math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 19

Speedup: 1.0×

• 50.4% 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, 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%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
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. Final simplification 100.0%

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

Alternative 2: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.999999999:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.999999999)
   (*
    (* (cos re) 0.5)
    (+ 2.0 (+ (* im im) (* (pow im 4.0) 0.08333333333333333))))
   (* 0.5 (+ (exp im) (exp (- im))))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.999999999) {
		tmp = (cos(re) * 0.5) * (2.0 + ((im * im) + (pow(im, 4.0) * 0.08333333333333333)));
	} else {
		tmp = 0.5 * (exp(im) + 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 (cos(re) <= 0.999999999d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + ((im * im) + ((im ** 4.0d0) * 0.08333333333333333d0)))
    else
        tmp = 0.5d0 * (exp(im) + exp(-im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.999999999) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + ((im * im) + (Math.pow(im, 4.0) * 0.08333333333333333)));
	} else {
		tmp = 0.5 * (Math.exp(im) + Math.exp(-im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.999999999:
		tmp = (math.cos(re) * 0.5) * (2.0 + ((im * im) + (math.pow(im, 4.0) * 0.08333333333333333)))
	else:
		tmp = 0.5 * (math.exp(im) + math.exp(-im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.999999999)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(Float64(im * im) + Float64((im ^ 4.0) * 0.08333333333333333))));
	else
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.999999999)
		tmp = (cos(re) * 0.5) * (2.0 + ((im * im) + ((im ^ 4.0) * 0.08333333333333333)));
	else
		tmp = 0.5 * (exp(im) + exp(-im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.999999999], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.999999999000000028

   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 88.6%

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

      1. unpow2 88.6%

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

      2. *-commutative 88.6%

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

   4. Simplified 88.6%

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

   if 0.999999999000000028 < (cos.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.999999999:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \end{array} \]

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+72}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -4500:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{elif}\;im \leq 1.9:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -4.5e+72)
   (* 0.041666666666666664 (* (cos re) (pow im 4.0)))
   (if (<= im -4500.0)
     (* 0.5 (+ (exp im) (exp (- im))))
     (if (<= im 1.9)
       (*
        (* (cos re) 0.5)
        (+ 2.0 (+ (* im im) (* (pow im 4.0) 0.08333333333333333))))
       (* (cos re) (fma 0.5 (exp im) 0.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -4.5e+72) {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	} else if (im <= -4500.0) {
		tmp = 0.5 * (exp(im) + exp(-im));
	} else if (im <= 1.9) {
		tmp = (cos(re) * 0.5) * (2.0 + ((im * im) + (pow(im, 4.0) * 0.08333333333333333)));
	} else {
		tmp = cos(re) * fma(0.5, exp(im), 0.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -4.5e+72)
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	elseif (im <= -4500.0)
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))));
	elseif (im <= 1.9)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(Float64(im * im) + Float64((im ^ 4.0) * 0.08333333333333333))));
	else
		tmp = Float64(cos(re) * fma(0.5, exp(im), 0.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -4.5e+72], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -4500.0], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.4999999999999998e72

   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 98.3%

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

      1. unpow2 98.3%

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

      2. *-commutative 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in im around inf 98.3%

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

      1. *-commutative 98.3%

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

   7. Simplified 98.3%

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

   if -4.4999999999999998e72 < im < -4500

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 72.7%

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

   if -4500 < im < 1.8999999999999999

   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 99.2%

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

      1. unpow2 99.2%

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

      2. *-commutative 99.2%

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

   4. Simplified 99.2%

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

   if 1.8999999999999999 < 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%

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

   5. Applied egg-rr 99.4%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+72}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -4500:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{elif}\;im \leq 1.9:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, 0\right)\\ \end{array} \]

Alternative 4: 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 5: 95.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{im} + e^{-im}\right)\\ t_1 := 0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -4500:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0285:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ (exp im) (exp (- im)))))
        (t_1 (* 0.041666666666666664 (* (cos re) (pow im 4.0)))))
   (if (<= im -4.5e+72)
     t_1
     (if (<= im -4500.0)
       t_0
       (if (<= im 0.0285)
         (* (* (cos re) 0.5) (+ 2.0 (* im im)))
         (if (<= im 4.2e+68) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(im) + exp(-im));
	double t_1 = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	double tmp;
	if (im <= -4.5e+72) {
		tmp = t_1;
	} else if (im <= -4500.0) {
		tmp = t_0;
	} else if (im <= 0.0285) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 4.2e+68) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(im) + exp(-im))
    t_1 = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    if (im <= (-4.5d+72)) then
        tmp = t_1
    else if (im <= (-4500.0d0)) then
        tmp = t_0
    else if (im <= 0.0285d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 4.2d+68) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(im) + Math.exp(-im));
	double t_1 = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	double tmp;
	if (im <= -4.5e+72) {
		tmp = t_1;
	} else if (im <= -4500.0) {
		tmp = t_0;
	} else if (im <= 0.0285) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 4.2e+68) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(im) + math.exp(-im))
	t_1 = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	tmp = 0
	if im <= -4.5e+72:
		tmp = t_1
	elif im <= -4500.0:
		tmp = t_0
	elif im <= 0.0285:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 4.2e+68:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))))
	t_1 = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)))
	tmp = 0.0
	if (im <= -4.5e+72)
		tmp = t_1;
	elseif (im <= -4500.0)
		tmp = t_0;
	elseif (im <= 0.0285)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 4.2e+68)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(im) + exp(-im));
	t_1 = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	tmp = 0.0;
	if (im <= -4.5e+72)
		tmp = t_1;
	elseif (im <= -4500.0)
		tmp = t_0;
	elseif (im <= 0.0285)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 4.2e+68)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.5e+72], t$95$1, If[LessEqual[im, -4500.0], t$95$0, If[LessEqual[im, 0.0285], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e+68], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.4999999999999998e72 or 4.20000000000000002e68 < 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 96.6%

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

      1. unpow2 96.6%

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

      2. *-commutative 96.6%

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

   4. Simplified 96.6%

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

   5. Taylor expanded in im around inf 96.6%

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

      1. *-commutative 96.6%

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

   7. Simplified 96.6%

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

   if -4.4999999999999998e72 < im < -4500 or 0.028500000000000001 < im < 4.20000000000000002e68

   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 78.5%

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

   if -4500 < im < 0.028500000000000001

   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 99.0%

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

      1. unpow2 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+72}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -4500:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{elif}\;im \leq 0.0285:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 6: 88.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.7 \lor \neg \left(im \leq 3.7\right):\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.7) (not (<= im 3.7)))
   (* 0.041666666666666664 (* (cos re) (pow im 4.0)))
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.7) || !(im <= 3.7)) {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	} else {
		tmp = (cos(re) * 0.5) * (2.0 + (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 <= (-3.7d0)) .or. (.not. (im <= 3.7d0))) then
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    else
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.7) || !(im <= 3.7)) {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	} else {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.7) or not (im <= 3.7):
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	else:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.7) || !(im <= 3.7))
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	else
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.7) || ~((im <= 3.7)))
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	else
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.7], N[Not[LessEqual[im, 3.7]], $MachinePrecision]], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.7000000000000002 or 3.7000000000000002 < 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 80.8%

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

      1. unpow2 80.8%

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

      2. *-commutative 80.8%

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

   4. Simplified 80.8%

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

   5. Taylor expanded in im around inf 80.8%

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

      1. *-commutative 80.8%

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

   7. Simplified 80.8%

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

   if -3.7000000000000002 < im < 3.7000000000000002

   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 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.7 \lor \neg \left(im \leq 3.7\right):\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 7: 68.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.8 \cdot 10^{+78} \lor \neg \left(im \leq 6.4 \cdot 10^{-9}\right):\\ \;\;\;\;0.5 \cdot \left(2 + {im}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.8e+78) (not (<= im 6.4e-9)))
   (* 0.5 (+ 2.0 (pow im 2.0)))
   (cos re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.8e+78) || !(im <= 6.4e-9)) {
		tmp = 0.5 * (2.0 + pow(im, 2.0));
	} else {
		tmp = cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-3.8d+78)) .or. (.not. (im <= 6.4d-9))) then
        tmp = 0.5d0 * (2.0d0 + (im ** 2.0d0))
    else
        tmp = cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.8e+78) || !(im <= 6.4e-9)) {
		tmp = 0.5 * (2.0 + Math.pow(im, 2.0));
	} else {
		tmp = Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.8e+78) or not (im <= 6.4e-9):
		tmp = 0.5 * (2.0 + math.pow(im, 2.0))
	else:
		tmp = math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.8e+78) || !(im <= 6.4e-9))
		tmp = Float64(0.5 * Float64(2.0 + (im ^ 2.0)));
	else
		tmp = cos(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.8e+78) || ~((im <= 6.4e-9)))
		tmp = 0.5 * (2.0 + (im ^ 2.0));
	else
		tmp = cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.8e+78], N[Not[LessEqual[im, 6.4e-9]], $MachinePrecision]], N[(0.5 * N[(2.0 + N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cos[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.7999999999999999e78 or 6.40000000000000023e-9 < 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 60.8%

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

      1. unpow2 60.8%

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

   4. Simplified 60.8%

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

   5. Taylor expanded in re around 0 47.0%

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

   if -3.7999999999999999e78 < im < 6.40000000000000023e-9

   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 89.3%

      \[\leadsto \color{blue}{\cos re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.8 \cdot 10^{+78} \lor \neg \left(im \leq 6.4 \cdot 10^{-9}\right):\\ \;\;\;\;0.5 \cdot \left(2 + {im}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re\\ \end{array} \]

Alternative 8: 76.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * 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. Taylor expanded in im around 0 76.0%

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

   1. unpow2 76.0%

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

4. Simplified 76.0%

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

5. Final simplification 76.0%

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

Alternative 9: 50.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. 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 49.7%

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

5. Final simplification 49.7%

   \[\leadsto \cos re \]

Alternative 10: 3.8% 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. Taylor expanded in re around 0 65.8%

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

4. Applied egg-rr 3.2%

   \[\leadsto 0.5 \cdot \color{blue}{-2} \]

5. Final simplification 3.2%

   \[\leadsto -1 \]

Alternative 11: 3.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -0.5

Julia

function code(re, im)
	return -0.5
end

MATLAB

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

Wolfram

code[re_, im_] := -0.5

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 65.8%

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

4. Applied egg-rr 3.2%

   \[\leadsto 0.5 \cdot \color{blue}{-1} \]

5. Final simplification 3.2%

   \[\leadsto -0.5 \]

Alternative 12: 6.5% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.0009765625 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0009765625)

C

double code(double re, double im) {
	return 0.0009765625;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0009765625d0
end function

Java

public static double code(double re, double im) {
	return 0.0009765625;
}

Python

def code(re, im):
	return 0.0009765625

Julia

function code(re, im)
	return 0.0009765625
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0009765625;
end

Wolfram

code[re_, im_] := 0.0009765625

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 65.8%

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

4. Applied egg-rr 6.8%

   \[\leadsto 0.5 \cdot \color{blue}{0.001953125} \]

5. Final simplification 6.8%

   \[\leadsto 0.0009765625 \]

Alternative 13: 6.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.0078125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0078125)

C

double code(double re, double im) {
	return 0.0078125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0078125d0
end function

Java

public static double code(double re, double im) {
	return 0.0078125;
}

Python

def code(re, im):
	return 0.0078125

Julia

function code(re, im)
	return 0.0078125
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0078125;
end

Wolfram

code[re_, im_] := 0.0078125

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 65.8%

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

4. Applied egg-rr 7.1%

   \[\leadsto 0.5 \cdot \color{blue}{0.015625} \]

5. Final simplification 7.1%

   \[\leadsto 0.0078125 \]

Alternative 14: 7.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0625)

C

double code(double re, double im) {
	return 0.0625;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0625d0
end function

Java

public static double code(double re, double im) {
	return 0.0625;
}

Python

def code(re, im):
	return 0.0625

Julia

function code(re, im)
	return 0.0625
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0625;
end

Wolfram

code[re_, im_] := 0.0625

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 65.8%

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

4. Applied egg-rr 7.6%

   \[\leadsto 0.5 \cdot \color{blue}{0.125} \]

5. Final simplification 7.6%

   \[\leadsto 0.0625 \]

Alternative 15: 7.6% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.125)

C

double code(double re, double im) {
	return 0.125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.125d0
end function

Java

public static double code(double re, double im) {
	return 0.125;
}

Python

def code(re, im):
	return 0.125

Julia

function code(re, im)
	return 0.125
end

MATLAB

function tmp = code(re, im)
	tmp = 0.125;
end

Wolfram

code[re_, im_] := 0.125

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 65.8%

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

4. Applied egg-rr 8.0%

   \[\leadsto 0.5 \cdot \color{blue}{0.25} \]

5. Final simplification 8.0%

   \[\leadsto 0.125 \]

Alternative 16: 8.0% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.25)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

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

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 65.8%

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

4. Applied egg-rr 8.2%

   \[\leadsto 0.5 \cdot \color{blue}{0.5} \]

5. Final simplification 8.2%

   \[\leadsto 0.25 \]

Alternative 17: 8.5% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.5

Julia

function code(re, im)
	return 0.5
end

MATLAB

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

Wolfram

code[re_, im_] := 0.5

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 65.8%

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

4. Applied egg-rr 8.7%

   \[\leadsto 0.5 \cdot \color{blue}{1} \]

5. Final simplification 8.7%

   \[\leadsto 0.5 \]

Alternative 18: 9.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.75)

C

double code(double re, double im) {
	return 0.75;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.75d0
end function

Java

public static double code(double re, double im) {
	return 0.75;
}

Python

def code(re, im):
	return 0.75

Julia

function code(re, im)
	return 0.75
end

MATLAB

function tmp = code(re, im)
	tmp = 0.75;
end

Wolfram

code[re_, im_] := 0.75

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 65.8%

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

4. Applied egg-rr 9.2%

   \[\leadsto 0.5 \cdot \color{blue}{1.5} \]

5. Final simplification 9.2%

   \[\leadsto 0.75 \]

Alternative 19: 28.5% 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. Taylor expanded in re around 0 65.8%

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

3. Taylor expanded in im around 0 29.7%

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

4. Final simplification 29.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023181 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))