = Float64(cos(re) + Float64(cos(re) * Float64(0.5 * Float64(im * im))));
	else
		tmp = Float64(cos(re) * Float64(0.001953125 + Float64(0.5 * exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.35)
		tmp = cos(re) + (cos(re) * (0.5 * (im * im)));
	else
		tmp = cos(re) * (0.001953125 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.35], N[(N[Cos[re], $MachinePrecision] + N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.001953125 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3500000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.5%

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

      1. unpow2 84.5%

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

   4. Simplified 84.5%

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

      1. +-commutative 84.5%

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

      2. distribute-lft-in 84.5%

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

      3. *-commutative 84.5%

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

      4. associate-*l* 84.5%

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

      5. *-commutative 84.5%

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

      6. associate-*r* 84.5%

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

      7. metadata-eval 84.5%

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

      8. *-un-lft-identity 84.5%

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

   6. Applied egg-rr 84.5%

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

   if 1.3500000000000001 < im

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 99.2%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around inf 99.2%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\cos re + \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 4: 88.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.35)
   (* (* (cos re) 0.5) (+ (* im im) 2.0))
   (* (cos re) (+ 0.001953125 (* 0.5 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		tmp = (cos(re) * 0.5) * ((im * im) + 2.0);
	} else {
		tmp = cos(re) * (0.001953125 + (0.5 * exp(im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.35d0) then
        tmp = (cos(re) * 0.5d0) * ((im * im) + 2.0d0)
    else
        tmp = cos(re) * (0.001953125d0 + (0.5d0 * exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		tmp = (Math.cos(re) * 0.5) * ((im * im) + 2.0);
	} else {
		tmp = Math.cos(re) * (0.001953125 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.35:
		tmp = (math.cos(re) * 0.5) * ((im * im) + 2.0)
	else:
		tmp = math.cos(re) * (0.001953125 + (0.5 * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.35)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(Float64(im * im) + 2.0));
	else
		tmp = Float64(cos(re) * Float64(0.001953125 + Float64(0.5 * exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.35)
		tmp = (cos(re) * 0.5) * ((im * im) + 2.0);
	else
		tmp = cos(re) * (0.001953125 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.35], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.001953125 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3500000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.5%

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

      1. unpow2 84.5%

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

   4. Simplified 84.5%

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

   if 1.3500000000000001 < im

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 99.2%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around inf 99.2%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 5: 85.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;0.001953125 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(0.501953125 + 0.5 \cdot im\right) + \left(im \cdot im\right) \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.2)
   (* (* (cos re) 0.5) (+ (* im im) 2.0))
   (if (<= im 1.3e+103)
     (+ 0.001953125 (* 0.5 (exp im)))
     (*
      (cos re)
      (+
       (+ 0.501953125 (* 0.5 im))
       (* (* im im) (+ (* im 0.08333333333333333) 0.25)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = (cos(re) * 0.5) * ((im * im) + 2.0);
	} else if (im <= 1.3e+103) {
		tmp = 0.001953125 + (0.5 * exp(im));
	} else {
		tmp = cos(re) * ((0.501953125 + (0.5 * im)) + ((im * im) * ((im * 0.08333333333333333) + 0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.2d0) then
        tmp = (cos(re) * 0.5d0) * ((im * im) + 2.0d0)
    else if (im <= 1.3d+103) then
        tmp = 0.001953125d0 + (0.5d0 * exp(im))
    else
        tmp = cos(re) * ((0.501953125d0 + (0.5d0 * im)) + ((im * im) * ((im * 0.08333333333333333d0) + 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = (Math.cos(re) * 0.5) * ((im * im) + 2.0);
	} else if (im <= 1.3e+103) {
		tmp = 0.001953125 + (0.5 * Math.exp(im));
	} else {
		tmp = Math.cos(re) * ((0.501953125 + (0.5 * im)) + ((im * im) * ((im * 0.08333333333333333) + 0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.2:
		tmp = (math.cos(re) * 0.5) * ((im * im) + 2.0)
	elif im <= 1.3e+103:
		tmp = 0.001953125 + (0.5 * math.exp(im))
	else:
		tmp = math.cos(re) * ((0.501953125 + (0.5 * im)) + ((im * im) * ((im * 0.08333333333333333) + 0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.2)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(Float64(im * im) + 2.0));
	elseif (im <= 1.3e+103)
		tmp = Float64(0.001953125 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.501953125 + Float64(0.5 * im)) + Float64(Float64(im * im) * Float64(Float64(im * 0.08333333333333333) + 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.2)
		tmp = (cos(re) * 0.5) * ((im * im) + 2.0);
	elseif (im <= 1.3e+103)
		tmp = 0.001953125 + (0.5 * exp(im));
	else
		tmp = cos(re) * ((0.501953125 + (0.5 * im)) + ((im * im) * ((im * 0.08333333333333333) + 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.2], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.3e+103], N[(0.001953125 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.501953125 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision] + N[(N[(im * im), $MachinePrecision] * N[(N[(im * 0.08333333333333333), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.20000000000000018

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.2%

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

      1. unpow2 84.2%

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

   4. Simplified 84.2%

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

   if 6.20000000000000018 < im < 1.3000000000000001e103

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 64.6%

      \[\leadsto \color{blue}{0.001953125 + 0.5 \cdot e^{im}} \]

   if 1.3000000000000001e103 < im

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + \left(0.25 \cdot \left({im}^{2} \cdot \cos re\right) + \left(0.5 \cdot \left(im \cdot \cos re\right) + 0.501953125 \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(0.5 \cdot \left(im \cdot \cos re\right) + 0.501953125 \cdot \cos re\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(im \cdot \cos re\right) + 0.501953125 \cdot \cos re\right) + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]

      3. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.501953125 \cdot \cos re + 0.5 \cdot \left(im \cdot \cos re\right)\right)} + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      4. associate-*r* 100.0%

         \[\leadsto \left(0.501953125 \cdot \cos re + \color{blue}{\left(0.5 \cdot im\right) \cdot \cos re}\right) + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(0.501953125 + 0.5 \cdot im\right)} + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      6. associate-*r* 100.0%

         \[\leadsto \cos re \cdot \left(0.501953125 + 0.5 \cdot im\right) + \left(\color{blue}{\left(0.08333333333333333 \cdot {im}^{3}\right) \cdot \cos re} + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      7. unpow2 100.0%

         \[\leadsto \cos re \cdot \left(0.501953125 + 0.5 \cdot im\right) + \left(\left(0.08333333333333333 \cdot {im}^{3}\right) \cdot \cos re + 0.25 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \cos re\right)\right) \]

      8. associate-*r* 100.0%

         \[\leadsto \cos re \cdot \left(0.501953125 + 0.5 \cdot im\right) + \left(\left(0.08333333333333333 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(0.25 \cdot \left(im \cdot im\right)\right) \cdot \cos re}\right) \]

      9. distribute-rgt-out 100.0%

         \[\leadsto \cos re \cdot \left(0.501953125 + 0.5 \cdot im\right) + \color{blue}{\cos re \cdot \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot \left(im \cdot im\right)\right)} \]

      10. distribute-lft-out 100.0%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.501953125 + 0.5 \cdot im\right) + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot \left(im \cdot im\right)\right)\right)} \]

      11. cube-mult 100.0%

          \[\leadsto \cos re \cdot \left(\left(0.501953125 + 0.5 \cdot im\right) + \left(0.08333333333333333 \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} + 0.25 \cdot \left(im \cdot im\right)\right)\right) \]

      12. associate-*r* 100.0%

          \[\leadsto \cos re \cdot \left(\left(0.501953125 + 0.5 \cdot im\right) + \left(\color{blue}{\left(0.08333333333333333 \cdot im\right) \cdot \left(im \cdot im\right)} + 0.25 \cdot \left(im \cdot im\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.501953125 + 0.5 \cdot im\right) + \left(im \cdot im\right) \cdot \left(0.08333333333333333 \cdot im + 0.25\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;0.001953125 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(0.501953125 + 0.5 \cdot im\right) + \left(im \cdot im\right) \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\\ \end{array} \]

Alternative 6: 69.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+154}:\\ \;\;\;\;0.001953125 + 0.5 \cdot e^{im}\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im + -0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.2)
   (cos re)
   (if (<= im 2.3e+154)
     (+ 0.001953125 (* 0.5 (exp im)))
     (if (<= im 5e+165)
       (* 0.5 (* im (+ im (* -0.5 (* im (* re re))))))
       (* 0.5 (fma im im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = cos(re);
	} else if (im <= 2.3e+154) {
		tmp = 0.001953125 + (0.5 * exp(im));
	} else if (im <= 5e+165) {
		tmp = 0.5 * (im * (im + (-0.5 * (im * (re * re)))));
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.2)
		tmp = cos(re);
	elseif (im <= 2.3e+154)
		tmp = Float64(0.001953125 + Float64(0.5 * exp(im)));
	elseif (im <= 5e+165)
		tmp = Float64(0.5 * Float64(im * Float64(im + Float64(-0.5 * Float64(im * Float64(re * re))))));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.2], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.3e+154], N[(0.001953125 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e+165], N[(0.5 * N[(im * N[(im + N[(-0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 6.20000000000000018

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.2%

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

      1. unpow2 84.2%

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

   4. Simplified 84.2%

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

   5. Taylor expanded in im around 0 64.6%

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

   if 6.20000000000000018 < im < 2.3e154

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 65.2%

      \[\leadsto \color{blue}{0.001953125 + 0.5 \cdot e^{im}} \]

   if 2.3e154 < im < 4.9999999999999997e165

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 100.0%

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

      1. unpow2 100.0%

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

   10. Simplified 100.0%

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

   if 4.9999999999999997e165 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 96.2%

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

      1. unpow2 96.2%

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

      2. +-commutative 96.2%

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

      3. fma-udef 96.2%

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

   7. Simplified 96.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+154}:\\ \;\;\;\;0.001953125 + 0.5 \cdot e^{im}\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im + -0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]

Alternative 7: 72.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.001953125 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.2)
   (cos re)
   (if (<= im 1.35e+154)
     (+ 0.001953125 (* 0.5 (exp im)))
     (* 0.5 (* im (* (cos re) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.001953125 + (0.5 * exp(im));
	} else {
		tmp = 0.5 * (im * (cos(re) * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.2d0) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = 0.001953125d0 + (0.5d0 * exp(im))
    else
        tmp = 0.5d0 * (im * (cos(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.001953125 + (0.5 * Math.exp(im));
	} else {
		tmp = 0.5 * (im * (Math.cos(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.2:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = 0.001953125 + (0.5 * math.exp(im))
	else:
		tmp = 0.5 * (im * (math.cos(re) * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.2)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(0.001953125 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(0.5 * Float64(im * Float64(cos(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.2)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = 0.001953125 + (0.5 * exp(im));
	else
		tmp = 0.5 * (im * (cos(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.2], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.001953125 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.20000000000000018

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.2%

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

      1. unpow2 84.2%

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

   4. Simplified 84.2%

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

   5. Taylor expanded in im around 0 64.6%

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

   if 6.20000000000000018 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 65.2%

      \[\leadsto \color{blue}{0.001953125 + 0.5 \cdot e^{im}} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.001953125 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 84.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;0.001953125 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.2)
   (* (* (cos re) 0.5) (+ (* im im) 2.0))
   (if (<= im 1.5e+154)
     (+ 0.001953125 (* 0.5 (exp im)))
     (* 0.5 (* im (* (cos re) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = (cos(re) * 0.5) * ((im * im) + 2.0);
	} else if (im <= 1.5e+154) {
		tmp = 0.001953125 + (0.5 * exp(im));
	} else {
		tmp = 0.5 * (im * (cos(re) * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.2d0) then
        tmp = (cos(re) * 0.5d0) * ((im * im) + 2.0d0)
    else if (im <= 1.5d+154) then
        tmp = 0.001953125d0 + (0.5d0 * exp(im))
    else
        tmp = 0.5d0 * (im * (cos(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = (Math.cos(re) * 0.5) * ((im * im) + 2.0);
	} else if (im <= 1.5e+154) {
		tmp = 0.001953125 + (0.5 * Math.exp(im));
	} else {
		tmp = 0.5 * (im * (Math.cos(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.2:
		tmp = (math.cos(re) * 0.5) * ((im * im) + 2.0)
	elif im <= 1.5e+154:
		tmp = 0.001953125 + (0.5 * math.exp(im))
	else:
		tmp = 0.5 * (im * (math.cos(re) * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.2)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(Float64(im * im) + 2.0));
	elseif (im <= 1.5e+154)
		tmp = Float64(0.001953125 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(0.5 * Float64(im * Float64(cos(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.2)
		tmp = (cos(re) * 0.5) * ((im * im) + 2.0);
	elseif (im <= 1.5e+154)
		tmp = 0.001953125 + (0.5 * exp(im));
	else
		tmp = 0.5 * (im * (cos(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.2], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.5e+154], N[(0.001953125 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.20000000000000018

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.2%

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

      1. unpow2 84.2%

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

   4. Simplified 84.2%

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

   if 6.20000000000000018 < im < 1.50000000000000013e154

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 65.2%

      \[\leadsto \color{blue}{0.001953125 + 0.5 \cdot e^{im}} \]

   if 1.50000000000000013e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;0.001953125 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 62.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 30500000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im + -0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 30500000000.0)
   (cos re)
   (if (<= im 1.35e+165)
     (* 0.5 (* im (+ im (* -0.5 (* im (* re re))))))
     (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 30500000000.0) {
		tmp = cos(re);
	} else if (im <= 1.35e+165) {
		tmp = 0.5 * (im * (im + (-0.5 * (im * (re * re)))));
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 30500000000.0)
		tmp = cos(re);
	elseif (im <= 1.35e+165)
		tmp = Float64(0.5 * Float64(im * Float64(im + Float64(-0.5 * Float64(im * Float64(re * re))))));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 30500000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+165], N[(0.5 * N[(im * N[(im + N[(-0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.05e10

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 81.8%

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

      1. unpow2 81.8%

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

   4. Simplified 81.8%

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

   5. Taylor expanded in im around 0 62.8%

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

   if 3.05e10 < im < 1.35e165

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 17.1%

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

      1. unpow2 17.1%

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

   4. Simplified 17.1%

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

   5. Taylor expanded in im around inf 17.1%

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

      1. unpow2 17.1%

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

      2. associate-*l* 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in re around 0 37.8%

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

      1. unpow2 37.8%

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

   10. Simplified 37.8%

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

   if 1.35e165 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 96.2%

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

      1. unpow2 96.2%

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

      2. +-commutative 96.2%

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

      3. fma-udef 96.2%

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

   7. Simplified 96.2%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 30500000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im + -0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]

Alternative 10: 62.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 100000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+165}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im + -0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 100000000000.0)
   (cos re)
   (if (<= im 1e+165)
     (* 0.5 (* im (+ im (* -0.5 (* im (* re re))))))
     (* im (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 100000000000.0) {
		tmp = cos(re);
	} else if (im <= 1e+165) {
		tmp = 0.5 * (im * (im + (-0.5 * (im * (re * re)))));
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 100000000000.0d0) then
        tmp = cos(re)
    else if (im <= 1d+165) then
        tmp = 0.5d0 * (im * (im + ((-0.5d0) * (im * (re * re)))))
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 100000000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 1e+165) {
		tmp = 0.5 * (im * (im + (-0.5 * (im * (re * re)))));
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 100000000000.0:
		tmp = math.cos(re)
	elif im <= 1e+165:
		tmp = 0.5 * (im * (im + (-0.5 * (im * (re * re)))))
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 100000000000.0)
		tmp = cos(re);
	elseif (im <= 1e+165)
		tmp = Float64(0.5 * Float64(im * Float64(im + Float64(-0.5 * Float64(im * Float64(re * re))))));
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 100000000000.0)
		tmp = cos(re);
	elseif (im <= 1e+165)
		tmp = 0.5 * (im * (im + (-0.5 * (im * (re * re)))));
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 100000000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1e+165], N[(0.5 * N[(im * N[(im + N[(-0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1e11

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 81.8%

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

      1. unpow2 81.8%

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

   4. Simplified 81.8%

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

   5. Taylor expanded in im around 0 62.8%

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

   if 1e11 < im < 9.99999999999999899e164

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 17.1%

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

      1. unpow2 17.1%

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

   4. Simplified 17.1%

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

   5. Taylor expanded in im around inf 17.1%

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

      1. unpow2 17.1%

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

      2. associate-*l* 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in re around 0 37.8%

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

      1. unpow2 37.8%

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

   10. Simplified 37.8%

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

   if 9.99999999999999899e164 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 96.2%

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

      1. unpow2 96.2%

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

      2. associate-*r* 96.2%

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

      3. *-commutative 96.2%

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

   10. Simplified 96.2%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 100000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+165}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im + -0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 11: 40.1% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.8 \cdot 10^{+58}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im + im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.8e+58)
   (+ 1.0 (* -0.5 (* re re)))
   (*
    0.5
    (*
     im
     (+
      im
      (* im (* (* re re) (+ -0.5 (* (* re re) 0.041666666666666664)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.8e+58) {
		tmp = 1.0 + (-0.5 * (re * re));
	} else {
		tmp = 0.5 * (im * (im + (im * ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7.8d+58) then
        tmp = 1.0d0 + ((-0.5d0) * (re * re))
    else
        tmp = 0.5d0 * (im * (im + (im * ((re * re) * ((-0.5d0) + ((re * re) * 0.041666666666666664d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.8e+58) {
		tmp = 1.0 + (-0.5 * (re * re));
	} else {
		tmp = 0.5 * (im * (im + (im * ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.8e+58:
		tmp = 1.0 + (-0.5 * (re * re))
	else:
		tmp = 0.5 * (im * (im + (im * ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.8e+58)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(re * re)));
	else
		tmp = Float64(0.5 * Float64(im * Float64(im + Float64(im * Float64(Float64(re * re) * Float64(-0.5 + Float64(Float64(re * re) * 0.041666666666666664)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.8e+58)
		tmp = 1.0 + (-0.5 * (re * re));
	else
		tmp = 0.5 * (im * (im + (im * ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.8e+58], N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(im + N[(im * N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.8000000000000002e58

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 76.7%

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

      1. unpow2 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in re around 0 37.1%

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

      1. +-commutative 37.1%

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

      2. unpow2 37.1%

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

      3. +-commutative 37.1%

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

      4. fma-udef 37.1%

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

      5. associate-*r* 37.1%

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

      6. unpow2 37.1%

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

      7. +-commutative 37.1%

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

      8. fma-udef 37.1%

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

      9. distribute-rgt-out 50.3%

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

      10. *-commutative 50.3%

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

      11. unpow2 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in im around 0 38.9%

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

      1. distribute-lft-in 38.9%

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

      2. metadata-eval 38.9%

         \[\leadsto \color{blue}{1} + 2 \cdot \left(-0.25 \cdot {re}^{2}\right) \]

      3. associate-*r* 38.9%

         \[\leadsto 1 + \color{blue}{\left(2 \cdot -0.25\right) \cdot {re}^{2}} \]

      4. metadata-eval 38.9%

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

      5. unpow2 38.9%

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

   10. Simplified 38.9%

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

   if 7.8000000000000002e58 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 70.3%

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

      1. unpow2 70.3%

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

   4. Simplified 70.3%

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

   5. Taylor expanded in im around inf 70.3%

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

      1. unpow2 70.3%

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

      2. associate-*l* 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in re around 0 44.6%

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

      1. *-commutative 44.6%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(im + \left(-0.5 \cdot \color{blue}{\left({re}^{2} \cdot im\right)} + 0.041666666666666664 \cdot \left(im \cdot {re}^{4}\right)\right)\right)\right) \]

      2. associate-*r* 44.6%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(im + \left(\color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot im} + 0.041666666666666664 \cdot \left(im \cdot {re}^{4}\right)\right)\right)\right) \]

      3. *-commutative 44.6%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(im + \left(\left(-0.5 \cdot {re}^{2}\right) \cdot im + 0.041666666666666664 \cdot \color{blue}{\left({re}^{4} \cdot im\right)}\right)\right)\right) \]

      4. associate-*r* 44.6%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(im + \left(\left(-0.5 \cdot {re}^{2}\right) \cdot im + \color{blue}{\left(0.041666666666666664 \cdot {re}^{4}\right) \cdot im}\right)\right)\right) \]

      5. distribute-rgt-out 46.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(im + \color{blue}{im \cdot \left(-0.5 \cdot {re}^{2} + 0.041666666666666664 \cdot {re}^{4}\right)}\right)\right) \]

      6. metadata-eval 46.9%

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

      7. pow-sqr 46.9%

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

      8. associate-*r* 46.9%

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

      9. distribute-rgt-out 60.5%

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

      10. unpow2 60.5%

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

      11. unpow2 60.5%

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

   10. Simplified 60.5%

       \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(im + im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.8 \cdot 10^{+58}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im + im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]

Alternative 12: 40.0% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im + -0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.2)
   (+ 1.0 (* -0.5 (* re re)))
   (if (<= im 2e+165)
     (* 0.5 (* im (+ im (* -0.5 (* im (* re re))))))
     (* im (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = 1.0 + (-0.5 * (re * re));
	} else if (im <= 2e+165) {
		tmp = 0.5 * (im * (im + (-0.5 * (im * (re * re)))));
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.2d0) then
        tmp = 1.0d0 + ((-0.5d0) * (re * re))
    else if (im <= 2d+165) then
        tmp = 0.5d0 * (im * (im + ((-0.5d0) * (im * (re * re)))))
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = 1.0 + (-0.5 * (re * re));
	} else if (im <= 2e+165) {
		tmp = 0.5 * (im * (im + (-0.5 * (im * (re * re)))));
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.2:
		tmp = 1.0 + (-0.5 * (re * re))
	elif im <= 2e+165:
		tmp = 0.5 * (im * (im + (-0.5 * (im * (re * re)))))
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.2)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(re * re)));
	elseif (im <= 2e+165)
		tmp = Float64(0.5 * Float64(im * Float64(im + Float64(-0.5 * Float64(im * Float64(re * re))))));
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.2)
		tmp = 1.0 + (-0.5 * (re * re));
	elseif (im <= 2e+165)
		tmp = 0.5 * (im * (im + (-0.5 * (im * (re * re)))));
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.2], N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2e+165], N[(0.5 * N[(im * N[(im + N[(-0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.20000000000000018

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.2%

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

      1. unpow2 84.2%

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

   4. Simplified 84.2%

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

   5. Taylor expanded in re around 0 38.5%

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

      1. +-commutative 38.5%

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

      2. unpow2 38.5%

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

      3. +-commutative 38.5%

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

      4. fma-udef 38.5%

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

      5. associate-*r* 38.5%

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

      6. unpow2 38.5%

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

      7. +-commutative 38.5%

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

      8. fma-udef 38.5%

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

      9. distribute-rgt-out 53.1%

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

      10. *-commutative 53.1%

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

      11. unpow2 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in im around 0 40.6%

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

      1. distribute-lft-in 40.6%

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

      2. metadata-eval 40.6%

         \[\leadsto \color{blue}{1} + 2 \cdot \left(-0.25 \cdot {re}^{2}\right) \]

      3. associate-*r* 40.6%

         \[\leadsto 1 + \color{blue}{\left(2 \cdot -0.25\right) \cdot {re}^{2}} \]

      4. metadata-eval 40.6%

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

      5. unpow2 40.6%

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

   10. Simplified 40.6%

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

   if 6.20000000000000018 < im < 1.9999999999999998e165

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 15.4%

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

      1. unpow2 15.4%

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

   4. Simplified 15.4%

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

   5. Taylor expanded in im around inf 15.4%

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

      1. unpow2 15.4%

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

      2. associate-*l* 15.4%

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

   7. Simplified 15.4%

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

   8. Taylor expanded in re around 0 32.6%

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

      1. unpow2 32.6%

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

   10. Simplified 32.6%

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

   if 1.9999999999999998e165 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 96.2%

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

      1. unpow2 96.2%

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

      2. associate-*r* 96.2%

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

      3. *-commutative 96.2%

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

   10. Simplified 96.2%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im + -0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 13: 38.5% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 10^{+165}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1e+165) (+ 1.0 (* -0.5 (* re re))) (* im (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1e+165) {
		tmp = 1.0 + (-0.5 * (re * re));
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1d+165) then
        tmp = 1.0d0 + ((-0.5d0) * (re * re))
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1e+165) {
		tmp = 1.0 + (-0.5 * (re * re));
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1e+165:
		tmp = 1.0 + (-0.5 * (re * re))
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1e+165)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(re * re)));
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1e+165)
		tmp = 1.0 + (-0.5 * (re * re));
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1e+165], N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.99999999999999899e164

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 72.8%

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

      1. unpow2 72.8%

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

   4. Simplified 72.8%

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

   5. Taylor expanded in re around 0 35.8%

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

      1. +-commutative 35.8%

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

      2. unpow2 35.8%

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

      3. +-commutative 35.8%

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

      4. fma-udef 35.8%

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

      5. associate-*r* 35.8%

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

      6. unpow2 35.8%

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

      7. +-commutative 35.8%

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

      8. fma-udef 35.8%

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

      9. distribute-rgt-out 49.7%

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

      10. *-commutative 49.7%

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

      11. unpow2 49.7%

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

   7. Simplified 49.7%

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

   8. Taylor expanded in im around 0 37.4%

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

      1. distribute-lft-in 37.4%

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

      2. metadata-eval 37.4%

         \[\leadsto \color{blue}{1} + 2 \cdot \left(-0.25 \cdot {re}^{2}\right) \]

      3. associate-*r* 37.4%

         \[\leadsto 1 + \color{blue}{\left(2 \cdot -0.25\right) \cdot {re}^{2}} \]

      4. metadata-eval 37.4%

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

      5. unpow2 37.4%

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

   10. Simplified 37.4%

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

   if 9.99999999999999899e164 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 96.2%

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

      1. unpow2 96.2%

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

      2. associate-*r* 96.2%

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

      3. *-commutative 96.2%

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

   10. Simplified 96.2%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10^{+165}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 14: 21.0% accurate, 61.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 75.6%

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

   1. unpow2 75.6%

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

4. Simplified 75.6%

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

5. Taylor expanded in im around inf 29.7%

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

   1. unpow2 29.7%

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

   2. associate-*r* 29.7%

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

   3. *-commutative 29.7%

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

7. Simplified 29.7%

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

8. Taylor expanded in re around 0 24.9%

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

   1. unpow2 24.9%

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

   2. associate-*r* 24.9%

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

   3. *-commutative 24.9%

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

10. Simplified 24.9%

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

11. Final simplification 24.9%

    \[\leadsto im \cdot \left(0.5 \cdot im\right) \]

Reproduce

herbie shell --seed 2023293 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))