math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.5s

Alternatives: 29

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   \[\leadsto \cos re \cdot \cosh im \]
8. Add Preprocessing

Alternative 2: 94.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\\ \mathbf{if}\;im \leq 235:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot t\_0\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + t\_0 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))
   (if (<= im 235.0)
     (* (cos re) (+ 1.0 (* im (* im (+ 0.5 (* im (* im t_0)))))))
     (if (<= im 4.6e+51)
       (cosh im)
       (* (cos re) (+ 1.0 (* t_0 (* im (* im (* im im))))))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double tmp;
	if (im <= 235.0) {
		tmp = cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * t_0))))));
	} else if (im <= 4.6e+51) {
		tmp = cosh(im);
	} else {
		tmp = cos(re) * (1.0 + (t_0 * (im * (im * (im * im)))));
	}
	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 <= 235.0d0) then
        tmp = cos(re) * (1.0d0 + (im * (im * (0.5d0 + (im * (im * t_0))))))
    else if (im <= 4.6d+51) then
        tmp = cosh(im)
    else
        tmp = cos(re) * (1.0d0 + (t_0 * (im * (im * (im * im)))))
    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 <= 235.0) {
		tmp = Math.cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * t_0))))));
	} else if (im <= 4.6e+51) {
		tmp = Math.cosh(im);
	} else {
		tmp = Math.cos(re) * (1.0 + (t_0 * (im * (im * (im * im)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889)
	tmp = 0
	if im <= 235.0:
		tmp = math.cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * t_0))))))
	elif im <= 4.6e+51:
		tmp = math.cosh(im)
	else:
		tmp = math.cos(re) * (1.0 + (t_0 * (im * (im * (im * im)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))
	tmp = 0.0
	if (im <= 235.0)
		tmp = Float64(cos(re) * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * t_0)))))));
	elseif (im <= 4.6e+51)
		tmp = cosh(im);
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(t_0 * Float64(im * Float64(im * Float64(im * im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	tmp = 0.0;
	if (im <= 235.0)
		tmp = cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * t_0))))));
	elseif (im <= 4.6e+51)
		tmp = cosh(im);
	else
		tmp = cos(re) * (1.0 + (t_0 * (im * (im * (im * im)))));
	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, 235.0], 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], If[LessEqual[im, 4.6e+51], N[Cosh[im], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(t$95$0 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 235

   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 93.2%

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

   if 235 < im < 4.6000000000000001e51

   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 86.7%

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

   if 4.6000000000000001e51 < 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 100.0%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 100.0%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7e-9)
   (cos re)
   (if (<= im 4.6e+51)
     (cosh im)
     (*
      (cos re)
      (+
       1.0
       (*
        (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))
        (* im (* im (* im im)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = cos(re);
	} else if (im <= 4.6e+51) {
		tmp = cosh(im);
	} else {
		tmp = cos(re) * (1.0 + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.7d-9) then
        tmp = cos(re)
    else if (im <= 4.6d+51) then
        tmp = cosh(im)
    else
        tmp = cos(re) * (1.0d0 + ((0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)) * (im * (im * (im * im)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = Math.cos(re);
	} else if (im <= 4.6e+51) {
		tmp = Math.cosh(im);
	} else {
		tmp = Math.cos(re) * (1.0 + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7e-9:
		tmp = math.cos(re)
	elif im <= 4.6e+51:
		tmp = math.cosh(im)
	else:
		tmp = math.cos(re) * (1.0 + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 4.6e+51)
		tmp = cosh(im);
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)) * Float64(im * Float64(im * Float64(im * im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 4.6e+51)
		tmp = cosh(im);
	else
		tmp = cos(re) * (1.0 + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7e-9], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4.6e+51], N[Cosh[im], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.7e-9

   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 63.1%

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

   5. Simplified 63.1%

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

   if 3.7e-9 < im < 4.6000000000000001e51

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   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 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   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 99.9%

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

   6. Applied egg-rr 99.9%

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

   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 82.4%

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

   if 4.6000000000000001e51 < 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 100.0%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 100.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7e-9)
   (cos re)
   (if (<= im 1.4e+77)
     (cosh im)
     (*
      (cos re)
      (+ 1.0 (* (* im im) (+ 0.5 (* 0.041666666666666664 (* im im)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = cos(re);
	} else if (im <= 1.4e+77) {
		tmp = cosh(im);
	} else {
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.7d-9) then
        tmp = cos(re)
    else if (im <= 1.4d+77) then
        tmp = cosh(im)
    else
        tmp = cos(re) * (1.0d0 + ((im * im) * (0.5d0 + (0.041666666666666664d0 * (im * im)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = Math.cos(re);
	} else if (im <= 1.4e+77) {
		tmp = Math.cosh(im);
	} else {
		tmp = Math.cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7e-9:
		tmp = math.cos(re)
	elif im <= 1.4e+77:
		tmp = math.cosh(im)
	else:
		tmp = math.cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.4e+77)
		tmp = cosh(im);
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.4e+77)
		tmp = cosh(im);
	else
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7e-9], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.4e+77], N[Cosh[im], $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.7e-9

   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 63.1%

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

   5. Simplified 63.1%

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

   if 3.7e-9 < im < 1.4e77

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   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 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   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 99.9%

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

   6. Applied egg-rr 99.9%

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

   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 73.1%

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

   if 1.4e77 < 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%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7e-9)
   (cos re)
   (if (<= im 1.4e+77)
     (cosh im)
     (* (cos re) (+ 1.0 (* im (* im (* 0.041666666666666664 (* im im)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = cos(re);
	} else if (im <= 1.4e+77) {
		tmp = cosh(im);
	} else {
		tmp = cos(re) * (1.0 + (im * (im * (0.041666666666666664 * (im * im)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.7d-9) then
        tmp = cos(re)
    else if (im <= 1.4d+77) then
        tmp = cosh(im)
    else
        tmp = cos(re) * (1.0d0 + (im * (im * (0.041666666666666664d0 * (im * im)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = Math.cos(re);
	} else if (im <= 1.4e+77) {
		tmp = Math.cosh(im);
	} else {
		tmp = Math.cos(re) * (1.0 + (im * (im * (0.041666666666666664 * (im * im)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7e-9:
		tmp = math.cos(re)
	elif im <= 1.4e+77:
		tmp = math.cosh(im)
	else:
		tmp = math.cos(re) * (1.0 + (im * (im * (0.041666666666666664 * (im * im)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.4e+77)
		tmp = cosh(im);
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(im * Float64(im * Float64(0.041666666666666664 * Float64(im * im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.4e+77)
		tmp = cosh(im);
	else
		tmp = cos(re) * (1.0 + (im * (im * (0.041666666666666664 * (im * im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7e-9], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.4e+77], N[Cosh[im], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.7e-9

   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 63.1%

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

   5. Simplified 63.1%

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

   if 3.7e-9 < im < 1.4e77

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   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 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   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 99.9%

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

   6. Applied egg-rr 99.9%

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

   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 73.1%

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

   if 1.4e77 < 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 im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}, 1\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7e-9)
   (cos re)
   (if (<= im 1.4e+77)
     (cosh im)
     (* (cos re) (* im (* im (* 0.041666666666666664 (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = cos(re);
	} else if (im <= 1.4e+77) {
		tmp = cosh(im);
	} else {
		tmp = cos(re) * (im * (im * (0.041666666666666664 * (im * im))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.7d-9) then
        tmp = cos(re)
    else if (im <= 1.4d+77) then
        tmp = cosh(im)
    else
        tmp = cos(re) * (im * (im * (0.041666666666666664d0 * (im * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = Math.cos(re);
	} else if (im <= 1.4e+77) {
		tmp = Math.cosh(im);
	} else {
		tmp = Math.cos(re) * (im * (im * (0.041666666666666664 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7e-9:
		tmp = math.cos(re)
	elif im <= 1.4e+77:
		tmp = math.cosh(im)
	else:
		tmp = math.cos(re) * (im * (im * (0.041666666666666664 * (im * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.4e+77)
		tmp = cosh(im);
	else
		tmp = Float64(cos(re) * Float64(im * Float64(im * Float64(0.041666666666666664 * Float64(im * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.4e+77)
		tmp = cosh(im);
	else
		tmp = cos(re) * (im * (im * (0.041666666666666664 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7e-9], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.4e+77], N[Cosh[im], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(im * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.7e-9

   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 63.1%

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

   5. Simplified 63.1%

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

   if 3.7e-9 < im < 1.4e77

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   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 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   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 99.9%

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

   6. Applied egg-rr 99.9%

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

   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 73.1%

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

   if 1.4e77 < 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 im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7e-9)
   (cos re)
   (if (<= im 1.35e+154) (cosh im) (* (cos re) (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = cosh(im);
	} else {
		tmp = cos(re) * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.7d-9) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = cosh(im)
    else
        tmp = cos(re) * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = Math.cosh(im);
	} else {
		tmp = Math.cos(re) * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7e-9:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = math.cosh(im)
	else:
		tmp = math.cos(re) * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = cosh(im);
	else
		tmp = Float64(cos(re) * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = cosh(im);
	else
		tmp = cos(re) * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7e-9], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[Cosh[im], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.7e-9

   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 63.1%

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

   5. Simplified 63.1%

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

   if 3.7e-9 < 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

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

   9. Simplified 71.1%

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

   if 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 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7e-9)
   (cos re)
   (if (<= im 2e+132)
     (cosh im)
     (*
      (* (* im im) (* im im))
      (+ 0.041666666666666664 (* (* re re) -0.020833333333333332))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = cos(re);
	} else if (im <= 2e+132) {
		tmp = cosh(im);
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.7d-9) then
        tmp = cos(re)
    else if (im <= 2d+132) then
        tmp = cosh(im)
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7e-9) {
		tmp = Math.cos(re);
	} else if (im <= 2e+132) {
		tmp = Math.cosh(im);
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7e-9:
		tmp = math.cos(re)
	elif im <= 2e+132:
		tmp = math.cosh(im)
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 2e+132)
		tmp = cosh(im);
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 2e+132)
		tmp = cosh(im);
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7e-9], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2e+132], N[Cosh[im], $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.7e-9

   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 63.1%

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

   5. Simplified 63.1%

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

   if 3.7e-9 < im < 1.99999999999999998e132

   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 72.2%

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

   if 1.99999999999999998e132 < 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 im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot \left({im}^{4} \cdot {re}^{2}\right) + \frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{24}} \cdot {im}^{4} \]

       2. +-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right)} \cdot {re}^{2} \]

       4. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \frac{-1}{48} \cdot \color{blue}{\left({im}^{4} \cdot {re}^{2}\right)} \]

       5. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \left({im}^{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{48}} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + {im}^{4} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)} \]

       7. distribute-lft-out N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       10. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{48}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{48}\right)\right)\right) \]

       19. *-lowering-*.f64 83.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{48}\right)\right)\right) \]

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\\ t_1 := 0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\\ t_2 := \left(im \cdot im\right) \cdot t\_1\\ \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+26}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\right) \cdot \left(t\_0 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{im \cdot \left(\left(im \cdot t\_1\right) \cdot t\_2\right) + -1}{t\_2 + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot t\_0\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))
        (t_1
         (+
          0.5
          (*
           (* im im)
           (+ 0.041666666666666664 (* im (* im 0.001388888888888889))))))
        (t_2 (* (* im im) t_1)))
   (if (<= im 3.7e-9)
     (cos re)
     (if (<= im 1.45e+26)
       (*
        (+ 1.0 (* (* re re) (+ -0.5 (* re (* re 0.041666666666666664)))))
        (+ (* t_0 (* im (* im (* im im)))) (+ 1.0 (* 0.5 (* im im)))))
       (if (<= im 4.6e+51)
         (/ (+ (* im (* (* im t_1) t_2)) -1.0) (+ t_2 -1.0))
         (*
          (+ 1.0 (* im (* im (+ 0.5 (* im (* im t_0))))))
          (+ 1.0 (* re (* re -0.5)))))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double t_1 = 0.5 + ((im * im) * (0.041666666666666664 + (im * (im * 0.001388888888888889))));
	double t_2 = (im * im) * t_1;
	double tmp;
	if (im <= 3.7e-9) {
		tmp = cos(re);
	} else if (im <= 1.45e+26) {
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_0 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))));
	} else if (im <= 4.6e+51) {
		tmp = ((im * ((im * t_1) * t_2)) + -1.0) / (t_2 + -1.0);
	} else {
		tmp = (1.0 + (im * (im * (0.5 + (im * (im * t_0)))))) * (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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)
    t_1 = 0.5d0 + ((im * im) * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0))))
    t_2 = (im * im) * t_1
    if (im <= 3.7d-9) then
        tmp = cos(re)
    else if (im <= 1.45d+26) then
        tmp = (1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * 0.041666666666666664d0))))) * ((t_0 * (im * (im * (im * im)))) + (1.0d0 + (0.5d0 * (im * im))))
    else if (im <= 4.6d+51) then
        tmp = ((im * ((im * t_1) * t_2)) + (-1.0d0)) / (t_2 + (-1.0d0))
    else
        tmp = (1.0d0 + (im * (im * (0.5d0 + (im * (im * t_0)))))) * (1.0d0 + (re * (re * (-0.5d0))))
    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 t_1 = 0.5 + ((im * im) * (0.041666666666666664 + (im * (im * 0.001388888888888889))));
	double t_2 = (im * im) * t_1;
	double tmp;
	if (im <= 3.7e-9) {
		tmp = Math.cos(re);
	} else if (im <= 1.45e+26) {
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_0 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))));
	} else if (im <= 4.6e+51) {
		tmp = ((im * ((im * t_1) * t_2)) + -1.0) / (t_2 + -1.0);
	} else {
		tmp = (1.0 + (im * (im * (0.5 + (im * (im * t_0)))))) * (1.0 + (re * (re * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889)
	t_1 = 0.5 + ((im * im) * (0.041666666666666664 + (im * (im * 0.001388888888888889))))
	t_2 = (im * im) * t_1
	tmp = 0
	if im <= 3.7e-9:
		tmp = math.cos(re)
	elif im <= 1.45e+26:
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_0 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))))
	elif im <= 4.6e+51:
		tmp = ((im * ((im * t_1) * t_2)) + -1.0) / (t_2 + -1.0)
	else:
		tmp = (1.0 + (im * (im * (0.5 + (im * (im * t_0)))))) * (1.0 + (re * (re * -0.5)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))
	t_1 = Float64(0.5 + Float64(Float64(im * im) * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889)))))
	t_2 = Float64(Float64(im * im) * t_1)
	tmp = 0.0
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.45e+26)
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * 0.041666666666666664))))) * Float64(Float64(t_0 * Float64(im * Float64(im * Float64(im * im)))) + Float64(1.0 + Float64(0.5 * Float64(im * im)))));
	elseif (im <= 4.6e+51)
		tmp = Float64(Float64(Float64(im * Float64(Float64(im * t_1) * t_2)) + -1.0) / Float64(t_2 + -1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * t_0)))))) * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	t_1 = 0.5 + ((im * im) * (0.041666666666666664 + (im * (im * 0.001388888888888889))));
	t_2 = (im * im) * t_1;
	tmp = 0.0;
	if (im <= 3.7e-9)
		tmp = cos(re);
	elseif (im <= 1.45e+26)
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_0 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))));
	elseif (im <= 4.6e+51)
		tmp = ((im * ((im * t_1) * t_2)) + -1.0) / (t_2 + -1.0);
	else
		tmp = (1.0 + (im * (im * (0.5 + (im * (im * t_0)))))) * (1.0 + (re * (re * -0.5)));
	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]}, Block[{t$95$1 = N[(0.5 + N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 + N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(im * im), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[im, 3.7e-9], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.45e+26], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(re * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.6e+51], N[(N[(N[(im * N[(N[(im * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 3.7e-9

   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 63.1%

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

   5. Simplified 63.1%

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

   if 3.7e-9 < im < 1.45e26

   1. Initial program 99.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   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 16.2%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 39.0%

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

   8. Simplified 39.0%

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

   if 1.45e26 < im < 4.6000000000000001e51

   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 6.3%

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

   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 5.6%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. flip-+ N/A

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

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

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

   10. Applied egg-rr 88.9%

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

   if 4.6000000000000001e51 < 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 100.0%

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

   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 75.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+26}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\right) \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{im \cdot \left(\left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\right) + -1}{\left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\\ t_1 := \left(im \cdot im\right) \cdot t\_0\\ \mathbf{if}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{im \cdot \left(\left(im \cdot t\_0\right) \cdot t\_1\right) + -1}{t\_1 + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          0.5
          (*
           (* im im)
           (+ 0.041666666666666664 (* im (* im 0.001388888888888889))))))
        (t_1 (* (* im im) t_0)))
   (if (<= im 4.6e+51)
     (/ (+ (* im (* (* im t_0) t_1)) -1.0) (+ t_1 -1.0))
     (*
      (+
       1.0
       (*
        im
        (*
         im
         (+
          0.5
          (*
           im
           (*
            im
            (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))))))
      (+ 1.0 (* re (* re -0.5)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 + ((im * im) * (0.041666666666666664 + (im * (im * 0.001388888888888889))));
	double t_1 = (im * im) * t_0;
	double tmp;
	if (im <= 4.6e+51) {
		tmp = ((im * ((im * t_0) * t_1)) + -1.0) / (t_1 + -1.0);
	} else {
		tmp = (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))) * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 + ((im * im) * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0))))
    t_1 = (im * im) * t_0
    if (im <= 4.6d+51) then
        tmp = ((im * ((im * t_0) * t_1)) + (-1.0d0)) / (t_1 + (-1.0d0))
    else
        tmp = (1.0d0 + (im * (im * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))) * (1.0d0 + (re * (re * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + ((im * im) * (0.041666666666666664 + (im * (im * 0.001388888888888889))));
	double t_1 = (im * im) * t_0;
	double tmp;
	if (im <= 4.6e+51) {
		tmp = ((im * ((im * t_0) * t_1)) + -1.0) / (t_1 + -1.0);
	} else {
		tmp = (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))) * (1.0 + (re * (re * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + ((im * im) * (0.041666666666666664 + (im * (im * 0.001388888888888889))))
	t_1 = (im * im) * t_0
	tmp = 0
	if im <= 4.6e+51:
		tmp = ((im * ((im * t_0) * t_1)) + -1.0) / (t_1 + -1.0)
	else:
		tmp = (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))) * (1.0 + (re * (re * -0.5)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(Float64(im * im) * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889)))))
	t_1 = Float64(Float64(im * im) * t_0)
	tmp = 0.0
	if (im <= 4.6e+51)
		tmp = Float64(Float64(Float64(im * Float64(Float64(im * t_0) * t_1)) + -1.0) / Float64(t_1 + -1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)))))))) * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + ((im * im) * (0.041666666666666664 + (im * (im * 0.001388888888888889))));
	t_1 = (im * im) * t_0;
	tmp = 0.0;
	if (im <= 4.6e+51)
		tmp = ((im * ((im * t_0) * t_1)) + -1.0) / (t_1 + -1.0);
	else
		tmp = (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))) * (1.0 + (re * (re * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 + N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[im, 4.6e+51], N[(N[(N[(im * N[(N[(im * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], N[(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[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if im < 4.6000000000000001e51

   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 86.8%

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

   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 55.7%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. flip-+ N/A

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

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

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

   10. Applied egg-rr 39.3%

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

   if 4.6000000000000001e51 < 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 100.0%

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

   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 75.5%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{im \cdot \left(\left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\right) + -1}{\left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\\ t_1 := \left(im \cdot im\right) \cdot t\_0\\ t_2 := 0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\\ t_3 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot t\_2\right)\right)\right)\\ \mathbf{if}\;im \leq 380:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\right) \cdot \left(t\_2 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;1 + im \cdot \frac{im \cdot \left(0.25 - \left(im \cdot im\right) \cdot \left(t\_0 \cdot t\_1\right)\right)}{0.5 - t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.041666666666666664 (* im (* im 0.001388888888888889))))
        (t_1 (* (* im im) t_0))
        (t_2 (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))
        (t_3 (+ 1.0 (* im (* im (+ 0.5 (* im (* im t_2))))))))
   (if (<= im 380.0)
     t_3
     (if (<= im 9.5e+30)
       (*
        (+ 1.0 (* (* re re) (+ -0.5 (* re (* re 0.041666666666666664)))))
        (+ (* t_2 (* im (* im (* im im)))) (+ 1.0 (* 0.5 (* im im)))))
       (if (<= im 4.6e+51)
         (+
          1.0
          (* im (/ (* im (- 0.25 (* (* im im) (* t_0 t_1)))) (- 0.5 t_1))))
         (* t_3 (+ 1.0 (* re (* re -0.5)))))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 + (im * (im * 0.001388888888888889));
	double t_1 = (im * im) * t_0;
	double t_2 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double t_3 = 1.0 + (im * (im * (0.5 + (im * (im * t_2)))));
	double tmp;
	if (im <= 380.0) {
		tmp = t_3;
	} else if (im <= 9.5e+30) {
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_2 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))));
	} else if (im <= 4.6e+51) {
		tmp = 1.0 + (im * ((im * (0.25 - ((im * im) * (t_0 * t_1)))) / (0.5 - t_1)));
	} else {
		tmp = t_3 * (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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.041666666666666664d0 + (im * (im * 0.001388888888888889d0))
    t_1 = (im * im) * t_0
    t_2 = 0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)
    t_3 = 1.0d0 + (im * (im * (0.5d0 + (im * (im * t_2)))))
    if (im <= 380.0d0) then
        tmp = t_3
    else if (im <= 9.5d+30) then
        tmp = (1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * 0.041666666666666664d0))))) * ((t_2 * (im * (im * (im * im)))) + (1.0d0 + (0.5d0 * (im * im))))
    else if (im <= 4.6d+51) then
        tmp = 1.0d0 + (im * ((im * (0.25d0 - ((im * im) * (t_0 * t_1)))) / (0.5d0 - t_1)))
    else
        tmp = t_3 * (1.0d0 + (re * (re * (-0.5d0))))
    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 t_1 = (im * im) * t_0;
	double t_2 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double t_3 = 1.0 + (im * (im * (0.5 + (im * (im * t_2)))));
	double tmp;
	if (im <= 380.0) {
		tmp = t_3;
	} else if (im <= 9.5e+30) {
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_2 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))));
	} else if (im <= 4.6e+51) {
		tmp = 1.0 + (im * ((im * (0.25 - ((im * im) * (t_0 * t_1)))) / (0.5 - t_1)));
	} else {
		tmp = t_3 * (1.0 + (re * (re * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.041666666666666664 + (im * (im * 0.001388888888888889))
	t_1 = (im * im) * t_0
	t_2 = 0.041666666666666664 + ((im * im) * 0.001388888888888889)
	t_3 = 1.0 + (im * (im * (0.5 + (im * (im * t_2)))))
	tmp = 0
	if im <= 380.0:
		tmp = t_3
	elif im <= 9.5e+30:
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_2 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))))
	elif im <= 4.6e+51:
		tmp = 1.0 + (im * ((im * (0.25 - ((im * im) * (t_0 * t_1)))) / (0.5 - t_1)))
	else:
		tmp = t_3 * (1.0 + (re * (re * -0.5)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889)))
	t_1 = Float64(Float64(im * im) * t_0)
	t_2 = Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))
	t_3 = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * t_2))))))
	tmp = 0.0
	if (im <= 380.0)
		tmp = t_3;
	elseif (im <= 9.5e+30)
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * 0.041666666666666664))))) * Float64(Float64(t_2 * Float64(im * Float64(im * Float64(im * im)))) + Float64(1.0 + Float64(0.5 * Float64(im * im)))));
	elseif (im <= 4.6e+51)
		tmp = Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.25 - Float64(Float64(im * im) * Float64(t_0 * t_1)))) / Float64(0.5 - t_1))));
	else
		tmp = Float64(t_3 * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.041666666666666664 + (im * (im * 0.001388888888888889));
	t_1 = (im * im) * t_0;
	t_2 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	t_3 = 1.0 + (im * (im * (0.5 + (im * (im * t_2)))));
	tmp = 0.0;
	if (im <= 380.0)
		tmp = t_3;
	elseif (im <= 9.5e+30)
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_2 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))));
	elseif (im <= 4.6e+51)
		tmp = 1.0 + (im * ((im * (0.25 - ((im * im) * (t_0 * t_1)))) / (0.5 - t_1)));
	else
		tmp = t_3 * (1.0 + (re * (re * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.041666666666666664 + N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 380.0], t$95$3, If[LessEqual[im, 9.5e+30], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(re * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.6e+51], N[(1.0 + N[(im * N[(N[(im * N[(0.25 - N[(N[(im * im), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 380

   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 93.2%

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

   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 59.7%

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

   if 380 < im < 9.5000000000000003e30

   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 4.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 51.5%

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

   8. Simplified 51.5%

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

   if 9.5000000000000003e30 < im < 4.6000000000000001e51

   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 6.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   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 5.8%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 72.2%

       \[\leadsto 1 + im \cdot \color{blue}{\frac{\left(0.25 - \left(im \cdot im\right) \cdot \left(\left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot im}{0.5 - \left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)}} \]

   if 4.6000000000000001e51 < 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 100.0%

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

   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 75.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 380:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\right) \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;1 + im \cdot \frac{im \cdot \left(0.25 - \left(im \cdot im\right) \cdot \left(\left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\right)}{0.5 - \left(im \cdot im\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot im\right)\\ t_1 := 0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\\ t_2 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot t\_1\right)\right)\right)\\ t_3 := im \cdot \left(im \cdot 0.001388888888888889\right)\\ \mathbf{if}\;im \leq 320:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\right) \cdot \left(t\_1 \cdot \left(im \cdot t\_0\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \frac{\left(im \cdot im\right) \cdot \left(7.233796296296296 \cdot 10^{-5} + t\_0 \cdot \left(t\_0 \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right)}{0.001736111111111111 + t\_3 \cdot \left(t\_3 - 0.041666666666666664\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im im)))
        (t_1 (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))
        (t_2 (+ 1.0 (* im (* im (+ 0.5 (* im (* im t_1)))))))
        (t_3 (* im (* im 0.001388888888888889))))
   (if (<= im 320.0)
     t_2
     (if (<= im 9.5e+30)
       (*
        (+ 1.0 (* (* re re) (+ -0.5 (* re (* re 0.041666666666666664)))))
        (+ (* t_1 (* im t_0)) (+ 1.0 (* 0.5 (* im im)))))
       (if (<= im 4.6e+51)
         (+
          1.0
          (*
           im
           (*
            im
            (+
             0.5
             (/
              (*
               (* im im)
               (+ 7.233796296296296e-5 (* t_0 (* t_0 2.6791838134430728e-9))))
              (+
               0.001736111111111111
               (* t_3 (- t_3 0.041666666666666664))))))))
         (* t_2 (+ 1.0 (* re (* re -0.5)))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * im);
	double t_1 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double t_2 = 1.0 + (im * (im * (0.5 + (im * (im * t_1)))));
	double t_3 = im * (im * 0.001388888888888889);
	double tmp;
	if (im <= 320.0) {
		tmp = t_2;
	} else if (im <= 9.5e+30) {
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_1 * (im * t_0)) + (1.0 + (0.5 * (im * im))));
	} else if (im <= 4.6e+51) {
		tmp = 1.0 + (im * (im * (0.5 + (((im * im) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_3 * (t_3 - 0.041666666666666664)))))));
	} else {
		tmp = t_2 * (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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = im * (im * im)
    t_1 = 0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)
    t_2 = 1.0d0 + (im * (im * (0.5d0 + (im * (im * t_1)))))
    t_3 = im * (im * 0.001388888888888889d0)
    if (im <= 320.0d0) then
        tmp = t_2
    else if (im <= 9.5d+30) then
        tmp = (1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * 0.041666666666666664d0))))) * ((t_1 * (im * t_0)) + (1.0d0 + (0.5d0 * (im * im))))
    else if (im <= 4.6d+51) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (((im * im) * (7.233796296296296d-5 + (t_0 * (t_0 * 2.6791838134430728d-9)))) / (0.001736111111111111d0 + (t_3 * (t_3 - 0.041666666666666664d0)))))))
    else
        tmp = t_2 * (1.0d0 + (re * (re * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * im);
	double t_1 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double t_2 = 1.0 + (im * (im * (0.5 + (im * (im * t_1)))));
	double t_3 = im * (im * 0.001388888888888889);
	double tmp;
	if (im <= 320.0) {
		tmp = t_2;
	} else if (im <= 9.5e+30) {
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_1 * (im * t_0)) + (1.0 + (0.5 * (im * im))));
	} else if (im <= 4.6e+51) {
		tmp = 1.0 + (im * (im * (0.5 + (((im * im) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_3 * (t_3 - 0.041666666666666664)))))));
	} else {
		tmp = t_2 * (1.0 + (re * (re * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * im)
	t_1 = 0.041666666666666664 + ((im * im) * 0.001388888888888889)
	t_2 = 1.0 + (im * (im * (0.5 + (im * (im * t_1)))))
	t_3 = im * (im * 0.001388888888888889)
	tmp = 0
	if im <= 320.0:
		tmp = t_2
	elif im <= 9.5e+30:
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_1 * (im * t_0)) + (1.0 + (0.5 * (im * im))))
	elif im <= 4.6e+51:
		tmp = 1.0 + (im * (im * (0.5 + (((im * im) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_3 * (t_3 - 0.041666666666666664)))))))
	else:
		tmp = t_2 * (1.0 + (re * (re * -0.5)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * im))
	t_1 = Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))
	t_2 = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * t_1))))))
	t_3 = Float64(im * Float64(im * 0.001388888888888889))
	tmp = 0.0
	if (im <= 320.0)
		tmp = t_2;
	elseif (im <= 9.5e+30)
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * 0.041666666666666664))))) * Float64(Float64(t_1 * Float64(im * t_0)) + Float64(1.0 + Float64(0.5 * Float64(im * im)))));
	elseif (im <= 4.6e+51)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(Float64(im * im) * Float64(7.233796296296296e-5 + Float64(t_0 * Float64(t_0 * 2.6791838134430728e-9)))) / Float64(0.001736111111111111 + Float64(t_3 * Float64(t_3 - 0.041666666666666664))))))));
	else
		tmp = Float64(t_2 * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * im);
	t_1 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	t_2 = 1.0 + (im * (im * (0.5 + (im * (im * t_1)))));
	t_3 = im * (im * 0.001388888888888889);
	tmp = 0.0;
	if (im <= 320.0)
		tmp = t_2;
	elseif (im <= 9.5e+30)
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_1 * (im * t_0)) + (1.0 + (0.5 * (im * im))));
	elseif (im <= 4.6e+51)
		tmp = 1.0 + (im * (im * (0.5 + (((im * im) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_3 * (t_3 - 0.041666666666666664)))))));
	else
		tmp = t_2 * (1.0 + (re * (re * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 320.0], t$95$2, If[LessEqual[im, 9.5e+30], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(re * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.6e+51], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(N[(im * im), $MachinePrecision] * N[(7.233796296296296e-5 + N[(t$95$0 * N[(t$95$0 * 2.6791838134430728e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.001736111111111111 + N[(t$95$3 * N[(t$95$3 - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 320

   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 93.2%

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

   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 59.7%

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

   if 320 < im < 9.5000000000000003e30

   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 4.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 51.5%

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

   8. Simplified 51.5%

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

   if 9.5000000000000003e30 < im < 4.6000000000000001e51

   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 6.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   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 5.8%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

      2. flip3-+ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \frac{{\frac{1}{24}}^{3} + {\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)}^{3}}{\color{blue}{\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right)}}\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\left(im \cdot im\right) \cdot \left({\frac{1}{24}}^{3} + {\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)}^{3}\right)}{\color{blue}{\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right)}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\left(im \cdot im\right) \cdot \left({\frac{1}{24}}^{3} + {\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)}^{3}\right)\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right)\right)}\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 58.7%

       \[\leadsto 1 + im \cdot \left(im \cdot \left(0.5 + \color{blue}{\frac{\left(im \cdot im\right) \cdot \left(7.233796296296296 \cdot 10^{-5} + \left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right)}{0.001736111111111111 + \left(im \cdot \left(im \cdot 0.001388888888888889\right)\right) \cdot \left(im \cdot \left(im \cdot 0.001388888888888889\right) - 0.041666666666666664\right)}}\right)\right) \]

   if 4.6000000000000001e51 < 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 100.0%

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

   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 75.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 320:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\right) \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \frac{\left(im \cdot im\right) \cdot \left(7.233796296296296 \cdot 10^{-5} + \left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right)}{0.001736111111111111 + \left(im \cdot \left(im \cdot 0.001388888888888889\right)\right) \cdot \left(im \cdot \left(im \cdot 0.001388888888888889\right) - 0.041666666666666664\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\\ \mathbf{if}\;im \leq 360:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot t\_0\right)\right)\right)\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\right) \cdot \left(t\_0 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))
   (if (<= im 360.0)
     (+ 1.0 (* im (* im (+ 0.5 (* im (* im t_0))))))
     (if (<= im 5.2e+134)
       (*
        (+ 1.0 (* (* re re) (+ -0.5 (* re (* re 0.041666666666666664)))))
        (+ (* t_0 (* im (* im (* im im)))) (+ 1.0 (* 0.5 (* im im)))))
       (*
        (* (* im im) (* im im))
        (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double tmp;
	if (im <= 360.0) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * t_0)))));
	} else if (im <= 5.2e+134) {
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_0 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	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 <= 360.0d0) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (im * (im * t_0)))))
    else if (im <= 5.2d+134) then
        tmp = (1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * 0.041666666666666664d0))))) * ((t_0 * (im * (im * (im * im)))) + (1.0d0 + (0.5d0 * (im * im))))
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    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 <= 360.0) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * t_0)))));
	} else if (im <= 5.2e+134) {
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_0 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889)
	tmp = 0
	if im <= 360.0:
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * t_0)))))
	elif im <= 5.2e+134:
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_0 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))))
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))
	tmp = 0.0
	if (im <= 360.0)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * t_0))))));
	elseif (im <= 5.2e+134)
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * 0.041666666666666664))))) * Float64(Float64(t_0 * Float64(im * Float64(im * Float64(im * im)))) + Float64(1.0 + Float64(0.5 * Float64(im * im)))));
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	tmp = 0.0;
	if (im <= 360.0)
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * t_0)))));
	elseif (im <= 5.2e+134)
		tmp = (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))) * ((t_0 * (im * (im * (im * im)))) + (1.0 + (0.5 * (im * im))));
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	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, 360.0], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.2e+134], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(re * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 360

   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 93.2%

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

   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 59.7%

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

   if 360 < im < 5.2000000000000003e134

   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 58.2%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 51.4%

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

   8. Simplified 51.4%

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

   if 5.2000000000000003e134 < 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 im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot \left({im}^{4} \cdot {re}^{2}\right) + \frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{24}} \cdot {im}^{4} \]

       2. +-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right)} \cdot {re}^{2} \]

       4. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \frac{-1}{48} \cdot \color{blue}{\left({im}^{4} \cdot {re}^{2}\right)} \]

       5. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \left({im}^{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{48}} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + {im}^{4} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)} \]

       7. distribute-lft-out N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       10. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{48}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{48}\right)\right)\right) \]

       19. *-lowering-*.f64 83.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{48}\right)\right)\right) \]

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.9% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{if}\;im \leq 420:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.18 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (*
           im
           (*
            im
            (+
             0.5
             (*
              im
              (*
               im
               (+
                0.041666666666666664
                (* (* im im) 0.001388888888888889))))))))))
   (if (<= im 420.0)
     t_0
     (if (<= im 1.18e+51)
       (*
        (+ 1.0 (* (* im im) (+ 0.5 (* 0.041666666666666664 (* im im)))))
        (+ 1.0 (* (* re re) (+ -0.5 (* re (* re 0.041666666666666664))))))
       (* t_0 (+ 1.0 (* re (* re -0.5))))))))

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 (im <= 420.0) {
		tmp = t_0;
	} else if (im <= 1.18e+51) {
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664)))));
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (im * (im * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))
    if (im <= 420.0d0) then
        tmp = t_0
    else if (im <= 1.18d+51) then
        tmp = (1.0d0 + ((im * im) * (0.5d0 + (0.041666666666666664d0 * (im * im))))) * (1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * 0.041666666666666664d0)))))
    else
        tmp = t_0 * (1.0d0 + (re * (re * (-0.5d0))))
    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 (im <= 420.0) {
		tmp = t_0;
	} else if (im <= 1.18e+51) {
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664)))));
	} else {
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	}
	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 im <= 420.0:
		tmp = t_0
	elif im <= 1.18e+51:
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664)))))
	else:
		tmp = t_0 * (1.0 + (re * (re * -0.5)))
	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 (im <= 420.0)
		tmp = t_0;
	elseif (im <= 1.18e+51)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))) * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * 0.041666666666666664))))));
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	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 (im <= 420.0)
		tmp = t_0;
	elseif (im <= 1.18e+51)
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664)))));
	else
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	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[im, 420.0], t$95$0, If[LessEqual[im, 1.18e+51], 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[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(re * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 420

   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 93.2%

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

   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 59.7%

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

   if 420 < im < 1.18e51

   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 4.2%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 35.7%

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

   8. Simplified 35.7%

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

   if 1.18e51 < 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 100.0%

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

   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 75.5%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.18 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.9% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{if}\;im \leq 550:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (*
           im
           (*
            im
            (+
             0.5
             (*
              im
              (*
               im
               (+
                0.041666666666666664
                (* (* im im) 0.001388888888888889))))))))))
   (if (<= im 550.0)
     t_0
     (if (<= im 1.25e+51)
       (*
        (+ 1.0 (* (* im im) (+ 0.5 (* 0.041666666666666664 (* im im)))))
        (* 0.041666666666666664 (* (* re re) (* re re))))
       (* t_0 (+ 1.0 (* re (* re -0.5))))))))

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 (im <= 550.0) {
		tmp = t_0;
	} else if (im <= 1.25e+51) {
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (0.041666666666666664 * ((re * re) * (re * re)));
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (im * (im * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))
    if (im <= 550.0d0) then
        tmp = t_0
    else if (im <= 1.25d+51) then
        tmp = (1.0d0 + ((im * im) * (0.5d0 + (0.041666666666666664d0 * (im * im))))) * (0.041666666666666664d0 * ((re * re) * (re * re)))
    else
        tmp = t_0 * (1.0d0 + (re * (re * (-0.5d0))))
    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 (im <= 550.0) {
		tmp = t_0;
	} else if (im <= 1.25e+51) {
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (0.041666666666666664 * ((re * re) * (re * re)));
	} else {
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	}
	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 im <= 550.0:
		tmp = t_0
	elif im <= 1.25e+51:
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (0.041666666666666664 * ((re * re) * (re * re)))
	else:
		tmp = t_0 * (1.0 + (re * (re * -0.5)))
	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 (im <= 550.0)
		tmp = t_0;
	elseif (im <= 1.25e+51)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))) * Float64(0.041666666666666664 * Float64(Float64(re * re) * Float64(re * re))));
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	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 (im <= 550.0)
		tmp = t_0;
	elseif (im <= 1.25e+51)
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (0.041666666666666664 * ((re * re) * (re * re)));
	else
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	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[im, 550.0], t$95$0, If[LessEqual[im, 1.25e+51], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 * N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 550

   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 93.2%

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

   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 59.7%

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

   if 550 < im < 1.25e51

   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 4.2%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 35.7%

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

   8. Simplified 35.7%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

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

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

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

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

       14. metadata-eval N/A

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

       15. pow-sqr N/A

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

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

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

       17. unpow2 N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 34.6%

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

   11. Simplified 34.6%

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

   if 1.25e51 < 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 100.0%

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

   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 75.5%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\\ \mathbf{if}\;im \leq 550:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+123}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          im
          (* im (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))))
   (if (<= im 550.0)
     (+ 1.0 (* im (* im (+ 0.5 t_0))))
     (if (<= im 4.6e+51)
       (*
        (+ 1.0 (* (* im im) (+ 0.5 (* 0.041666666666666664 (* im im)))))
        (* 0.041666666666666664 (* (* re re) (* re re))))
       (if (<= im 4.1e+123)
         (+ 1.0 (* (* im im) t_0))
         (*
          (* (* im im) (* im im))
          (+ 0.041666666666666664 (* (* re re) -0.020833333333333332))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)));
	double tmp;
	if (im <= 550.0) {
		tmp = 1.0 + (im * (im * (0.5 + t_0)));
	} else if (im <= 4.6e+51) {
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (0.041666666666666664 * ((re * re) * (re * re)));
	} else if (im <= 4.1e+123) {
		tmp = 1.0 + ((im * im) * t_0);
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))
    if (im <= 550.0d0) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + t_0)))
    else if (im <= 4.6d+51) then
        tmp = (1.0d0 + ((im * im) * (0.5d0 + (0.041666666666666664d0 * (im * im))))) * (0.041666666666666664d0 * ((re * re) * (re * re)))
    else if (im <= 4.1d+123) then
        tmp = 1.0d0 + ((im * im) * t_0)
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)));
	double tmp;
	if (im <= 550.0) {
		tmp = 1.0 + (im * (im * (0.5 + t_0)));
	} else if (im <= 4.6e+51) {
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (0.041666666666666664 * ((re * re) * (re * re)));
	} else if (im <= 4.1e+123) {
		tmp = 1.0 + ((im * im) * t_0);
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))
	tmp = 0
	if im <= 550.0:
		tmp = 1.0 + (im * (im * (0.5 + t_0)))
	elif im <= 4.6e+51:
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (0.041666666666666664 * ((re * re) * (re * re)))
	elif im <= 4.1e+123:
		tmp = 1.0 + ((im * im) * t_0)
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))
	tmp = 0.0
	if (im <= 550.0)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + t_0))));
	elseif (im <= 4.6e+51)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))) * Float64(0.041666666666666664 * Float64(Float64(re * re) * Float64(re * re))));
	elseif (im <= 4.1e+123)
		tmp = Float64(1.0 + Float64(Float64(im * im) * t_0));
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)));
	tmp = 0.0;
	if (im <= 550.0)
		tmp = 1.0 + (im * (im * (0.5 + t_0)));
	elseif (im <= 4.6e+51)
		tmp = (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im))))) * (0.041666666666666664 * ((re * re) * (re * re)));
	elseif (im <= 4.1e+123)
		tmp = 1.0 + ((im * im) * t_0);
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 550.0], N[(1.0 + N[(im * N[(im * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.6e+51], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 * N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.1e+123], N[(1.0 + N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 550

   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 93.2%

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

   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 59.7%

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

   if 550 < im < 4.6000000000000001e51

   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 4.2%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 35.7%

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

   8. Simplified 35.7%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

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

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

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

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

       14. metadata-eval N/A

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

       15. pow-sqr N/A

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

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

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

       17. unpow2 N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 34.6%

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

   11. Simplified 34.6%

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

   if 4.6000000000000001e51 < im < 4.09999999999999989e123

   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 100.0%

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

   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 66.7%

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

   9. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 66.7%

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

   11. Simplified 66.7%

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

   12. Taylor expanded in im around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

   14. Simplified 66.7%

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

   if 4.09999999999999989e123 < 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 im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot \left({im}^{4} \cdot {re}^{2}\right) + \frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{24}} \cdot {im}^{4} \]

       2. +-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right)} \cdot {re}^{2} \]

       4. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \frac{-1}{48} \cdot \color{blue}{\left({im}^{4} \cdot {re}^{2}\right)} \]

       5. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \left({im}^{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{48}} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + {im}^{4} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)} \]

       7. distribute-lft-out N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       10. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{48}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{48}\right)\right)\right) \]

       19. *-lowering-*.f64 83.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{48}\right)\right)\right) \]

   11. Simplified 83.9%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+123}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\\ \mathbf{if}\;im \leq 320:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+123}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          im
          (* im (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))))
   (if (<= im 320.0)
     (+ 1.0 (* im (* im (+ 0.5 t_0))))
     (if (<= im 2.2e+51)
       (*
        (* 0.5 (* im im))
        (+ 1.0 (* re (* re (+ -0.5 (* 0.041666666666666664 (* re re)))))))
       (if (<= im 3.2e+123)
         (+ 1.0 (* (* im im) t_0))
         (*
          (* (* im im) (* im im))
          (+ 0.041666666666666664 (* (* re re) -0.020833333333333332))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)));
	double tmp;
	if (im <= 320.0) {
		tmp = 1.0 + (im * (im * (0.5 + t_0)));
	} else if (im <= 2.2e+51) {
		tmp = (0.5 * (im * im)) * (1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re))))));
	} else if (im <= 3.2e+123) {
		tmp = 1.0 + ((im * im) * t_0);
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))
    if (im <= 320.0d0) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + t_0)))
    else if (im <= 2.2d+51) then
        tmp = (0.5d0 * (im * im)) * (1.0d0 + (re * (re * ((-0.5d0) + (0.041666666666666664d0 * (re * re))))))
    else if (im <= 3.2d+123) then
        tmp = 1.0d0 + ((im * im) * t_0)
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)));
	double tmp;
	if (im <= 320.0) {
		tmp = 1.0 + (im * (im * (0.5 + t_0)));
	} else if (im <= 2.2e+51) {
		tmp = (0.5 * (im * im)) * (1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re))))));
	} else if (im <= 3.2e+123) {
		tmp = 1.0 + ((im * im) * t_0);
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))
	tmp = 0
	if im <= 320.0:
		tmp = 1.0 + (im * (im * (0.5 + t_0)))
	elif im <= 2.2e+51:
		tmp = (0.5 * (im * im)) * (1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re))))))
	elif im <= 3.2e+123:
		tmp = 1.0 + ((im * im) * t_0)
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))
	tmp = 0.0
	if (im <= 320.0)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + t_0))));
	elseif (im <= 2.2e+51)
		tmp = Float64(Float64(0.5 * Float64(im * im)) * Float64(1.0 + Float64(re * Float64(re * Float64(-0.5 + Float64(0.041666666666666664 * Float64(re * re)))))));
	elseif (im <= 3.2e+123)
		tmp = Float64(1.0 + Float64(Float64(im * im) * t_0));
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)));
	tmp = 0.0;
	if (im <= 320.0)
		tmp = 1.0 + (im * (im * (0.5 + t_0)));
	elseif (im <= 2.2e+51)
		tmp = (0.5 * (im * im)) * (1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re))))));
	elseif (im <= 3.2e+123)
		tmp = 1.0 + ((im * im) * t_0);
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 320.0], N[(1.0 + N[(im * N[(im * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.2e+51], N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(re * N[(-0.5 + N[(0.041666666666666664 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.2e+123], N[(1.0 + N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 320

   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 93.2%

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

   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 59.7%

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

   if 320 < im < 2.19999999999999992e51

   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 3.6%

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

   5. Simplified 3.6%

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

   6. Taylor expanded in im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 3.6%

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

   8. Simplified 3.6%

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

   9. Taylor expanded in re around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 29.2%

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

   11. Simplified 29.2%

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

   if 2.19999999999999992e51 < im < 3.20000000000000005e123

   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 100.0%

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

   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 66.7%

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

   9. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 66.7%

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

   11. Simplified 66.7%

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

   12. Taylor expanded in im around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

   14. Simplified 66.7%

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

   if 3.20000000000000005e123 < 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 im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot \left({im}^{4} \cdot {re}^{2}\right) + \frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{24}} \cdot {im}^{4} \]

       2. +-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right)} \cdot {re}^{2} \]

       4. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \frac{-1}{48} \cdot \color{blue}{\left({im}^{4} \cdot {re}^{2}\right)} \]

       5. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \left({im}^{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{48}} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + {im}^{4} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)} \]

       7. distribute-lft-out N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       10. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{48}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{48}\right)\right)\right) \]

       19. *-lowering-*.f64 83.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{48}\right)\right)\right) \]

   11. Simplified 83.9%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 320:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+123}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 360:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+122}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 360.0)
   (+ 1.0 (* im (* im (+ 0.5 (* im (* im 0.041666666666666664))))))
   (if (<= im 4.2e+51)
     (*
      (* 0.5 (* im im))
      (+ 1.0 (* re (* re (+ -0.5 (* 0.041666666666666664 (* re re)))))))
     (if (<= im 8.2e+122)
       (+
        1.0
        (*
         (* im im)
         (*
          im
          (* im (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))))
       (*
        (* (* im im) (* im im))
        (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 360.0) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} else if (im <= 4.2e+51) {
		tmp = (0.5 * (im * im)) * (1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re))))));
	} else if (im <= 8.2e+122) {
		tmp = 1.0 + ((im * im) * (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 360.0d0) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    else if (im <= 4.2d+51) then
        tmp = (0.5d0 * (im * im)) * (1.0d0 + (re * (re * ((-0.5d0) + (0.041666666666666664d0 * (re * re))))))
    else if (im <= 8.2d+122) then
        tmp = 1.0d0 + ((im * im) * (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 360.0) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} else if (im <= 4.2e+51) {
		tmp = (0.5 * (im * im)) * (1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re))))));
	} else if (im <= 8.2e+122) {
		tmp = 1.0 + ((im * im) * (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 360.0:
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))))
	elif im <= 4.2e+51:
		tmp = (0.5 * (im * im)) * (1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re))))))
	elif im <= 8.2e+122:
		tmp = 1.0 + ((im * im) * (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 360.0)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	elseif (im <= 4.2e+51)
		tmp = Float64(Float64(0.5 * Float64(im * im)) * Float64(1.0 + Float64(re * Float64(re * Float64(-0.5 + Float64(0.041666666666666664 * Float64(re * re)))))));
	elseif (im <= 8.2e+122)
		tmp = Float64(1.0 + Float64(Float64(im * im) * Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))));
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 360.0)
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	elseif (im <= 4.2e+51)
		tmp = (0.5 * (im * im)) * (1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re))))));
	elseif (im <= 8.2e+122)
		tmp = 1.0 + ((im * im) * (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))));
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 360.0], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e+51], N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(re * N[(-0.5 + N[(0.041666666666666664 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.2e+122], N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 360

   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 93.2%

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

   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 59.7%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 53.8%

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

   11. Simplified 53.8%

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

   if 360 < im < 4.2000000000000002e51

   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 3.6%

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

   5. Simplified 3.6%

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

   6. Taylor expanded in im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 3.6%

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

   8. Simplified 3.6%

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

   9. Taylor expanded in re around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 29.2%

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

   11. Simplified 29.2%

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

   if 4.2000000000000002e51 < im < 8.2000000000000004e122

   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 100.0%

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

   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 66.7%

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

   9. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 66.7%

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

   11. Simplified 66.7%

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

   12. Taylor expanded in im around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

   14. Simplified 66.7%

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

   if 8.2000000000000004e122 < 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 im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot \left({im}^{4} \cdot {re}^{2}\right) + \frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{24}} \cdot {im}^{4} \]

       2. +-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right)} \cdot {re}^{2} \]

       4. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \frac{-1}{48} \cdot \color{blue}{\left({im}^{4} \cdot {re}^{2}\right)} \]

       5. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \left({im}^{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{48}} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + {im}^{4} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)} \]

       7. distribute-lft-out N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       10. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{48}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{48}\right)\right)\right) \]

       19. *-lowering-*.f64 83.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{48}\right)\right)\right) \]

   11. Simplified 83.9%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 360:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+122}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 410:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;1 + re \cdot \left(re \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+123}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 410.0)
   (+ 1.0 (* im (* im (+ 0.5 (* im (* im 0.041666666666666664))))))
   (if (<= im 4.6e+51)
     (+ 1.0 (* re (* re (+ -0.5 (* 0.041666666666666664 (* re re))))))
     (if (<= im 1.4e+123)
       (+
        1.0
        (*
         (* im im)
         (*
          im
          (* im (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))))
       (*
        (* (* im im) (* im im))
        (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 410.0) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} else if (im <= 4.6e+51) {
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	} else if (im <= 1.4e+123) {
		tmp = 1.0 + ((im * im) * (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 410.0d0) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    else if (im <= 4.6d+51) then
        tmp = 1.0d0 + (re * (re * ((-0.5d0) + (0.041666666666666664d0 * (re * re)))))
    else if (im <= 1.4d+123) then
        tmp = 1.0d0 + ((im * im) * (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 410.0) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} else if (im <= 4.6e+51) {
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	} else if (im <= 1.4e+123) {
		tmp = 1.0 + ((im * im) * (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 410.0:
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))))
	elif im <= 4.6e+51:
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))))
	elif im <= 1.4e+123:
		tmp = 1.0 + ((im * im) * (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 410.0)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	elseif (im <= 4.6e+51)
		tmp = Float64(1.0 + Float64(re * Float64(re * Float64(-0.5 + Float64(0.041666666666666664 * Float64(re * re))))));
	elseif (im <= 1.4e+123)
		tmp = Float64(1.0 + Float64(Float64(im * im) * Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))));
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 410.0)
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	elseif (im <= 4.6e+51)
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	elseif (im <= 1.4e+123)
		tmp = 1.0 + ((im * im) * (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))));
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 410.0], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.6e+51], N[(1.0 + N[(re * N[(re * N[(-0.5 + N[(0.041666666666666664 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+123], N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 410

   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 93.2%

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

   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 59.7%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 53.8%

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

   11. Simplified 53.8%

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

   if 410 < im < 4.6000000000000001e51

   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 4.2%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 35.7%

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

   8. Simplified 35.7%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   10. 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. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 28.8%

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

   11. Simplified 28.8%

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

   if 4.6000000000000001e51 < im < 1.40000000000000006e123

   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 100.0%

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

   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 66.7%

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

   9. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 66.7%

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

   11. Simplified 66.7%

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

   12. Taylor expanded in im around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

   14. Simplified 66.7%

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

   if 1.40000000000000006e123 < 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 im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot \left({im}^{4} \cdot {re}^{2}\right) + \frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{24}} \cdot {im}^{4} \]

       2. +-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right)} \cdot {re}^{2} \]

       4. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \frac{-1}{48} \cdot \color{blue}{\left({im}^{4} \cdot {re}^{2}\right)} \]

       5. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \left({im}^{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{48}} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + {im}^{4} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)} \]

       7. distribute-lft-out N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       10. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{48}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{48}\right)\right)\right) \]

       19. *-lowering-*.f64 83.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{48}\right)\right)\right) \]

   11. Simplified 83.9%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 20: 58.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 950:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+51}:\\ \;\;\;\;1 + re \cdot \left(re \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+123}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 950.0)
   (+ 1.0 (* im (* im (+ 0.5 (* im (* im 0.041666666666666664))))))
   (if (<= im 3e+51)
     (+ 1.0 (* re (* re (+ -0.5 (* 0.041666666666666664 (* re re))))))
     (if (<= im 5e+123)
       (+ 1.0 (* im (* im (* im (* im (* (* im im) 0.001388888888888889))))))
       (*
        (* (* im im) (* im im))
        (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 950.0) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} else if (im <= 3e+51) {
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	} else if (im <= 5e+123) {
		tmp = 1.0 + (im * (im * (im * (im * ((im * im) * 0.001388888888888889)))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 950.0d0) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    else if (im <= 3d+51) then
        tmp = 1.0d0 + (re * (re * ((-0.5d0) + (0.041666666666666664d0 * (re * re)))))
    else if (im <= 5d+123) then
        tmp = 1.0d0 + (im * (im * (im * (im * ((im * im) * 0.001388888888888889d0)))))
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 950.0) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} else if (im <= 3e+51) {
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	} else if (im <= 5e+123) {
		tmp = 1.0 + (im * (im * (im * (im * ((im * im) * 0.001388888888888889)))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 950.0:
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))))
	elif im <= 3e+51:
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))))
	elif im <= 5e+123:
		tmp = 1.0 + (im * (im * (im * (im * ((im * im) * 0.001388888888888889)))))
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 950.0)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	elseif (im <= 3e+51)
		tmp = Float64(1.0 + Float64(re * Float64(re * Float64(-0.5 + Float64(0.041666666666666664 * Float64(re * re))))));
	elseif (im <= 5e+123)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(im * Float64(im * Float64(Float64(im * im) * 0.001388888888888889))))));
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 950.0)
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	elseif (im <= 3e+51)
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	elseif (im <= 5e+123)
		tmp = 1.0 + (im * (im * (im * (im * ((im * im) * 0.001388888888888889)))));
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 950.0], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3e+51], N[(1.0 + N[(re * N[(re * N[(-0.5 + N[(0.041666666666666664 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e+123], N[(1.0 + N[(im * N[(im * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 950

   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 93.2%

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

   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 59.7%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 53.8%

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

   11. Simplified 53.8%

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

   if 950 < im < 3e51

   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 4.2%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 35.7%

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

   8. Simplified 35.7%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   10. 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. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {re}^{2}}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2}\right)}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 28.8%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 28.8%

       \[\leadsto \color{blue}{1 + re \cdot \left(re \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)} \]

   if 3e51 < im < 4.99999999999999974e123

   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 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   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 66.7%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{720} \cdot {im}^{4}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{720} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{720}} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 66.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 66.7%

       \[\leadsto 1 + im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)}\right) \]

   if 4.99999999999999974e123 < 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 im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\cos re} \]

      2. *-commutative N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \cos re \cdot \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \cos re \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \cos re \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot \left({im}^{4} \cdot {re}^{2}\right) + \frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{24}} \cdot {im}^{4} \]

       2. +-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right)} \cdot {re}^{2} \]

       4. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \frac{-1}{48} \cdot \color{blue}{\left({im}^{4} \cdot {re}^{2}\right)} \]

       5. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \left({im}^{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{48}} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + {im}^{4} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)} \]

       7. distribute-lft-out N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       10. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       16. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)}\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{48}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{48}\right)\right)\right) \]

       19. *-lowering-*.f64 83.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{48}\right)\right)\right) \]

   11. Simplified 83.9%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 21: 56.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{if}\;im \leq 400:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+75}:\\ \;\;\;\;1 + re \cdot \left(re \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+139}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* im (* im (+ 0.5 (* im (* im 0.041666666666666664))))))))
   (if (<= im 400.0)
     t_0
     (if (<= im 6e+75)
       (+ 1.0 (* re (* re (+ -0.5 (* 0.041666666666666664 (* re re))))))
       (if (<= im 1.02e+139) t_0 (* (* im im) (+ 0.5 (* (* re re) -0.25))))))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	double tmp;
	if (im <= 400.0) {
		tmp = t_0;
	} else if (im <= 6e+75) {
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	} else if (im <= 1.02e+139) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (im * (im * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    if (im <= 400.0d0) then
        tmp = t_0
    else if (im <= 6d+75) then
        tmp = 1.0d0 + (re * (re * ((-0.5d0) + (0.041666666666666664d0 * (re * re)))))
    else if (im <= 1.02d+139) then
        tmp = t_0
    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 t_0 = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	double tmp;
	if (im <= 400.0) {
		tmp = t_0;
	} else if (im <= 6e+75) {
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	} else if (im <= 1.02e+139) {
		tmp = t_0;
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))))
	tmp = 0
	if im <= 400.0:
		tmp = t_0
	elif im <= 6e+75:
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))))
	elif im <= 1.02e+139:
		tmp = t_0
	else:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))))
	tmp = 0.0
	if (im <= 400.0)
		tmp = t_0;
	elseif (im <= 6e+75)
		tmp = Float64(1.0 + Float64(re * Float64(re * Float64(-0.5 + Float64(0.041666666666666664 * Float64(re * re))))));
	elseif (im <= 1.02e+139)
		tmp = t_0;
	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)
	t_0 = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	tmp = 0.0;
	if (im <= 400.0)
		tmp = t_0;
	elseif (im <= 6e+75)
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	elseif (im <= 1.02e+139)
		tmp = t_0;
	else
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	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 * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 400.0], t$95$0, If[LessEqual[im, 6e+75], N[(1.0 + N[(re * N[(re * N[(-0.5 + N[(0.041666666666666664 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.02e+139], t$95$0, N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 400 or 6e75 < im < 1.02e139

   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 93.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   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 60.6%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot im\right)}\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 54.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 54.6%

       \[\leadsto 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \color{blue}{\left(im \cdot 0.041666666666666664\right)}\right)\right) \]

   if 400 < im < 6e75

   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 5.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {re}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\color{blue}{im}, im\right)\right)\right)\right), 1\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(re \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      13. *-lowering-*.f64 36.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), 1\right)\right) \]

   8. Simplified 36.6%

      \[\leadsto \color{blue}{\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   10. 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. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{24} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {re}^{2}}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2}\right)}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 27.9%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 27.9%

       \[\leadsto \color{blue}{1 + re \cdot \left(re \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)} \]

   if 1.02e139 < 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 96.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 96.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\cos re} \]

      2. *-commutative N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

      7. *-lowering-*.f64 96.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   8. Simplified 96.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({im}^{2} \cdot {re}^{2}\right) + \frac{1}{2} \cdot {im}^{2}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{4} \cdot {im}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{2}} \cdot {im}^{2} \]

       2. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot {im}^{2} + \color{blue}{\left(\frac{-1}{4} \cdot {im}^{2}\right) \cdot {re}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + \color{blue}{\left(\frac{-1}{4} \cdot {im}^{2}\right)} \cdot {re}^{2} \]

       4. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{\left({im}^{2} \cdot {re}^{2}\right)} \]

       5. *-commutative N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + \left({im}^{2} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{4}} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + {im}^{2} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{4}\right)} \]

       7. distribute-lft-out N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{4}\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{4}\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \frac{-1}{4}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \frac{-1}{4}\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({re}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{4}\right)\right)\right) \]

       14. *-lowering-*.f64 82.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right)\right) \]

   11. Simplified 82.8%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 51.5% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{if}\;im \leq 760000:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+139}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (+ 0.5 (* (* re re) -0.25)))))
   (if (<= im 760000.0)
     (+ 1.0 (* 0.5 (* im im)))
     (if (<= im 2.15e+77)
       t_0
       (if (<= im 1.02e+139)
         (* im (* im (* 0.041666666666666664 (* im im))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (0.5 + ((re * re) * -0.25));
	double tmp;
	if (im <= 760000.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if (im <= 2.15e+77) {
		tmp = t_0;
	} else if (im <= 1.02e+139) {
		tmp = im * (im * (0.041666666666666664 * (im * im)));
	} 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 = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    if (im <= 760000.0d0) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else if (im <= 2.15d+77) then
        tmp = t_0
    else if (im <= 1.02d+139) then
        tmp = im * (im * (0.041666666666666664d0 * (im * im)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (0.5 + ((re * re) * -0.25));
	double tmp;
	if (im <= 760000.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if (im <= 2.15e+77) {
		tmp = t_0;
	} else if (im <= 1.02e+139) {
		tmp = im * (im * (0.041666666666666664 * (im * im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (0.5 + ((re * re) * -0.25))
	tmp = 0
	if im <= 760000.0:
		tmp = 1.0 + (0.5 * (im * im))
	elif im <= 2.15e+77:
		tmp = t_0
	elif im <= 1.02e+139:
		tmp = im * (im * (0.041666666666666664 * (im * im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)))
	tmp = 0.0
	if (im <= 760000.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	elseif (im <= 2.15e+77)
		tmp = t_0;
	elseif (im <= 1.02e+139)
		tmp = Float64(im * Float64(im * Float64(0.041666666666666664 * Float64(im * im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (0.5 + ((re * re) * -0.25));
	tmp = 0.0;
	if (im <= 760000.0)
		tmp = 1.0 + (0.5 * (im * im));
	elseif (im <= 2.15e+77)
		tmp = t_0;
	elseif (im <= 1.02e+139)
		tmp = im * (im * (0.041666666666666664 * (im * im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 760000.0], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.15e+77], t$95$0, If[LessEqual[im, 1.02e+139], N[(im * N[(im * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 7.6e5

   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 79.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 79.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   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 47.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   8. Simplified 47.9%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   if 7.6e5 < im < 2.14999999999999996e77 or 1.02e139 < 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 54.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 54.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\cos re} \]

      2. *-commutative N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

      7. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({im}^{2} \cdot {re}^{2}\right) + \frac{1}{2} \cdot {im}^{2}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{4} \cdot {im}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{2}} \cdot {im}^{2} \]

       2. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot {im}^{2} + \color{blue}{\left(\frac{-1}{4} \cdot {im}^{2}\right) \cdot {re}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + \color{blue}{\left(\frac{-1}{4} \cdot {im}^{2}\right)} \cdot {re}^{2} \]

       4. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{\left({im}^{2} \cdot {re}^{2}\right)} \]

       5. *-commutative N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + \left({im}^{2} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{4}} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + {im}^{2} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{4}\right)} \]

       7. distribute-lft-out N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{4}\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{4}\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \frac{-1}{4}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \frac{-1}{4}\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({re}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{4}\right)\right)\right) \]

       14. *-lowering-*.f64 51.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right)\right) \]

   11. Simplified 51.9%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)} \]

   if 2.14999999999999996e77 < im < 1.02e139

   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 im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\cos re} \]

      2. *-commutative N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \cos re \cdot \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \cos re \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \cos re \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{im}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot im\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right) \]

       14. *-lowering-*.f64 72.7%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right) \]

   11. Simplified 72.7%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 760000:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+77}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+139}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 57.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 380:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+51}:\\ \;\;\;\;1 + re \cdot \left(re \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 380.0)
   (+ 1.0 (* im (* im (+ 0.5 (* im (* im 0.041666666666666664))))))
   (if (<= im 1.05e+51)
     (+ 1.0 (* re (* re (+ -0.5 (* 0.041666666666666664 (* re re))))))
     (*
      (* (* im im) (* im im))
      (+ 0.041666666666666664 (* (* re re) -0.020833333333333332))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 380.0) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} else if (im <= 1.05e+51) {
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 380.0d0) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    else if (im <= 1.05d+51) then
        tmp = 1.0d0 + (re * (re * ((-0.5d0) + (0.041666666666666664d0 * (re * re)))))
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 380.0) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} else if (im <= 1.05e+51) {
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 380.0:
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))))
	elif im <= 1.05e+51:
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))))
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 380.0)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	elseif (im <= 1.05e+51)
		tmp = Float64(1.0 + Float64(re * Float64(re * Float64(-0.5 + Float64(0.041666666666666664 * Float64(re * re))))));
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 380.0)
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	elseif (im <= 1.05e+51)
		tmp = 1.0 + (re * (re * (-0.5 + (0.041666666666666664 * (re * re)))));
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 380.0], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.05e+51], N[(1.0 + N[(re * N[(re * N[(-0.5 + N[(0.041666666666666664 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 380

   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 93.2%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   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 59.7%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot im\right)}\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 53.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 53.8%

       \[\leadsto 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \color{blue}{\left(im \cdot 0.041666666666666664\right)}\right)\right) \]

   if 380 < im < 1.0500000000000001e51

   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 4.2%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {re}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\color{blue}{im}, im\right)\right)\right)\right), 1\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \left(re \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      13. *-lowering-*.f64 35.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), 1\right)\right) \]

   8. Simplified 35.7%

      \[\leadsto \color{blue}{\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   10. 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. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{24} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {re}^{2}}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2}\right)}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 28.8%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 28.8%

       \[\leadsto \color{blue}{1 + re \cdot \left(re \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)} \]

   if 1.0500000000000001e51 < 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 83.2%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\cos re} \]

      2. *-commutative N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \cos re \cdot \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \cos re \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \cos re \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 83.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   8. Simplified 83.2%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot \left({im}^{4} \cdot {re}^{2}\right) + \frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{24}} \cdot {im}^{4} \]

       2. +-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{4} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right) \cdot {re}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{48} \cdot {im}^{4}\right)} \cdot {re}^{2} \]

       4. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \frac{-1}{48} \cdot \color{blue}{\left({im}^{4} \cdot {re}^{2}\right)} \]

       5. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + \left({im}^{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{48}} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} + {im}^{4} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)} \]

       7. distribute-lft-out N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       10. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + {re}^{2} \cdot \frac{-1}{48}\right)\right) \]

       16. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2} \cdot \frac{-1}{48}\right)}\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{48}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{48}\right)\right)\right) \]

       19. *-lowering-*.f64 71.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{48}\right)\right)\right) \]

   11. Simplified 71.8%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 55.8% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.1 \cdot 10^{+86}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.1e+86)
   (+ 1.0 (* im (* im (+ 0.5 (* im (* im 0.041666666666666664))))))
   (* (* im im) (+ 0.5 (* (* re re) -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.1e+86) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} 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 (re <= 3.1d+86) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    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 (re <= 3.1e+86) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.1e+86:
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))))
	else:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.1e+86)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	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 (re <= 3.1e+86)
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	else
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.1e+86], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.1000000000000002e86

   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 89.4%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   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 64.6%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot im\right)}\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 58.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 58.4%

       \[\leadsto 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \color{blue}{\left(im \cdot 0.041666666666666664\right)}\right)\right) \]

   if 3.1000000000000002e86 < re

   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 71.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 71.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\cos re} \]

      2. *-commutative N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot {im}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

      7. *-lowering-*.f64 17.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   8. Simplified 17.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({im}^{2} \cdot {re}^{2}\right) + \frac{1}{2} \cdot {im}^{2}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{4} \cdot {im}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{2}} \cdot {im}^{2} \]

       2. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot {im}^{2} + \color{blue}{\left(\frac{-1}{4} \cdot {im}^{2}\right) \cdot {re}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + \color{blue}{\left(\frac{-1}{4} \cdot {im}^{2}\right)} \cdot {re}^{2} \]

       4. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{\left({im}^{2} \cdot {re}^{2}\right)} \]

       5. *-commutative N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + \left({im}^{2} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{4}} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{2} \cdot \frac{1}{2} + {im}^{2} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{4}\right)} \]

       7. distribute-lft-out N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{4}\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{4}\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \frac{-1}{4}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \frac{-1}{4}\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({re}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{4}\right)\right)\right) \]

       14. *-lowering-*.f64 17.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right)\right) \]

   11. Simplified 17.6%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 51.2% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 235.0)
   (+ 1.0 (* 0.5 (* im im)))
   (* im (* im (* 0.041666666666666664 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 235.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = im * (im * (0.041666666666666664 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 235.0d0) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = im * (im * (0.041666666666666664d0 * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 235.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = im * (im * (0.041666666666666664 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 235.0:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = im * (im * (0.041666666666666664 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 235.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(im * Float64(im * Float64(0.041666666666666664 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 235.0)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = im * (im * (0.041666666666666664 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 235.0], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 235

   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 79.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 79.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   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 47.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   8. Simplified 47.9%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   if 235 < 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 64.7%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\cos re} \]

      2. *-commutative N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \cos re \cdot \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \cos re \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \cos re \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 64.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   8. Simplified 64.7%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{im}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot im\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right) \]

       14. *-lowering-*.f64 45.4%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right) \]

   11. Simplified 45.4%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 47.7% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.15 \cdot 10^{+139}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.15e+139) (+ 1.0 (* 0.5 (* im im))) (+ 1.0 (* re (* re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.15e+139) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 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 (re <= 2.15d+139) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = 1.0d0 + (re * (re * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.15e+139) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 1.0 + (re * (re * -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.15e+139:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = 1.0 + (re * (re * -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.15e+139)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(1.0 + Float64(re * Float64(re * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.15e+139)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = 1.0 + (re * (re * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.15e+139], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.1499999999999999e139

   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 70.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 70.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   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 48.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   8. Simplified 48.2%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   if 2.1499999999999999e139 < re

   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 61.2%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 61.2%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \frac{-1}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \frac{-1}{2}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \frac{-1}{2}\right)}\right)\right) \]

      6. *-lowering-*.f64 18.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   8. Simplified 18.0%

      \[\leadsto \color{blue}{1 + re \cdot \left(re \cdot -0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 37.9% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 235.0) 1.0 (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 235.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 235.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 235.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 235.0:
		tmp = 1.0
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 235.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 235.0)
		tmp = 1.0;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 235.0], 1.0, N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 235

   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 62.9%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 62.9%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 35.1%

      \[\leadsto \color{blue}{1} \]

   if 235 < 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 77.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   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 52.7%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 33.1%

       \[\leadsto 1 + im \cdot \left(im \cdot \color{blue}{0.5}\right) \]

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {im}^{2}} \]
   13. 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 33.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

   14. Simplified 33.1%

       \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 47.1% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot \left(im \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ 1.0 (* 0.5 (* im im))))

C

double code(double re, double im) {
	return 1.0 + (0.5 * (im * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + (0.5d0 * (im * im))
end function

Java

public static double code(double re, double im) {
	return 1.0 + (0.5 * (im * im));
}

Python

def code(re, im):
	return 1.0 + (0.5 * (im * im))

Julia

function code(re, im)
	return Float64(1.0 + Float64(0.5 * Float64(im * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + (0.5 * (im * im));
end

Wolfram

code[re_, im_] := N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. 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 71.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

5. Simplified 71.1%

   \[\leadsto \color{blue}{\cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

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 44.2%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

8. Simplified 44.2%

   \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]
9. Add Preprocessing

Alternative 29: 28.6% 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 47.9%

      \[\leadsto \mathsf{cos.f64}\left(re\right) \]

5. Simplified 47.9%

   \[\leadsto \color{blue}{\cos re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 26.9%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))