math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.6s

Alternatives: 18

Speedup: 1.0×

• 49.6% 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. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. +-commutative 100.0%

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

   5. distribute-rgt-in 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. cos-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-neg 100.0%

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

   11. remove-double-neg 100.0%

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

   12. exp-neg 100.0%

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

   13. associate-*l/ 100.0%

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

   14. 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. Add Preprocessing
5. Add Preprocessing

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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 3: 75.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (cos(re) * 0.5) * (exp(im) + (1.0 - 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) + (1.0d0 - im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 76.5%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
4. Step-by-step derivation

   1. neg-mul-1 76.5%

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

   2. unsub-neg 76.5%

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

5. Simplified 76.5%

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

6. Final simplification 76.5%

   \[\leadsto \left(\cos re \cdot 0.5\right) \cdot \left(e^{im} + \left(1 - im\right)\right) \]
7. Add Preprocessing

Alternative 4: 74.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 76.5%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
4. Step-by-step derivation

   1. neg-mul-1 76.5%

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

   2. unsub-neg 76.5%

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

5. Simplified 76.5%

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

6. Taylor expanded in im around 0 75.5%

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

7. Final simplification 75.5%

   \[\leadsto \left(\cos re \cdot 0.5\right) \cdot \left(e^{im} + 1\right) \]
8. Add Preprocessing

Alternative 5: 85.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ \mathbf{if}\;im \leq 3.4:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)))
   (if (<= im 3.4)
     (* t_0 (+ (- 1.0 im) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 1.05e+103)
       (+ 0.5 (* 0.5 (exp im)))
       (*
        t_0
        (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * 0.5;
	double tmp;
	if (im <= 3.4) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.05e+103) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	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) :: tmp
    t_0 = cos(re) * 0.5d0
    if (im <= 3.4d0) then
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 1.05d+103) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = t_0 * (2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * 0.5;
	double tmp;
	if (im <= 3.4) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.05e+103) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * 0.5
	tmp = 0
	if im <= 3.4:
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 1.05e+103:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * 0.5)
	tmp = 0.0
	if (im <= 3.4)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 1.05e+103)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(t_0 * Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * 0.5;
	tmp = 0.0;
	if (im <= 3.4)
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 1.05e+103)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[im, 3.4], N[(t$95$0 * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.05e+103], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.39999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 69.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 69.3%

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

      2. unsub-neg 69.3%

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

   5. Simplified 69.3%

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

   6. Taylor expanded in im around 0 82.0%

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

   if 3.39999999999999991 < im < 1.0500000000000001e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   7. Taylor expanded in re around 0 75.0%

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

      1. distribute-lft-in 75.0%

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

      2. metadata-eval 75.0%

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

   9. Simplified 75.0%

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

   if 1.0500000000000001e103 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.4:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right) - im\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.4)
   (* 0.5 (* (cos re) (- (+ (* im (+ 1.0 (* 0.5 im))) 2.0) im)))
   (if (<= im 1.05e+103)
     (+ 0.5 (* 0.5 (exp im)))
     (*
      (* (cos re) 0.5)
      (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.4) {
		tmp = 0.5 * (cos(re) * (((im * (1.0 + (0.5 * im))) + 2.0) - im));
	} else if (im <= 1.05e+103) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = (cos(re) * 0.5) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.4d0) then
        tmp = 0.5d0 * (cos(re) * (((im * (1.0d0 + (0.5d0 * im))) + 2.0d0) - im))
    else if (im <= 1.05d+103) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.4) {
		tmp = 0.5 * (Math.cos(re) * (((im * (1.0 + (0.5 * im))) + 2.0) - im));
	} else if (im <= 1.05e+103) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.4:
		tmp = 0.5 * (math.cos(re) * (((im * (1.0 + (0.5 * im))) + 2.0) - im))
	elif im <= 1.05e+103:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.4)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(Float64(im * Float64(1.0 + Float64(0.5 * im))) + 2.0) - im)));
	elseif (im <= 1.05e+103)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.4)
		tmp = 0.5 * (cos(re) * (((im * (1.0 + (0.5 * im))) + 2.0) - im));
	elseif (im <= 1.05e+103)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = (cos(re) * 0.5) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.4], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.05e+103], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.39999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 69.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 69.3%

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

      2. unsub-neg 69.3%

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

   5. Simplified 69.3%

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

   6. Taylor expanded in im around 0 82.0%

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

   7. Taylor expanded in re around inf 82.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\right)} \]

   if 3.39999999999999991 < im < 1.0500000000000001e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   7. Taylor expanded in re around 0 75.0%

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

      1. distribute-lft-in 75.0%

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

      2. metadata-eval 75.0%

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

   9. Simplified 75.0%

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

   if 1.0500000000000001e103 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right) - im\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + 0.5 \cdot im\right) + 2\\ \mathbf{if}\;im \leq 1.1:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(t\_0 - im\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* im (+ 1.0 (* 0.5 im))) 2.0)))
   (if (<= im 1.1)
     (* 0.5 (* (cos re) (- t_0 im)))
     (if (<= im 1.9e+154)
       (* 0.5 (- (+ (exp im) 1.0) im))
       (* (* (cos re) 0.5) t_0)))))

C

double code(double re, double im) {
	double t_0 = (im * (1.0 + (0.5 * im))) + 2.0;
	double tmp;
	if (im <= 1.1) {
		tmp = 0.5 * (cos(re) * (t_0 - im));
	} else if (im <= 1.9e+154) {
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	} else {
		tmp = (cos(re) * 0.5) * t_0;
	}
	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) :: tmp
    t_0 = (im * (1.0d0 + (0.5d0 * im))) + 2.0d0
    if (im <= 1.1d0) then
        tmp = 0.5d0 * (cos(re) * (t_0 - im))
    else if (im <= 1.9d+154) then
        tmp = 0.5d0 * ((exp(im) + 1.0d0) - im)
    else
        tmp = (cos(re) * 0.5d0) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * (1.0 + (0.5 * im))) + 2.0;
	double tmp;
	if (im <= 1.1) {
		tmp = 0.5 * (Math.cos(re) * (t_0 - im));
	} else if (im <= 1.9e+154) {
		tmp = 0.5 * ((Math.exp(im) + 1.0) - im);
	} else {
		tmp = (Math.cos(re) * 0.5) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * (1.0 + (0.5 * im))) + 2.0
	tmp = 0
	if im <= 1.1:
		tmp = 0.5 * (math.cos(re) * (t_0 - im))
	elif im <= 1.9e+154:
		tmp = 0.5 * ((math.exp(im) + 1.0) - im)
	else:
		tmp = (math.cos(re) * 0.5) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * Float64(1.0 + Float64(0.5 * im))) + 2.0)
	tmp = 0.0
	if (im <= 1.1)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(t_0 - im)));
	elseif (im <= 1.9e+154)
		tmp = Float64(0.5 * Float64(Float64(exp(im) + 1.0) - im));
	else
		tmp = Float64(Float64(cos(re) * 0.5) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * (1.0 + (0.5 * im))) + 2.0;
	tmp = 0.0;
	if (im <= 1.1)
		tmp = 0.5 * (cos(re) * (t_0 - im));
	elseif (im <= 1.9e+154)
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	else
		tmp = (cos(re) * 0.5) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[im, 1.1], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(t$95$0 - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(0.5 * N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.1000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 69.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 69.3%

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

      2. unsub-neg 69.3%

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

   5. Simplified 69.3%

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

   6. Taylor expanded in im around 0 82.0%

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

   7. Taylor expanded in re around inf 82.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\right)} \]

   if 1.1000000000000001 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 75.0%

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

   if 1.8999999999999999e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right) - im\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.4:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.4)
   (cos re)
   (if (<= im 1.9e+154)
     (* 0.5 (- (+ (exp im) 1.0) im))
     (* (* (cos re) 0.5) (+ (* im (+ 1.0 (* 0.5 im))) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.4) {
		tmp = cos(re);
	} else if (im <= 1.9e+154) {
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	} else {
		tmp = (cos(re) * 0.5) * ((im * (1.0 + (0.5 * im))) + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.4d0) then
        tmp = cos(re)
    else if (im <= 1.9d+154) then
        tmp = 0.5d0 * ((exp(im) + 1.0d0) - im)
    else
        tmp = (cos(re) * 0.5d0) * ((im * (1.0d0 + (0.5d0 * im))) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.4) {
		tmp = Math.cos(re);
	} else if (im <= 1.9e+154) {
		tmp = 0.5 * ((Math.exp(im) + 1.0) - im);
	} else {
		tmp = (Math.cos(re) * 0.5) * ((im * (1.0 + (0.5 * im))) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.4:
		tmp = math.cos(re)
	elif im <= 1.9e+154:
		tmp = 0.5 * ((math.exp(im) + 1.0) - im)
	else:
		tmp = (math.cos(re) * 0.5) * ((im * (1.0 + (0.5 * im))) + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.4)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = Float64(0.5 * Float64(Float64(exp(im) + 1.0) - im));
	else
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(Float64(im * Float64(1.0 + Float64(0.5 * im))) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.4)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	else
		tmp = (cos(re) * 0.5) * ((im * (1.0 + (0.5 * im))) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.4], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(0.5 * N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.39999999999999991

   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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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. Add Preprocessing

   5. Taylor expanded in im around 0 68.7%

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

   if 2.39999999999999991 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 75.0%

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

   if 1.8999999999999999e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.4:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		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 <= 1.35d0) 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 <= 1.35) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.35:
		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 <= 1.35)
		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 <= 1.35)
		tmp = cos(re);
	else
		tmp = 0.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.35], 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 < 1.3500000000000001

   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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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. Add Preprocessing

   5. Taylor expanded in im around 0 68.7%

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

   if 1.3500000000000001 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   7. Taylor expanded in re around 0 78.3%

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

      1. distribute-lft-in 78.3%

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

      2. metadata-eval 78.3%

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

   9. Simplified 78.3%

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

Alternative 10: 63.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+57}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.22e+57)
   (cos re)
   (* 0.5 (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.22e+57) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.22d+57) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.22e+57) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.22e+57:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.22e+57)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.22e+57)
		tmp = cos(re);
	else
		tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.22e+57], N[Cos[re], $MachinePrecision], N[(0.5 * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.22e57

   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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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. Add Preprocessing

   5. Taylor expanded in im around 0 65.8%

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

   if 1.22e57 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   7. Taylor expanded in im around 0 87.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 87.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   9. Simplified 87.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]

   10. Taylor expanded in re around 0 69.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+57}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.6% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))

C

double code(double re, double im) {
	return 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
}

Python

def code(re, im):
	return 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))

Julia

function code(re, im)
	return Float64(0.5 * Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Add Preprocessing

3. Taylor expanded in im around 0 76.5%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
4. Step-by-step derivation

   1. neg-mul-1 76.5%

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

   2. unsub-neg 76.5%

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

5. Simplified 76.5%

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

6. Taylor expanded in im around 0 75.5%

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

7. Taylor expanded in im around 0 68.8%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative 68.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

9. Simplified 68.8%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]

10. Taylor expanded in re around 0 47.5%

    \[\leadsto \color{blue}{0.5 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]

11. Final simplification 47.5%

    \[\leadsto 0.5 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \]
12. Add Preprocessing

Alternative 12: 47.9% accurate, 23.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right) - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (- (+ (* im (+ 1.0 (* 0.5 im))) 2.0) im)))

C

double code(double re, double im) {
	return 0.5 * (((im * (1.0 + (0.5 * im))) + 2.0) - im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (((im * (1.0d0 + (0.5d0 * im))) + 2.0d0) - im)
end function

Java

public static double code(double re, double im) {
	return 0.5 * (((im * (1.0 + (0.5 * im))) + 2.0) - im);
}

Python

def code(re, im):
	return 0.5 * (((im * (1.0 + (0.5 * im))) + 2.0) - im)

Julia

function code(re, im)
	return Float64(0.5 * Float64(Float64(Float64(im * Float64(1.0 + Float64(0.5 * im))) + 2.0) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (((im * (1.0 + (0.5 * im))) + 2.0) - im);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[(N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 76.5%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
4. Step-by-step derivation

   1. neg-mul-1 76.5%

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

   2. unsub-neg 76.5%

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

5. Simplified 76.5%

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

6. Taylor expanded in im around 0 75.9%

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

7. Taylor expanded in re around 0 49.6%

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

8. Final simplification 49.6%

   \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right) - im\right) \]
9. Add Preprocessing

Alternative 13: 32.1% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 160:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 - re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 160.0) 1.0 (- 2.0 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 160.0) {
		tmp = 1.0;
	} else {
		tmp = 2.0 - (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 160.0d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 - (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 160.0) {
		tmp = 1.0;
	} else {
		tmp = 2.0 - (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 160.0:
		tmp = 1.0
	else:
		tmp = 2.0 - (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 160.0)
		tmp = 1.0;
	else
		tmp = Float64(2.0 - Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 160.0)
		tmp = 1.0;
	else
		tmp = 2.0 - (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 160.0], 1.0, N[(2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 160

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 63.5%

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

   4. Taylor expanded in im around 0 51.0%

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

      1. +-commutative 51.0%

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

      2. unpow2 51.0%

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

      3. fma-define 51.0%

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

   6. Simplified 51.0%

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

   7. Taylor expanded in im around 0 40.3%

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

   if 160 < 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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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. Add Preprocessing

   6. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\cos re + \cos re} \]
   7. Step-by-step derivation

      1. count-2 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in re around 0 7.7%

      \[\leadsto \color{blue}{2 + -1 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. mul-1-neg 7.7%

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

       2. unsub-neg 7.7%

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

   11. Simplified 7.7%

       \[\leadsto \color{blue}{2 - {re}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 7.7%

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

   13. Applied egg-rr 7.7%

       \[\leadsto 2 - \color{blue}{re \cdot re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 28.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 67.0%

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

4. Taylor expanded in im around 0 49.7%

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

   1. +-commutative 49.7%

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

   2. unpow2 49.7%

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

   3. fma-define 49.7%

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

6. Simplified 49.7%

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

7. Taylor expanded in im around 0 31.4%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Alternative 15: 8.7% 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. Add Preprocessing

3. Taylor expanded in im around 0 76.5%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
4. Step-by-step derivation

   1. neg-mul-1 76.5%

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

   2. unsub-neg 76.5%

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

5. Simplified 76.5%

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

7. Applied egg-rr 8.9%

   \[\leadsto \color{blue}{0.5} \]
8. Add Preprocessing

Alternative 16: 8.1% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.25)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

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

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 76.5%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
4. Step-by-step derivation

   1. neg-mul-1 76.5%

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

   2. unsub-neg 76.5%

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

5. Simplified 76.5%

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

7. Applied egg-rr 8.3%

   \[\leadsto \color{blue}{0.25} \]
8. Add Preprocessing

Alternative 17: 3.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.0)

C

double code(double re, double im) {
	return -1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.0d0
end function

Java

public static double code(double re, double im) {
	return -1.0;
}

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0;
end

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 76.5%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
4. Step-by-step derivation

   1. neg-mul-1 76.5%

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

   2. unsub-neg 76.5%

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

5. Simplified 76.5%

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

7. Applied egg-rr 3.8%

   \[\leadsto \color{blue}{-1} \]
8. Add Preprocessing

Alternative 18: 3.4% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -2.0)

C

double code(double re, double im) {
	return -2.0;
}

Fortran

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

Java

public static double code(double re, double im) {
	return -2.0;
}

Python

def code(re, im):
	return -2.0

Julia

function code(re, im)
	return -2.0
end

MATLAB

function tmp = code(re, im)
	tmp = -2.0;
end

Wolfram

code[re_, im_] := -2.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 76.5%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
4. Step-by-step derivation

   1. neg-mul-1 76.5%

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

   2. unsub-neg 76.5%

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

5. Simplified 76.5%

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

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{-2} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024118 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))