math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.7s

Alternatives: 20

Speedup: 1.5×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \cosh im \cdot \cos re \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(cosh(im) * cos(re))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Cosh[im], $MachinePrecision] * N[Cos[re], $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. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\cos re}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right), \cos \color{blue}{re}\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \cos re\right) \]

   6. cosh-undef N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \cos re\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \cos \color{blue}{re}\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \cos re\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \cos \color{blue}{re}\right) \]

   10. cosh-lowering-cosh.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \cos re\right) \]

   11. cos-lowering-cos.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{cos.f64}\left(re\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
5. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \color{blue}{\cos re}\right) \]

   2. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\cosh im, \cos \color{blue}{re}\right) \]

   3. cosh-lowering-cosh.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \cos \color{blue}{re}\right) \]

   4. cos-lowering-cos.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{cos.f64}\left(re\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]
7. Add Preprocessing

Alternative 2: 94.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\\ \mathbf{if}\;im \leq 0.45:\\ \;\;\;\;\cos re \cdot \left(t\_0 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot t\_0\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))
   (if (<= im 0.45)
     (* (cos re) (+ (* t_0 (* im (* im (* im im)))) (+ 1.0 (* (* im im) 0.5))))
     (if (<= im 7e+51)
       (* (cosh im) (+ 1.0 (* (* re re) -0.5)))
       (* (cos re) (+ 1.0 (* im (* im (+ 0.5 (* im (* im t_0)))))))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double tmp;
	if (im <= 0.45) {
		tmp = cos(re) * ((t_0 * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5)));
	} else if (im <= 7e+51) {
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * 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 = 0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)
    if (im <= 0.45d0) then
        tmp = cos(re) * ((t_0 * (im * (im * (im * im)))) + (1.0d0 + ((im * im) * 0.5d0)))
    else if (im <= 7d+51) then
        tmp = cosh(im) * (1.0d0 + ((re * re) * (-0.5d0)))
    else
        tmp = cos(re) * (1.0d0 + (im * (im * (0.5d0 + (im * (im * t_0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double tmp;
	if (im <= 0.45) {
		tmp = Math.cos(re) * ((t_0 * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5)));
	} else if (im <= 7e+51) {
		tmp = Math.cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = Math.cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * t_0))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889)
	tmp = 0
	if im <= 0.45:
		tmp = math.cos(re) * ((t_0 * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5)))
	elif im <= 7e+51:
		tmp = math.cosh(im) * (1.0 + ((re * re) * -0.5))
	else:
		tmp = math.cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * t_0))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))
	tmp = 0.0
	if (im <= 0.45)
		tmp = Float64(cos(re) * Float64(Float64(t_0 * Float64(im * Float64(im * Float64(im * im)))) + Float64(1.0 + Float64(Float64(im * im) * 0.5))));
	elseif (im <= 7e+51)
		tmp = Float64(cosh(im) * Float64(1.0 + Float64(Float64(re * re) * -0.5)));
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * t_0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	tmp = 0.0;
	if (im <= 0.45)
		tmp = cos(re) * ((t_0 * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5)));
	elseif (im <= 7e+51)
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	else
		tmp = cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * t_0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.45], N[(N[Cos[re], $MachinePrecision] * N[(N[(t$95$0 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7e+51], N[(N[Cosh[im], $MachinePrecision] * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.450000000000000011

   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

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 94.2%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   if 0.450000000000000011 < im < 7e51

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right), \cos \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \cos re\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \cos \color{blue}{re}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \cos re\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \cos \color{blue}{re}\right) \]

      10. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \cos re\right) \]

      11. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{cos.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \color{blue}{\cos re}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\cosh im, \cos \color{blue}{re}\right) \]

      3. cosh-lowering-cosh.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \cos \color{blue}{re}\right) \]

      4. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{cos.f64}\left(re\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right)\right) \]

      5. *-lowering-*.f64 92.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right)\right) \]

   9. Simplified 92.3%

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

   if 7e51 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right), \cos \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \cos re\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \cos \color{blue}{re}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \cos re\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \cos \color{blue}{re}\right) \]

      10. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \cos re\right) \]

      11. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{cos.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}, \mathsf{cos.f64}\left(re\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \cdot \cos re \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.45:\\ \;\;\;\;\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{if}\;im \leq 0.62:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (cos re)
          (+
           1.0
           (*
            im
            (*
             im
             (+
              0.5
              (*
               im
               (*
                im
                (+
                 0.041666666666666664
                 (* (* im im) 0.001388888888888889)))))))))))
   (if (<= im 0.62)
     t_0
     (if (<= im 7e+51) (* (cosh im) (+ 1.0 (* (* re re) -0.5))) t_0))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))));
	double tmp;
	if (im <= 0.62) {
		tmp = t_0;
	} else if (im <= 7e+51) {
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = 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 = cos(re) * (1.0d0 + (im * (im * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0))))))))
    if (im <= 0.62d0) then
        tmp = t_0
    else if (im <= 7d+51) then
        tmp = cosh(im) * (1.0d0 + ((re * re) * (-0.5d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))));
	double tmp;
	if (im <= 0.62) {
		tmp = t_0;
	} else if (im <= 7e+51) {
		tmp = Math.cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))))
	tmp = 0
	if im <= 0.62:
		tmp = t_0
	elif im <= 7e+51:
		tmp = math.cosh(im) * (1.0 + ((re * re) * -0.5))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)))))))))
	tmp = 0.0
	if (im <= 0.62)
		tmp = t_0;
	elseif (im <= 7e+51)
		tmp = Float64(cosh(im) * Float64(1.0 + Float64(Float64(re * re) * -0.5)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))));
	tmp = 0.0;
	if (im <= 0.62)
		tmp = t_0;
	elseif (im <= 7e+51)
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.62], t$95$0, If[LessEqual[im, 7e+51], N[(N[Cosh[im], $MachinePrecision] * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.619999999999999996 or 7e51 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right), \cos \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \cos re\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \cos \color{blue}{re}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \cos re\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \cos \color{blue}{re}\right) \]

      10. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \cos re\right) \]

      11. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{cos.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}, \mathsf{cos.f64}\left(re\right)\right) \]

   7. Simplified 95.4%

      \[\leadsto \color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \cdot \cos re \]

   if 0.619999999999999996 < im < 7e51

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right), \cos \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \cos re\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \cos \color{blue}{re}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \cos re\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \cos \color{blue}{re}\right) \]

      10. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \cos re\right) \]

      11. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{cos.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \color{blue}{\cos re}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\cosh im, \cos \color{blue}{re}\right) \]

      3. cosh-lowering-cosh.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \cos \color{blue}{re}\right) \]

      4. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{cos.f64}\left(re\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right)\right) \]

      5. *-lowering-*.f64 92.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right)\right) \]

   9. Simplified 92.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.62:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{if}\;im \leq 0.12:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+76}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (cos re)
          (+ 1.0 (* (* im im) (+ 0.5 (* 0.041666666666666664 (* im im))))))))
   (if (<= im 0.12)
     t_0
     (if (<= im 3.6e+76) (* (cosh im) (+ 1.0 (* (* re re) -0.5))) t_0))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))));
	double tmp;
	if (im <= 0.12) {
		tmp = t_0;
	} else if (im <= 3.6e+76) {
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = 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 = cos(re) * (1.0d0 + ((im * im) * (0.5d0 + (0.041666666666666664d0 * (im * im)))))
    if (im <= 0.12d0) then
        tmp = t_0
    else if (im <= 3.6d+76) then
        tmp = cosh(im) * (1.0d0 + ((re * re) * (-0.5d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))));
	double tmp;
	if (im <= 0.12) {
		tmp = t_0;
	} else if (im <= 3.6e+76) {
		tmp = Math.cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))))
	tmp = 0
	if im <= 0.12:
		tmp = t_0
	elif im <= 3.6e+76:
		tmp = math.cosh(im) * (1.0 + ((re * re) * -0.5))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))))
	tmp = 0.0
	if (im <= 0.12)
		tmp = t_0;
	elseif (im <= 3.6e+76)
		tmp = Float64(cosh(im) * Float64(1.0 + Float64(Float64(re * re) * -0.5)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))));
	tmp = 0.0;
	if (im <= 0.12)
		tmp = t_0;
	elseif (im <= 3.6e+76)
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.12], t$95$0, If[LessEqual[im, 3.6e+76], N[(N[Cosh[im], $MachinePrecision] * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.12 or 3.6000000000000003e76 < 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

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      7. unpow2 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) \cdot \cos re \]

      8. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \cos \color{blue}{re} \]

      9. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) \cdot \cos re \]

      10. distribute-rgt-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + im \cdot \left(\frac{1}{2} \cdot im\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \color{blue}{\left(1 + im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + \color{blue}{1}\right)\right) \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   if 0.12 < im < 3.6000000000000003e76

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right), \cos \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \cos re\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \cos \color{blue}{re}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \cos re\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \cos \color{blue}{re}\right) \]

      10. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \cos re\right) \]

      11. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{cos.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \color{blue}{\cos re}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\cosh im, \cos \color{blue}{re}\right) \]

      3. cosh-lowering-cosh.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \cos \color{blue}{re}\right) \]

      4. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{cos.f64}\left(re\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right)\right) \]

      5. *-lowering-*.f64 89.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right)\right) \]

   9. Simplified 89.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.12:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+76}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \mathbf{if}\;im \leq 0.028:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+127}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (+ 1.0 (* (* im im) 0.5)))))
   (if (<= im 0.028)
     t_0
     (if (<= im 8e+127)
       (cosh im)
       (if (<= im 1.35e+154)
         (*
          (+ 1.0 (* (* im im) (+ 0.5 (* 0.041666666666666664 (* im im)))))
          (+ 1.0 (* re (* re -0.5))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (1.0 + ((im * im) * 0.5));
	double tmp;
	if (im <= 0.028) {
		tmp = t_0;
	} else if (im <= 8e+127) {
		tmp = cosh(im);
	} else if (im <= 1.35e+154) {
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = 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 = cos(re) * (1.0d0 + ((im * im) * 0.5d0))
    if (im <= 0.028d0) then
        tmp = t_0
    else if (im <= 8d+127) then
        tmp = cosh(im)
    else if (im <= 1.35d+154) then
        tmp = (1.0d0 + ((im * im) * (0.5d0 + (0.041666666666666664d0 * (im * im))))) * (1.0d0 + (re * (re * (-0.5d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (1.0 + ((im * im) * 0.5));
	double tmp;
	if (im <= 0.028) {
		tmp = t_0;
	} else if (im <= 8e+127) {
		tmp = Math.cosh(im);
	} else if (im <= 1.35e+154) {
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (1.0 + ((im * im) * 0.5))
	tmp = 0
	if im <= 0.028:
		tmp = t_0
	elif im <= 8e+127:
		tmp = math.cosh(im)
	elif im <= 1.35e+154:
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + (re * (re * -0.5)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * 0.5)))
	tmp = 0.0
	if (im <= 0.028)
		tmp = t_0;
	elseif (im <= 8e+127)
		tmp = cosh(im);
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))) * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (1.0 + ((im * im) * 0.5));
	tmp = 0.0;
	if (im <= 0.028)
		tmp = t_0;
	elseif (im <= 8e+127)
		tmp = cosh(im);
	elseif (im <= 1.35e+154)
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + (re * (re * -0.5)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.028], t$95$0, If[LessEqual[im, 8e+127], N[Cosh[im], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0280000000000000006 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 82.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 82.3%

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

   if 0.0280000000000000006 < im < 7.99999999999999964e127

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right), \cos \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \cos re\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \cos \color{blue}{re}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \cos re\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \cos \color{blue}{re}\right) \]

      10. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \cos re\right) \]

      11. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{cos.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \color{blue}{\cos re}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\cosh im, \cos \color{blue}{re}\right) \]

      3. cosh-lowering-cosh.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \cos \color{blue}{re}\right) \]

      4. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{cos.f64}\left(re\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \color{blue}{1}\right) \]
   8. Step-by-step derivation

   9. Simplified 80.0%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   if 7.99999999999999964e127 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      7. unpow2 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) \cdot \cos re \]

      8. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \cos \color{blue}{re} \]

      9. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) \cdot \cos re \]

      10. distribute-rgt-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + im \cdot \left(\frac{1}{2} \cdot im\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \color{blue}{\left(1 + im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + \color{blue}{1}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot -0.5\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.028:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+127}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \mathbf{if}\;im \leq 0.0155:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (+ 1.0 (* (* im im) 0.5)))))
   (if (<= im 0.0155)
     t_0
     (if (<= im 1.35e+154) (* (cosh im) (+ 1.0 (* (* re re) -0.5))) t_0))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (1.0 + ((im * im) * 0.5));
	double tmp;
	if (im <= 0.0155) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = 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 = cos(re) * (1.0d0 + ((im * im) * 0.5d0))
    if (im <= 0.0155d0) then
        tmp = t_0
    else if (im <= 1.35d+154) then
        tmp = cosh(im) * (1.0d0 + ((re * re) * (-0.5d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (1.0 + ((im * im) * 0.5));
	double tmp;
	if (im <= 0.0155) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = Math.cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (1.0 + ((im * im) * 0.5))
	tmp = 0
	if im <= 0.0155:
		tmp = t_0
	elif im <= 1.35e+154:
		tmp = math.cosh(im) * (1.0 + ((re * re) * -0.5))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * 0.5)))
	tmp = 0.0
	if (im <= 0.0155)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = Float64(cosh(im) * Float64(1.0 + Float64(Float64(re * re) * -0.5)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (1.0 + ((im * im) * 0.5));
	tmp = 0.0;
	if (im <= 0.0155)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.0155], t$95$0, If[LessEqual[im, 1.35e+154], N[(N[Cosh[im], $MachinePrecision] * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0155 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 82.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 82.3%

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

   if 0.0155 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right), \cos \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \cos re\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \cos \color{blue}{re}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \cos re\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \cos \color{blue}{re}\right) \]

      10. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \cos re\right) \]

      11. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{cos.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \color{blue}{\cos re}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\cosh im, \cos \color{blue}{re}\right) \]

      3. cosh-lowering-cosh.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \cos \color{blue}{re}\right) \]

      4. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{cos.f64}\left(re\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right)\right) \]

      5. *-lowering-*.f64 86.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right)\right) \]

   9. Simplified 86.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0155:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2e-5)
   (cos re)
   (if (<= im 2e+126)
     (cosh im)
     (*
      (* (* im im) (* im im))
      (+
       0.041666666666666664
       (*
        (+
         -0.5
         (*
          re
          (* re (+ 0.041666666666666664 (* (* re re) -0.001388888888888889)))))
        (* 0.041666666666666664 (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2e-5) {
		tmp = cos(re);
	} else if (im <= 2e+126) {
		tmp = cosh(im);
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))) * (0.041666666666666664 * (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 <= 2d-5) then
        tmp = cos(re)
    else if (im <= 2d+126) then
        tmp = cosh(im)
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + (((-0.5d0) + (re * (re * (0.041666666666666664d0 + ((re * re) * (-0.001388888888888889d0)))))) * (0.041666666666666664d0 * (re * re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2e-5) {
		tmp = Math.cos(re);
	} else if (im <= 2e+126) {
		tmp = Math.cosh(im);
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))) * (0.041666666666666664 * (re * re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2e-5:
		tmp = math.cos(re)
	elif im <= 2e+126:
		tmp = math.cosh(im)
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))) * (0.041666666666666664 * (re * re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2e-5)
		tmp = cos(re);
	elseif (im <= 2e+126)
		tmp = cosh(im);
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(-0.5 + Float64(re * Float64(re * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.001388888888888889))))) * Float64(0.041666666666666664 * Float64(re * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2e-5)
		tmp = cos(re);
	elseif (im <= 2e+126)
		tmp = cosh(im);
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))) * (0.041666666666666664 * (re * re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2e-5], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2e+126], N[Cosh[im], $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(-0.5 + N[(re * N[(re * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.00000000000000016e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 61.6%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 61.6%

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

   if 2.00000000000000016e-5 < im < 1.99999999999999985e126

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right), \cos \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \cos re\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \cos \color{blue}{re}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \cos re\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \cos \color{blue}{re}\right) \]

      10. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \cos re\right) \]

      11. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{cos.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \color{blue}{\cos re}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\cosh im, \cos \color{blue}{re}\right) \]

      3. cosh-lowering-cosh.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \cos \color{blue}{re}\right) \]

      4. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{cos.f64}\left(re\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \color{blue}{1}\right) \]
   8. Step-by-step derivation

   9. Simplified 80.8%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   if 1.99999999999999985e126 < 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

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      7. unpow2 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) \cdot \cos re \]

      8. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \cos \color{blue}{re} \]

      9. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) \cdot \cos re \]

      10. distribute-rgt-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + im \cdot \left(\frac{1}{2} \cdot im\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \color{blue}{\left(1 + im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + \color{blue}{1}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), 1\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), 1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), 1\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      20. *-lowering-*.f64 81.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

   8. Simplified 81.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({im}^{4} \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{24}} \]

       2. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right) \cdot \frac{1}{24}\right)} \]

       3. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right) \]

       12. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(1 \cdot \frac{1}{24} + \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right) \cdot \frac{1}{24}}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \frac{1}{24}\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right) \cdot \frac{1}{24}\right)}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) \cdot {re}^{2}\right) \cdot \frac{1}{24}\right)\right)\right) \]

   11. Simplified 81.6%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.2e-5)
   (cos re)
   (*
    (+
     (*
      (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))
      (* im (* im (* im im))))
     (+ 1.0 (* (* im im) 0.5)))
    (+
     1.0
     (*
      re
      (*
       re
       (+
        -0.5
        (*
         re
         (*
          re
          (+ 0.041666666666666664 (* (* re re) -0.001388888888888889)))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.2e-5) {
		tmp = cos(re);
	} else {
		tmp = (((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5))) * (1.0 + (re * (re * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.2d-5) then
        tmp = cos(re)
    else
        tmp = (((0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)) * (im * (im * (im * im)))) + (1.0d0 + ((im * im) * 0.5d0))) * (1.0d0 + (re * (re * ((-0.5d0) + (re * (re * (0.041666666666666664d0 + ((re * re) * (-0.001388888888888889d0)))))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.2e-5) {
		tmp = Math.cos(re);
	} else {
		tmp = (((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5))) * (1.0 + (re * (re * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.2e-5:
		tmp = math.cos(re)
	else:
		tmp = (((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5))) * (1.0 + (re * (re * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.2e-5)
		tmp = cos(re);
	else
		tmp = Float64(Float64(Float64(Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)) * Float64(im * Float64(im * Float64(im * im)))) + Float64(1.0 + Float64(Float64(im * im) * 0.5))) * Float64(1.0 + Float64(re * Float64(re * Float64(-0.5 + Float64(re * Float64(re * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.001388888888888889)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.2e-5)
		tmp = cos(re);
	else
		tmp = (((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5))) * (1.0 + (re * (re * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.2e-5], N[Cos[re], $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(re * N[(-0.5 + N[(re * N[(re * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.1999999999999999e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 61.6%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 61.6%

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

   if 2.1999999999999999e-5 < 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

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 80.7%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \color{blue}{\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \color{blue}{\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 66.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 66.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\right)} \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\\ \mathbf{if}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(t\_0 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))
   (if (<= re 1.5e+154)
     (*
      (+ (* t_0 (* im (* im (* im im)))) (+ 1.0 (* (* im im) 0.5)))
      (+
       1.0
       (*
        re
        (*
         re
         (+
          -0.5
          (*
           re
           (*
            re
            (+ 0.041666666666666664 (* (* re re) -0.001388888888888889)))))))))
     (+ 1.0 (* im (* im (+ 0.5 (* im (* im t_0)))))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double tmp;
	if (re <= 1.5e+154) {
		tmp = ((t_0 * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5))) * (1.0 + (re * (re * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))))));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 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 = 0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)
    if (re <= 1.5d+154) then
        tmp = ((t_0 * (im * (im * (im * im)))) + (1.0d0 + ((im * im) * 0.5d0))) * (1.0d0 + (re * (re * ((-0.5d0) + (re * (re * (0.041666666666666664d0 + ((re * re) * (-0.001388888888888889d0)))))))))
    else
        tmp = 1.0d0 + (im * (im * (0.5d0 + (im * (im * t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double tmp;
	if (re <= 1.5e+154) {
		tmp = ((t_0 * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5))) * (1.0 + (re * (re * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))))));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * t_0)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889)
	tmp = 0
	if re <= 1.5e+154:
		tmp = ((t_0 * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5))) * (1.0 + (re * (re * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))))))
	else:
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * t_0)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))
	tmp = 0.0
	if (re <= 1.5e+154)
		tmp = Float64(Float64(Float64(t_0 * Float64(im * Float64(im * Float64(im * im)))) + Float64(1.0 + Float64(Float64(im * im) * 0.5))) * Float64(1.0 + Float64(re * Float64(re * Float64(-0.5 + Float64(re * Float64(re * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.001388888888888889)))))))));
	else
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	tmp = 0.0;
	if (re <= 1.5e+154)
		tmp = ((t_0 * (im * (im * (im * im)))) + (1.0 + ((im * im) * 0.5))) * (1.0 + (re * (re * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))))));
	else
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 1.5e+154], N[(N[(N[(t$95$0 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(re * N[(-0.5 + N[(re * N[(re * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < 1.50000000000000013e154

   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

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 90.9%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \color{blue}{\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \color{blue}{\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 65.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 65.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\right)} \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right) \]

   if 1.50000000000000013e154 < re

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 89.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \color{blue}{\frac{1}{2} \cdot {im}^{2}}\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{4} + \left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right) + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4} + \frac{1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)} + \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{\color{blue}{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right) + \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2} + \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right) + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left({im}^{4} \cdot \frac{1}{720}\right) \cdot {im}^{2} + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\left({im}^{4} \cdot \frac{1}{720} + \frac{1}{2}\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{\left(2 \cdot 2\right)} \cdot \frac{1}{720} + \frac{1}{2}\right)\right)\right) \]

      13. pow-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left(\left({im}^{2} \cdot {im}^{2}\right) \cdot \frac{1}{720} + \frac{1}{2}\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \frac{1}{720}\right) + \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}\right)\right)\right) \]

   8. Simplified 35.6%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{if}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;t\_0 \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (*
           im
           (*
            im
            (+
             0.5
             (*
              im
              (*
               im
               (+
                0.041666666666666664
                (* (* im im) 0.001388888888888889))))))))))
   (if (<= re 1.5e+154) (* t_0 (+ 1.0 (* re (* re -0.5)))) t_0)))

C

double code(double re, double im) {
	double t_0 = 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	double tmp;
	if (re <= 1.5e+154) {
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = 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 = 1.0d0 + (im * (im * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))
    if (re <= 1.5d+154) then
        tmp = t_0 * (1.0d0 + (re * (re * (-0.5d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	double tmp;
	if (re <= 1.5e+154) {
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))
	tmp = 0
	if re <= 1.5e+154:
		tmp = t_0 * (1.0 + (re * (re * -0.5)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))))))
	tmp = 0.0
	if (re <= 1.5e+154)
		tmp = Float64(t_0 * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	tmp = 0.0;
	if (re <= 1.5e+154)
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 1.5e+154], N[(t$95$0 * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if re < 1.50000000000000013e154

   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

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 90.9%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{2} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]

   8. Simplified 65.4%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \]

   if 1.50000000000000013e154 < re

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 89.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \color{blue}{\frac{1}{2} \cdot {im}^{2}}\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{4} + \left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right) + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4} + \frac{1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)} + \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{\color{blue}{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right) + \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2} + \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right) + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left({im}^{4} \cdot \frac{1}{720}\right) \cdot {im}^{2} + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\left({im}^{4} \cdot \frac{1}{720} + \frac{1}{2}\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{\left(2 \cdot 2\right)} \cdot \frac{1}{720} + \frac{1}{2}\right)\right)\right) \]

      13. pow-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left(\left({im}^{2} \cdot {im}^{2}\right) \cdot \frac{1}{720} + \frac{1}{2}\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \frac{1}{720}\right) + \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}\right)\right)\right) \]

   8. Simplified 35.6%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8 \cdot 10^{+128}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8e+128)
   (+
    1.0
    (*
     im
     (*
      im
      (+
       0.5
       (*
        im
        (* im (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))))))
   (*
    (+ 1.0 (* (* im im) (+ 0.5 (* 0.041666666666666664 (* im im)))))
    (+ 1.0 (* re (* re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8e+128) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + (re * (re * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 8d+128) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))
    else
        tmp = (1.0d0 + ((im * im) * (0.5d0 + (0.041666666666666664d0 * (im * im))))) * (1.0d0 + (re * (re * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8e+128) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + (re * (re * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8e+128:
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))
	else:
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + (re * (re * -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8e+128)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))) * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8e+128)
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	else
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + (re * (re * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8e+128], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.0000000000000006e128

   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

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \color{blue}{\frac{1}{2} \cdot {im}^{2}}\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{4} + \left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right) + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4} + \frac{1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)} + \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{\color{blue}{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right) + \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2} + \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right) + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left({im}^{4} \cdot \frac{1}{720}\right) \cdot {im}^{2} + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\left({im}^{4} \cdot \frac{1}{720} + \frac{1}{2}\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{\left(2 \cdot 2\right)} \cdot \frac{1}{720} + \frac{1}{2}\right)\right)\right) \]

      13. pow-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left(\left({im}^{2} \cdot {im}^{2}\right) \cdot \frac{1}{720} + \frac{1}{2}\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \frac{1}{720}\right) + \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}\right)\right)\right) \]

   8. Simplified 55.7%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   if 8.0000000000000006e128 < 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

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      7. unpow2 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) \cdot \cos re \]

      8. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \cos \color{blue}{re} \]

      9. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) \cdot \cos re \]

      10. distribute-rgt-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + im \cdot \left(\frac{1}{2} \cdot im\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \color{blue}{\left(1 + im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + \color{blue}{1}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      6. *-lowering-*.f64 81.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 81.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot -0.5\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8 \cdot 10^{+128}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.5% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.1e+25)
   (+ 1.0 (* im (* im (+ 0.5 (* 0.041666666666666664 (* im im))))))
   (if (<= re 1.5e+154)
     (* (* im im) (+ 0.5 (* (* re re) -0.25)))
     (+ 1.0 (* (* re re) (+ -0.5 (* re (* re 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.1e+25) {
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	} else if (re <= 1.5e+154) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.1d+25) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (0.041666666666666664d0 * (im * im)))))
    else if (re <= 1.5d+154) then
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = 1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.1e+25) {
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	} else if (re <= 1.5e+154) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.1e+25:
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))))
	elif re <= 1.5e+154:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.1e+25)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))));
	elseif (re <= 1.5e+154)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.1e+25)
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	elseif (re <= 1.5e+154)
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	else
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.1e+25], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.5e+154], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(re * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 2.0999999999999999e25

   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

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      7. unpow2 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) \cdot \cos re \]

      8. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \cos \color{blue}{re} \]

      9. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) \cdot \cos re \]

      10. distribute-rgt-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + im \cdot \left(\frac{1}{2} \cdot im\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \color{blue}{\left(1 + im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + \color{blue}{1}\right)\right) \]

   5. Simplified 84.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 62.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 62.6%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   if 2.0999999999999999e25 < re < 1.50000000000000013e154

   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

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 80.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 80.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      20. *-lowering-*.f64 30.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 30.3%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\right)} \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. *-lowering-*.f64 26.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 26.9%

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

   12. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       7. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(1 \cdot \frac{1}{2} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{2}}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)} \cdot \frac{1}{2}\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left({re}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

       15. metadata-eval 27.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right)\right) \]

   14. Simplified 27.3%

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

   if 1.50000000000000013e154 < re

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 52.7%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 52.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {re}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \color{blue}{\left(re \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 34.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 34.4%

      \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.5% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.1e+25)
   (+ 1.0 (* im (* im (+ 0.5 (* 0.041666666666666664 (* im im))))))
   (if (<= re 1.5e+154)
     (* (* im im) (+ 0.5 (* (* re re) -0.25)))
     (* 0.041666666666666664 (* (* re re) (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.1e+25) {
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	} else if (re <= 1.5e+154) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 0.041666666666666664 * ((re * re) * (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 (re <= 2.1d+25) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (0.041666666666666664d0 * (im * im)))))
    else if (re <= 1.5d+154) then
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = 0.041666666666666664d0 * ((re * re) * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.1e+25) {
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	} else if (re <= 1.5e+154) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 0.041666666666666664 * ((re * re) * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.1e+25:
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))))
	elif re <= 1.5e+154:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = 0.041666666666666664 * ((re * re) * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.1e+25)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))));
	elseif (re <= 1.5e+154)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = Float64(0.041666666666666664 * Float64(Float64(re * re) * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.1e+25)
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	elseif (re <= 1.5e+154)
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	else
		tmp = 0.041666666666666664 * ((re * re) * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.1e+25], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.5e+154], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 2.0999999999999999e25

   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

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\cos re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\cos re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re} \]

      7. unpow2 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) \cdot \cos re \]

      8. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \cos \color{blue}{re} \]

      9. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) \cdot \cos re \]

      10. distribute-rgt-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + im \cdot \left(\frac{1}{2} \cdot im\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \color{blue}{\left(1 + im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + \color{blue}{1}\right)\right) \]

   5. Simplified 84.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 62.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 62.6%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   if 2.0999999999999999e25 < re < 1.50000000000000013e154

   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

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 80.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 80.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      20. *-lowering-*.f64 30.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 30.3%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\right)} \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. *-lowering-*.f64 26.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 26.9%

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

   12. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       7. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(1 \cdot \frac{1}{2} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{2}}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)} \cdot \frac{1}{2}\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left({re}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

       15. metadata-eval 27.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right)\right) \]

   14. Simplified 27.3%

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

   if 1.50000000000000013e154 < re

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 52.7%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 52.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {re}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \color{blue}{\left(re \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 34.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 34.4%

      \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {re}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({re}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \color{blue}{{re}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left({\color{blue}{re}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({\color{blue}{re}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       8. *-lowering-*.f64 34.4%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   11. Simplified 34.4%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.0% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(re \cdot \left(-0.5 + \left(im \cdot im\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.1e+25)
   (+ 1.0 (* (* im im) 0.5))
   (if (<= re 1.5e+154)
     (* re (* re (+ -0.5 (* (* im im) -0.25))))
     (* 0.041666666666666664 (* (* re re) (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.1e+25) {
		tmp = 1.0 + ((im * im) * 0.5);
	} else if (re <= 1.5e+154) {
		tmp = re * (re * (-0.5 + ((im * im) * -0.25)));
	} else {
		tmp = 0.041666666666666664 * ((re * re) * (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 (re <= 2.1d+25) then
        tmp = 1.0d0 + ((im * im) * 0.5d0)
    else if (re <= 1.5d+154) then
        tmp = re * (re * ((-0.5d0) + ((im * im) * (-0.25d0))))
    else
        tmp = 0.041666666666666664d0 * ((re * re) * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.1e+25) {
		tmp = 1.0 + ((im * im) * 0.5);
	} else if (re <= 1.5e+154) {
		tmp = re * (re * (-0.5 + ((im * im) * -0.25)));
	} else {
		tmp = 0.041666666666666664 * ((re * re) * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.1e+25:
		tmp = 1.0 + ((im * im) * 0.5)
	elif re <= 1.5e+154:
		tmp = re * (re * (-0.5 + ((im * im) * -0.25)))
	else:
		tmp = 0.041666666666666664 * ((re * re) * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.1e+25)
		tmp = Float64(1.0 + Float64(Float64(im * im) * 0.5));
	elseif (re <= 1.5e+154)
		tmp = Float64(re * Float64(re * Float64(-0.5 + Float64(Float64(im * im) * -0.25))));
	else
		tmp = Float64(0.041666666666666664 * Float64(Float64(re * re) * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.1e+25)
		tmp = 1.0 + ((im * im) * 0.5);
	elseif (re <= 1.5e+154)
		tmp = re * (re * (-0.5 + ((im * im) * -0.25)));
	else
		tmp = 0.041666666666666664 * ((re * re) * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.1e+25], N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.5e+154], N[(re * N[(re * N[(-0.5 + N[(N[(im * im), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 2.0999999999999999e25

   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

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 73.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 73.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

      4. *-lowering-*.f64 54.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   8. Simplified 54.0%

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

   if 2.0999999999999999e25 < re < 1.50000000000000013e154

   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

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 80.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 80.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      20. *-lowering-*.f64 30.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 30.3%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\right)} \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. *-lowering-*.f64 26.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 26.9%

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

   12. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)}\right)\right) \]

       8. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} \cdot 1 + \color{blue}{\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + \color{blue}{\frac{-1}{2}} \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)\right)\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left({im}^{2} \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{-1}{2}\right)}\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{-1}{4}\right)\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

       19. metadata-eval 26.9%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{4}\right)\right)\right)\right) \]

   14. Simplified 26.9%

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

   if 1.50000000000000013e154 < re

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 52.7%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 52.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {re}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \color{blue}{\left(re \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 34.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 34.4%

      \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {re}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({re}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \color{blue}{{re}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left({\color{blue}{re}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({\color{blue}{re}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       8. *-lowering-*.f64 34.4%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   11. Simplified 34.4%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(re \cdot \left(-0.5 + \left(im \cdot im\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.9% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (+
  1.0
  (*
   im
   (*
    im
    (+
     0.5
     (*
      im
      (* im (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))))))))

C

double code(double re, double im) {
	return 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + (im * (im * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))
end function

Java

public static double code(double re, double im) {
	return 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
}

Python

def code(re, im):
	return 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))

Julia

function code(re, im)
	return Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))))))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
end

Wolfram

code[re_, im_] := N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $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

   \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

   2. +-commutative N/A

      \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

   3. distribute-lft-in N/A

      \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

   4. associate-+l+ N/A

      \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

5. Simplified 90.8%

   \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \color{blue}{\frac{1}{2} \cdot {im}^{2}}\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{4} + \left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right) + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4} + \frac{1}{2} \cdot {im}^{2}\right)}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)} + \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{\color{blue}{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

   6. pow-sqr N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right) + \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2} + \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}} + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right) + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left({im}^{4} \cdot \frac{1}{720}\right) \cdot {im}^{2} + \color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right)\right) \]

   11. distribute-rgt-out N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\left({im}^{4} \cdot \frac{1}{720} + \frac{1}{2}\right)}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{\left(2 \cdot 2\right)} \cdot \frac{1}{720} + \frac{1}{2}\right)\right)\right) \]

   13. pow-sqr N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left(\left({im}^{2} \cdot {im}^{2}\right) \cdot \frac{1}{720} + \frac{1}{2}\right)\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \frac{1}{720}\right) + \frac{1}{2}\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}\right)\right)\right) \]

8. Simplified 60.0%

   \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]
9. Add Preprocessing

Alternative 16: 49.2% accurate, 19.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 280000000:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 280000000.0)
   (+ 1.0 (* (* im im) 0.5))
   (* (* im im) (+ 0.5 (* (* re re) -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 280000000.0) {
		tmp = 1.0 + ((im * im) * 0.5);
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 280000000.0d0) then
        tmp = 1.0d0 + ((im * im) * 0.5d0)
    else
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 280000000.0) {
		tmp = 1.0 + ((im * im) * 0.5);
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 280000000.0:
		tmp = 1.0 + ((im * im) * 0.5)
	else:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 280000000.0)
		tmp = Float64(1.0 + Float64(Float64(im * im) * 0.5));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 280000000.0)
		tmp = 1.0 + ((im * im) * 0.5);
	else
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 280000000.0], N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.8e8

   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

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 77.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 77.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

      4. *-lowering-*.f64 45.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   8. Simplified 45.7%

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

   if 2.8e8 < 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

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 59.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 59.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      20. *-lowering-*.f64 52.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 52.4%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\right)} \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. *-lowering-*.f64 50.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 50.9%

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

   12. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       7. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(1 \cdot \frac{1}{2} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{2}}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)} \cdot \frac{1}{2}\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left({re}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

       15. metadata-eval 50.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right)\right) \]

   14. Simplified 50.9%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 280000000:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.0% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.7 \cdot 10^{+135}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.7e+135)
   (+ 1.0 (* (* im im) 0.5))
   (* 0.041666666666666664 (* (* re re) (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.7e+135) {
		tmp = 1.0 + ((im * im) * 0.5);
	} else {
		tmp = 0.041666666666666664 * ((re * re) * (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 (re <= 2.7d+135) then
        tmp = 1.0d0 + ((im * im) * 0.5d0)
    else
        tmp = 0.041666666666666664d0 * ((re * re) * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.7e+135) {
		tmp = 1.0 + ((im * im) * 0.5);
	} else {
		tmp = 0.041666666666666664 * ((re * re) * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.7e+135:
		tmp = 1.0 + ((im * im) * 0.5)
	else:
		tmp = 0.041666666666666664 * ((re * re) * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.7e+135)
		tmp = Float64(1.0 + Float64(Float64(im * im) * 0.5));
	else
		tmp = Float64(0.041666666666666664 * Float64(Float64(re * re) * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.7e+135)
		tmp = 1.0 + ((im * im) * 0.5);
	else
		tmp = 0.041666666666666664 * ((re * re) * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.7e+135], N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.69999999999999985e135

   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

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 74.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 74.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

      4. *-lowering-*.f64 50.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   8. Simplified 50.8%

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

   if 2.69999999999999985e135 < re

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 54.2%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 54.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {re}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \color{blue}{\left(re \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 34.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 34.4%

      \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {re}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({re}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \color{blue}{{re}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left({\color{blue}{re}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({\color{blue}{re}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       8. *-lowering-*.f64 34.4%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   11. Simplified 34.4%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.7 \cdot 10^{+135}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.8% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.45:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1.45) 1.0 (* (* im im) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.45) {
		tmp = 1.0;
	} else {
		tmp = (im * im) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.45d0) then
        tmp = 1.0d0
    else
        tmp = (im * im) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.45) {
		tmp = 1.0;
	} else {
		tmp = (im * im) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.45:
		tmp = 1.0
	else:
		tmp = (im * im) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.45)
		tmp = 1.0;
	else
		tmp = Float64(Float64(im * im) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.45)
		tmp = 1.0;
	else
		tmp = (im * im) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.45], 1.0, N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.44999999999999996

   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

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

      1. cos-lowering-cos.f64 61.5%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 61.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 33.4%

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

   if 1.44999999999999996 < 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

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 56.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 56.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

      4. *-lowering-*.f64 47.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   8. Simplified 47.8%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right) \]

       3. *-lowering-*.f64 47.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

   11. Simplified 47.8%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.45:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.8% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. unpow2 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

   4. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

   5. *-commutative N/A

      \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   6. *-commutative N/A

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

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

   16. *-lowering-*.f64 73.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

5. Simplified 73.5%

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

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

   4. *-lowering-*.f64 46.9%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

8. Simplified 46.9%

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

9. Final simplification 46.9%

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

Alternative 20: 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. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. cos-lowering-cos.f64 47.1%

      \[\leadsto \mathsf{cos.f64}\left(re\right) \]

5. Simplified 47.1%

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

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 25.9%

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

Reproduce

herbie shell --seed 2024140 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))