math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 14

Speedup: 1.0×

• 49.4% 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

\[\begin{array}{l} \\ \cos re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (cos re) (+ (* 0.5 (/ 1.0 (exp im))) (* 0.5 (exp im)))))

C

double code(double re, double im) {
	return cos(re) * ((0.5 * (1.0 / exp(im))) + (0.5 * exp(im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(re) * ((0.5d0 * (1.0d0 / exp(im))) + (0.5d0 * exp(im)))
end function

Java

public static double code(double re, double im) {
	return Math.cos(re) * ((0.5 * (1.0 / Math.exp(im))) + (0.5 * Math.exp(im)));
}

Python

def code(re, im):
	return math.cos(re) * ((0.5 * (1.0 / math.exp(im))) + (0.5 * math.exp(im)))

Julia

function code(re, im)
	return Float64(cos(re) * Float64(Float64(0.5 * Float64(1.0 / exp(im))) + Float64(0.5 * exp(im))))
end

MATLAB

function tmp = code(re, im)
	tmp = cos(re) * ((0.5 * (1.0 / exp(im))) + (0.5 * exp(im)));
end

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(1.0 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. distribute-lft-in 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-def 100.0%

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

   9. exp-neg 100.0%

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

   10. associate-*l/ 100.0%

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

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around inf 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \cos re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\right) \]

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: 86.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.8:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.8)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 1.2e+77)
     (+ 0.5 (* 0.5 (exp im)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.8) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.2e+77) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.8d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.2d+77) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.8) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.2e+77) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.8:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.2e+77:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.8)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.2e+77)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.8)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.2e+77)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.8], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+77], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.79999999999999982

   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 83.7%

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

      1. unpow2 83.7%

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

   4. Simplified 83.7%

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

   if 6.79999999999999982 < im < 1.1999999999999999e77

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around inf 100.0%

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

   5. Taylor expanded in re around 0 64.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}} \]

   6. Taylor expanded in im around 0 64.7%

      \[\leadsto 0.5 \cdot \color{blue}{1} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around inf 64.7%

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

   if 1.1999999999999999e77 < 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%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \cos re\right)} \]

   7. Simplified 100.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.8:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 4: 86.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0148:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0148)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 1.2e+77)
     (* 0.5 (+ (exp im) (exp (- im))))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0148) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.2e+77) {
		tmp = 0.5 * (exp(im) + exp(-im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0148d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.2d+77) then
        tmp = 0.5d0 * (exp(im) + exp(-im))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0148) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.2e+77) {
		tmp = 0.5 * (Math.exp(im) + Math.exp(-im));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0148:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.2e+77:
		tmp = 0.5 * (math.exp(im) + math.exp(-im))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0148)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.2e+77)
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0148)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.2e+77)
		tmp = 0.5 * (exp(im) + exp(-im));
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0148], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+77], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.014800000000000001

   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 83.9%

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

      1. unpow2 83.9%

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

   4. Simplified 83.9%

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

   if 0.014800000000000001 < im < 1.1999999999999999e77

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 66.6%

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

   if 1.1999999999999999e77 < 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%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \cos re\right)} \]

   7. Simplified 100.0%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0148:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 5: 75.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. distribute-lft-in 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-def 100.0%

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

   9. exp-neg 100.0%

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

   10. associate-*l/ 100.0%

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

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around inf 100.0%

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

5. Taylor expanded in im around 0 74.0%

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

6. Final simplification 74.0%

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

Alternative 6: 85.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.7 \lor \neg \left(im \leq 1.3 \cdot 10^{+152}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 2.7) (not (<= im 1.3e+152)))
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (+ 0.5 (* 0.5 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 2.7) || !(im <= 1.3e+152)) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = 0.5 + (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 <= 2.7d0) .or. (.not. (im <= 1.3d+152))) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else
        tmp = 0.5d0 + (0.5d0 * exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 2.7) || !(im <= 1.3e+152)) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = 0.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 2.7) or not (im <= 1.3e+152):
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = 0.5 + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 2.7) || !(im <= 1.3e+152))
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 2.7) || ~((im <= 1.3e+152)))
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	else
		tmp = 0.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 2.7], N[Not[LessEqual[im, 1.3e+152]], $MachinePrecision]], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.7000000000000002 or 1.3e152 < 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 85.3%

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

      1. unpow2 85.3%

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

   4. Simplified 85.3%

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

   if 2.7000000000000002 < im < 1.3e152

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around inf 100.0%

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

   5. Taylor expanded in re around 0 69.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}} \]

   6. Taylor expanded in im around 0 69.0%

      \[\leadsto 0.5 \cdot \color{blue}{1} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around inf 69.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7 \lor \neg \left(im \leq 1.3 \cdot 10^{+152}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \]

Alternative 7: 72.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.0)
   (cos re)
   (if (<= im 1.3e+152)
     (+ 0.5 (* 0.5 (exp im)))
     (* 0.5 (* (cos re) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.0) {
		tmp = cos(re);
	} else if (im <= 1.3e+152) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = 0.5 * (cos(re) * (im * 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 <= 3.0d0) then
        tmp = cos(re)
    else if (im <= 1.3d+152) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = 0.5d0 * (cos(re) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.3e+152) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = 0.5 * (Math.cos(re) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.0:
		tmp = math.cos(re)
	elif im <= 1.3e+152:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = 0.5 * (math.cos(re) * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.0)
		tmp = cos(re);
	elseif (im <= 1.3e+152)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.0)
		tmp = cos(re);
	elseif (im <= 1.3e+152)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = 0.5 * (cos(re) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.3e+152], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 67.0%

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

   if 3 < im < 1.3e152

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around inf 100.0%

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

   5. Taylor expanded in re around 0 69.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}} \]

   6. Taylor expanded in im around 0 69.0%

      \[\leadsto 0.5 \cdot \color{blue}{1} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around inf 69.0%

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

   if 1.3e152 < 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 94.7%

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

      1. unpow2 94.7%

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

   4. Simplified 94.7%

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

   5. Taylor expanded in im around inf 94.7%

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

      1. *-commutative 94.7%

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

      2. unpow2 94.7%

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

   7. Simplified 94.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 65.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{+67}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.2e+14)
   (cos re)
   (if (<= im 5.3e+67)
     (* (* im im) (+ 0.5 (* (* re re) -0.25)))
     (* 0.041666666666666664 (pow im 4.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.2e+14) {
		tmp = cos(re);
	} else if (im <= 5.3e+67) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 0.041666666666666664 * 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 <= 9.2d+14) then
        tmp = cos(re)
    else if (im <= 5.3d+67) then
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = 0.041666666666666664d0 * (im ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.2e+14) {
		tmp = Math.cos(re);
	} else if (im <= 5.3e+67) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 0.041666666666666664 * Math.pow(im, 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.2e+14:
		tmp = math.cos(re)
	elif im <= 5.3e+67:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = 0.041666666666666664 * math.pow(im, 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.2e+14)
		tmp = cos(re);
	elseif (im <= 5.3e+67)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = Float64(0.041666666666666664 * (im ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.2e+14)
		tmp = cos(re);
	elseif (im <= 5.3e+67)
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	else
		tmp = 0.041666666666666664 * (im ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.2e+14], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5.3e+67], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 9.2e14

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 66.1%

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

   if 9.2e14 < im < 5.3e67

   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 3.9%

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

      1. unpow2 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around inf 3.9%

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

      1. *-commutative 3.9%

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

      2. unpow2 3.9%

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

   7. Simplified 3.9%

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

   8. Taylor expanded in re around 0 50.6%

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

      1. unpow2 50.6%

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

      2. associate-*l* 50.6%

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

      3. +-commutative 50.6%

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

      4. associate-*r* 50.6%

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

      5. associate-*l* 50.6%

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

      6. unpow2 50.6%

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

      7. distribute-rgt-out 50.6%

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

      8. unpow2 50.6%

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

      9. *-commutative 50.6%

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

      10. unpow2 50.6%

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

   10. Simplified 50.6%

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

   if 5.3e67 < 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 96.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 96.0%

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

      2. *-commutative 96.0%

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

   4. Simplified 96.0%

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

   5. Taylor expanded in im around inf 96.0%

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

      1. *-commutative 96.0%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \cos re\right)} \]

   7. Simplified 96.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]

   8. Taylor expanded in re around 0 70.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{+67}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \end{array} \]

Alternative 9: 69.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.0) (cos re) (+ 0.5 (* 0.5 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.0) {
		tmp = cos(re);
	} else {
		tmp = 0.5 + (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 <= 3.0d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 + (0.5d0 * exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.0:
		tmp = math.cos(re)
	else:
		tmp = 0.5 + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.0)
		tmp = cos(re);
	else
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.0)
		tmp = cos(re);
	else
		tmp = 0.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.0], N[Cos[re], $MachinePrecision], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 67.0%

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

   if 3 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around inf 100.0%

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

   5. Taylor expanded in re around 0 70.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}} \]

   6. Taylor expanded in im around 0 70.5%

      \[\leadsto 0.5 \cdot \color{blue}{1} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around inf 70.5%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \]

Alternative 10: 62.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7e+14) (cos re) (* (* im im) (+ 0.5 (* (* re re) -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7e+14) {
		tmp = cos(re);
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -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 <= 7d+14) then
        tmp = cos(re)
    else
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7e+14) {
		tmp = Math.cos(re);
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7e+14:
		tmp = math.cos(re)
	else:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7e+14)
		tmp = cos(re);
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7e+14)
		tmp = cos(re);
	else
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7e+14], N[Cos[re], $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7e14

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 66.1%

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

   if 7e14 < 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 54.6%

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

      1. unpow2 54.6%

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

   4. Simplified 54.6%

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

   5. Taylor expanded in im around inf 54.6%

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

      1. *-commutative 54.6%

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

      2. unpow2 54.6%

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

   7. Simplified 54.6%

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

   8. Taylor expanded in re around 0 20.1%

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

      1. unpow2 20.1%

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

      2. associate-*l* 20.1%

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

      3. +-commutative 20.1%

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

      4. associate-*r* 20.1%

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

      5. associate-*l* 20.1%

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

      6. unpow2 20.1%

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

      7. distribute-rgt-out 61.5%

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

      8. unpow2 61.5%

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

      9. *-commutative 61.5%

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

      10. unpow2 61.5%

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

   10. Simplified 61.5%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 11: 49.2% accurate, 23.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.5e+15)
   (* 0.5 (+ 2.0 (* im im)))
   (* (* im im) (+ 0.5 (* (* re re) -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.5e+15) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -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 <= 5.5d+15) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.5e+15) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.5e+15:
		tmp = 0.5 * (2.0 + (im * im))
	else:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.5e+15)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.5e+15)
		tmp = 0.5 * (2.0 + (im * im));
	else
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.5e+15], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.5e15

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around inf 100.0%

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

   5. Taylor expanded in re around 0 58.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}} \]
   6. Step-by-step derivation

      1. distribute-lft-out 58.5%

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

      2. rec-exp 58.5%

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

   7. Applied egg-rr 58.5%

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

   8. Taylor expanded in im around 0 44.4%

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

      1. unpow2 82.5%

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

   10. Simplified 44.4%

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

   if 5.5e15 < 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 54.6%

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

      1. unpow2 54.6%

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

   4. Simplified 54.6%

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

   5. Taylor expanded in im around inf 54.6%

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

      1. *-commutative 54.6%

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

      2. unpow2 54.6%

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

   7. Simplified 54.6%

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

   8. Taylor expanded in re around 0 20.1%

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

      1. unpow2 20.1%

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

      2. associate-*l* 20.1%

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

      3. +-commutative 20.1%

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

      4. associate-*r* 20.1%

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

      5. associate-*l* 20.1%

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

      6. unpow2 20.1%

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

      7. distribute-rgt-out 61.5%

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

      8. unpow2 61.5%

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

      9. *-commutative 61.5%

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

      10. unpow2 61.5%

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

   10. Simplified 61.5%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 12: 38.0% accurate, 43.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1.4) 1.0 (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.4d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.4:
		tmp = 1.0
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.4)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.4)
		tmp = 1.0;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.4], 1.0, N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3999999999999999

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around inf 100.0%

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

   5. Taylor expanded in re around 0 57.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}} \]

   6. Taylor expanded in im around 0 31.0%

      \[\leadsto 0.5 \cdot \color{blue}{1} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around 0 31.6%

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

   if 1.3999999999999999 < 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 52.1%

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

      1. unpow2 52.1%

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

   4. Simplified 52.1%

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

   5. Taylor expanded in im around inf 52.1%

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

      1. *-commutative 52.1%

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

      2. unpow2 52.1%

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

   7. Simplified 52.1%

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

   8. Taylor expanded in re around 0 39.5%

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

      1. unpow2 39.5%

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

   10. Simplified 39.5%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 13: 47.5% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(2 + im \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (+ 2.0 (* im im))))

C

double code(double re, double im) {
	return 0.5 * (2.0 + (im * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (2.0d0 + (im * im))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (2.0 + (im * im));
}

Python

def code(re, im):
	return 0.5 * (2.0 + (im * im))

Julia

function code(re, im)
	return Float64(0.5 * Float64(2.0 + Float64(im * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (2.0 + (im * im));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(2.0 + N[(im * 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. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. distribute-lft-in 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-def 100.0%

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

   9. exp-neg 100.0%

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

   10. associate-*l/ 100.0%

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

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around inf 100.0%

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

5. Taylor expanded in re around 0 60.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}} \]
6. Step-by-step derivation

   1. distribute-lft-out 60.9%

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

   2. rec-exp 60.9%

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

7. Applied egg-rr 60.9%

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

8. Taylor expanded in im around 0 43.7%

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

   1. unpow2 76.2%

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

10. Simplified 43.7%

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

11. Final simplification 43.7%

    \[\leadsto 0.5 \cdot \left(2 + im \cdot im\right) \]

Alternative 14: 28.4% 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. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. distribute-lft-in 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-def 100.0%

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

   9. exp-neg 100.0%

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

   10. associate-*l/ 100.0%

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

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around inf 100.0%

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

5. Taylor expanded in re around 0 60.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}} \]

6. Taylor expanded in im around 0 40.4%

   \[\leadsto 0.5 \cdot \color{blue}{1} + 0.5 \cdot e^{im} \]

7. Taylor expanded in im around 0 24.7%

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

8. Final simplification 24.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))