math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.2s

Alternatives: 19

Speedup: 1.5×

• 48.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. exp-neg N/A

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

   12. associate-*l/ N/A

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

   13. metadata-eval N/A

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

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

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

   15. exp-lowering-exp.f64 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

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

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

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

   3. div-inv N/A

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

   4. exp-neg N/A

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

   5. distribute-lft-out N/A

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

   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%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
7. 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%

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

8. Applied egg-rr 100.0%

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

Alternative 2: 87.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.031:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.031)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 7.2e+51)
     (*
      (cosh im)
      (+ 1.0 (* (* re re) (+ -0.5 (* (* re re) 0.041666666666666664)))))
     (*
      (cos re)
      (+
       1.0
       (*
        (* im im)
        (+
         0.5
         (*
          im
          (*
           im
           (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.031) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 7.2e+51) {
		tmp = cosh(im) * (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))));
	} else {
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.031d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 7.2d+51) then
        tmp = cosh(im) * (1.0d0 + ((re * re) * ((-0.5d0) + ((re * re) * 0.041666666666666664d0))))
    else
        tmp = cos(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.031) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 7.2e+51) {
		tmp = Math.cosh(im) * (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))));
	} else {
		tmp = Math.cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.031:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 7.2e+51:
		tmp = math.cosh(im) * (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))))
	else:
		tmp = math.cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.031)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 7.2e+51)
		tmp = Float64(cosh(im) * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(Float64(re * re) * 0.041666666666666664)))));
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.031)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 7.2e+51)
		tmp = cosh(im) * (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))));
	else
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.031], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e+51], N[(N[Cosh[im], $MachinePrecision] * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.031

   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

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 87.4%

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

   5. Simplified 87.4%

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

   if 0.031 < im < 7.20000000000000022e51

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative N/A

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

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

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

      3. div-inv N/A

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

      4. exp-neg N/A

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

      5. distribute-lft-out N/A

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

      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%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   7. 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%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in re around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-commutative N/A

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

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

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 92.5%

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

   11. Simplified 92.5%

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

   if 7.20000000000000022e51 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.031:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0225:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0225)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 2.6e+77)
     (*
      (cosh im)
      (+ 1.0 (* (* re re) (+ -0.5 (* (* re re) 0.041666666666666664)))))
     (*
      (cos re)
      (+ 1.0 (* (* im im) (+ 0.5 (* im (* im 0.041666666666666664)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0225) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 2.6e+77) {
		tmp = cosh(im) * (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))));
	} else {
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0225d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 2.6d+77) then
        tmp = cosh(im) * (1.0d0 + ((re * re) * ((-0.5d0) + ((re * re) * 0.041666666666666664d0))))
    else
        tmp = cos(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0225) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 2.6e+77) {
		tmp = Math.cosh(im) * (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))));
	} else {
		tmp = Math.cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0225:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 2.6e+77:
		tmp = math.cosh(im) * (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))))
	else:
		tmp = math.cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0225)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 2.6e+77)
		tmp = Float64(cosh(im) * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(Float64(re * re) * 0.041666666666666664)))));
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0225)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 2.6e+77)
		tmp = cosh(im) * (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664))));
	else
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0225], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(N[Cosh[im], $MachinePrecision] * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.022499999999999999

   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

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 87.4%

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

   5. Simplified 87.4%

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

   if 0.022499999999999999 < im < 2.6000000000000002e77

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative N/A

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

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

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

      3. div-inv N/A

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

      4. exp-neg N/A

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

      5. distribute-lft-out N/A

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

      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%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   7. 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%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in re around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-commutative N/A

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

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

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 86.0%

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

   11. Simplified 86.0%

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

   if 2.6000000000000002e77 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-out N/A

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

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

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

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

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

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

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

   7. Simplified 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0225:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0275:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0275)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 2.6e+77)
     (cosh im)
     (*
      (cos re)
      (+ 1.0 (* (* im im) (+ 0.5 (* im (* im 0.041666666666666664)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0275) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 2.6e+77) {
		tmp = cosh(im);
	} else {
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0275d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 2.6d+77) then
        tmp = cosh(im)
    else
        tmp = cos(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0275) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 2.6e+77) {
		tmp = Math.cosh(im);
	} else {
		tmp = Math.cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0275:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 2.6e+77:
		tmp = math.cosh(im)
	else:
		tmp = math.cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0275)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 2.6e+77)
		tmp = cosh(im);
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0275)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 2.6e+77)
		tmp = cosh(im);
	else
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0275], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], N[Cosh[im], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0275000000000000001

   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

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 87.4%

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

   5. Simplified 87.4%

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

   if 0.0275000000000000001 < im < 2.6000000000000002e77

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative N/A

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

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

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

      3. div-inv N/A

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

      4. exp-neg N/A

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

      5. distribute-lft-out N/A

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

      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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 85.6%

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

       1. *-lft-identity N/A

          \[\leadsto \cosh im \cdot 1 \]

       2. *-rgt-identity N/A

          \[\leadsto \cosh im \]

       3. cosh-lowering-cosh.f64 85.6%

          \[\leadsto \mathsf{cosh.f64}\left(im\right) \]

   11. Applied egg-rr 85.6%

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

   if 2.6000000000000002e77 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-out N/A

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

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

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

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

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

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

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

   7. Simplified 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0275:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{if}\;im \leq 0.0145:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ 2.0 (* im im)))))
   (if (<= im 0.0145) t_0 (if (<= im 1.35e+154) (cosh im) t_0))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (2.0 + (im * im));
	double tmp;
	if (im <= 0.0145) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = cosh(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 = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    if (im <= 0.0145d0) then
        tmp = t_0
    else if (im <= 1.35d+154) then
        tmp = cosh(im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	double tmp;
	if (im <= 0.0145) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = Math.cosh(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.cos(re) * 0.5) * (2.0 + (im * im))
	tmp = 0
	if im <= 0.0145:
		tmp = t_0
	elif im <= 1.35e+154:
		tmp = math.cosh(im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)))
	tmp = 0.0
	if (im <= 0.0145)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = cosh(im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (cos(re) * 0.5) * (2.0 + (im * im));
	tmp = 0.0;
	if (im <= 0.0145)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = cosh(im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.0145], t$95$0, If[LessEqual[im, 1.35e+154], N[Cosh[im], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0145000000000000007 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 88.9%

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

   5. Simplified 88.9%

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

   if 0.0145000000000000007 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative N/A

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

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

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

      3. div-inv N/A

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

      4. exp-neg N/A

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

      5. distribute-lft-out N/A

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

      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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 83.9%

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

       1. *-lft-identity N/A

          \[\leadsto \cosh im \cdot 1 \]

       2. *-rgt-identity N/A

          \[\leadsto \cosh im \]

       3. cosh-lowering-cosh.f64 83.9%

          \[\leadsto \mathsf{cosh.f64}\left(im\right) \]

   11. Applied egg-rr 83.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0145:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00047:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 0.00047) (cos re) (cosh im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00047) {
		tmp = cos(re);
	} else {
		tmp = cosh(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 <= 0.00047d0) then
        tmp = cos(re)
    else
        tmp = cosh(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00047) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.cosh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00047:
		tmp = math.cos(re)
	else:
		tmp = math.cosh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00047)
		tmp = cos(re);
	else
		tmp = cosh(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00047)
		tmp = cos(re);
	else
		tmp = cosh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00047], N[Cos[re], $MachinePrecision], N[Cosh[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.69999999999999986e-4

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 73.6%

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

   7. Simplified 73.6%

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

   if 4.69999999999999986e-4 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative N/A

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

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

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

      3. div-inv N/A

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

      4. exp-neg N/A

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

      5. distribute-lft-out N/A

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

      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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 80.9%

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

       1. *-lft-identity N/A

          \[\leadsto \cosh im \cdot 1 \]

       2. *-rgt-identity N/A

          \[\leadsto \cosh im \]

       3. cosh-lowering-cosh.f64 80.9%

          \[\leadsto \mathsf{cosh.f64}\left(im\right) \]

   11. Applied egg-rr 80.9%

       \[\leadsto \color{blue}{\cosh im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\\ t_1 := 1 + \left(re \cdot re\right) \cdot -0.5\\ t_2 := re \cdot \left(re \cdot re\right)\\ \mathbf{if}\;im \leq 0.00034:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{+27}:\\ \;\;\;\;\left(1 + t\_2 \cdot \left(t\_2 \cdot -0.125\right)\right) \cdot \left(\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot t\_1\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+63}:\\ \;\;\;\;1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot \left(t\_0 \cdot t\_0\right)\right)}{0.5 - im \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* im (+ 0.041666666666666664 (* im (* im 0.001388888888888889)))))
        (t_1 (+ 1.0 (* (* re re) -0.5)))
        (t_2 (* re (* re re))))
   (if (<= im 0.00034)
     (cos re)
     (if (<= im 4.7e+27)
       (*
        (+ 1.0 (* t_2 (* t_2 -0.125)))
        (*
         (+ 1.0 (* (* im im) (+ 0.5 (* im (* im 0.041666666666666664)))))
         t_1))
       (if (<= im 5e+63)
         (+
          1.0
          (/
           (* (* im im) (- 0.25 (* (* im im) (* t_0 t_0))))
           (- 0.5 (* im t_0))))
         (* t_1 (* im (* im (* (* im im) 0.041666666666666664)))))))))

C

double code(double re, double im) {
	double t_0 = im * (0.041666666666666664 + (im * (im * 0.001388888888888889)));
	double t_1 = 1.0 + ((re * re) * -0.5);
	double t_2 = re * (re * re);
	double tmp;
	if (im <= 0.00034) {
		tmp = cos(re);
	} else if (im <= 4.7e+27) {
		tmp = (1.0 + (t_2 * (t_2 * -0.125))) * ((1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664))))) * t_1);
	} else if (im <= 5e+63) {
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * (t_0 * t_0)))) / (0.5 - (im * t_0)));
	} else {
		tmp = t_1 * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	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 = im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))
    t_1 = 1.0d0 + ((re * re) * (-0.5d0))
    t_2 = re * (re * re)
    if (im <= 0.00034d0) then
        tmp = cos(re)
    else if (im <= 4.7d+27) then
        tmp = (1.0d0 + (t_2 * (t_2 * (-0.125d0)))) * ((1.0d0 + ((im * im) * (0.5d0 + (im * (im * 0.041666666666666664d0))))) * t_1)
    else if (im <= 5d+63) then
        tmp = 1.0d0 + (((im * im) * (0.25d0 - ((im * im) * (t_0 * t_0)))) / (0.5d0 - (im * t_0)))
    else
        tmp = t_1 * (im * (im * ((im * im) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.041666666666666664 + (im * (im * 0.001388888888888889)));
	double t_1 = 1.0 + ((re * re) * -0.5);
	double t_2 = re * (re * re);
	double tmp;
	if (im <= 0.00034) {
		tmp = Math.cos(re);
	} else if (im <= 4.7e+27) {
		tmp = (1.0 + (t_2 * (t_2 * -0.125))) * ((1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664))))) * t_1);
	} else if (im <= 5e+63) {
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * (t_0 * t_0)))) / (0.5 - (im * t_0)));
	} else {
		tmp = t_1 * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))
	t_1 = 1.0 + ((re * re) * -0.5)
	t_2 = re * (re * re)
	tmp = 0
	if im <= 0.00034:
		tmp = math.cos(re)
	elif im <= 4.7e+27:
		tmp = (1.0 + (t_2 * (t_2 * -0.125))) * ((1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664))))) * t_1)
	elif im <= 5e+63:
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * (t_0 * t_0)))) / (0.5 - (im * t_0)))
	else:
		tmp = t_1 * (im * (im * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889))))
	t_1 = Float64(1.0 + Float64(Float64(re * re) * -0.5))
	t_2 = Float64(re * Float64(re * re))
	tmp = 0.0
	if (im <= 0.00034)
		tmp = cos(re);
	elseif (im <= 4.7e+27)
		tmp = Float64(Float64(1.0 + Float64(t_2 * Float64(t_2 * -0.125))) * Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))) * t_1));
	elseif (im <= 5e+63)
		tmp = Float64(1.0 + Float64(Float64(Float64(im * im) * Float64(0.25 - Float64(Float64(im * im) * Float64(t_0 * t_0)))) / Float64(0.5 - Float64(im * t_0))));
	else
		tmp = Float64(t_1 * Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.041666666666666664 + (im * (im * 0.001388888888888889)));
	t_1 = 1.0 + ((re * re) * -0.5);
	t_2 = re * (re * re);
	tmp = 0.0;
	if (im <= 0.00034)
		tmp = cos(re);
	elseif (im <= 4.7e+27)
		tmp = (1.0 + (t_2 * (t_2 * -0.125))) * ((1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664))))) * t_1);
	elseif (im <= 5e+63)
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * (t_0 * t_0)))) / (0.5 - (im * t_0)));
	else
		tmp = t_1 * (im * (im * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.041666666666666664 + N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.00034], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4.7e+27], N[(N[(1.0 + N[(t$95$2 * N[(t$95$2 * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e+63], N[(1.0 + N[(N[(N[(im * im), $MachinePrecision] * N[(0.25 - N[(N[(im * im), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 3.4e-4

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 73.6%

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

   7. Simplified 73.6%

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

   if 3.4e-4 < im < 4.69999999999999976e27

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-out N/A

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

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

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

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

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

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

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

   7. Simplified 4.3%

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

   8. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 1.9%

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

   10. Simplified 1.9%

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

   12. Applied egg-rr 1.9%

       \[\leadsto \color{blue}{\left(1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot -0.125\right)\right) \cdot \left(\frac{1}{1 + \left(re \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot 0.25 - -0.5\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)} \]

   13. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right) \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 57.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right) \]

   15. Simplified 57.9%

       \[\leadsto \left(1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot -0.125\right)\right) \cdot \left(\color{blue}{\left(1 + \left(re \cdot re\right) \cdot -0.5\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right) \]

   if 4.69999999999999976e27 < im < 5.00000000000000011e63

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 21.2%

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

   7. Simplified 21.2%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   9. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 21.2%

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

   10. Simplified 21.2%

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

       1. associate-*r* N/A

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

       2. flip-+ N/A

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

       3. associate-*r/ N/A

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

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

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

   12. Applied egg-rr 92.7%

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

   if 5.00000000000000011e63 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-out N/A

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

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

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

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

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

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

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

   7. Simplified 93.8%

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

   8. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 77.5%

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

   10. Simplified 77.5%

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

   11. Taylor expanded in im around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 77.5%

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

   13. Simplified 77.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00034:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{+27}:\\ \;\;\;\;\left(1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot -0.125\right)\right) \cdot \left(\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+63}:\\ \;\;\;\;1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot \left(\left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right) \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\right)}{0.5 - im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\\ \mathbf{if}\;im \leq 5 \cdot 10^{+63}:\\ \;\;\;\;1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot \left(t\_0 \cdot t\_0\right)\right)}{0.5 - im \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* im (+ 0.041666666666666664 (* im (* im 0.001388888888888889))))))
   (if (<= im 5e+63)
     (+
      1.0
      (/ (* (* im im) (- 0.25 (* (* im im) (* t_0 t_0)))) (- 0.5 (* im t_0))))
     (*
      (+ 1.0 (* (* re re) -0.5))
      (* im (* im (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double t_0 = im * (0.041666666666666664 + (im * (im * 0.001388888888888889)));
	double tmp;
	if (im <= 5e+63) {
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * (t_0 * t_0)))) / (0.5 - (im * t_0)));
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	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 * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))
    if (im <= 5d+63) then
        tmp = 1.0d0 + (((im * im) * (0.25d0 - ((im * im) * (t_0 * t_0)))) / (0.5d0 - (im * t_0)))
    else
        tmp = (1.0d0 + ((re * re) * (-0.5d0))) * (im * (im * ((im * im) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.041666666666666664 + (im * (im * 0.001388888888888889)));
	double tmp;
	if (im <= 5e+63) {
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * (t_0 * t_0)))) / (0.5 - (im * t_0)));
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))
	tmp = 0
	if im <= 5e+63:
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * (t_0 * t_0)))) / (0.5 - (im * t_0)))
	else:
		tmp = (1.0 + ((re * re) * -0.5)) * (im * (im * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889))))
	tmp = 0.0
	if (im <= 5e+63)
		tmp = Float64(1.0 + Float64(Float64(Float64(im * im) * Float64(0.25 - Float64(Float64(im * im) * Float64(t_0 * t_0)))) / Float64(0.5 - Float64(im * t_0))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * -0.5)) * Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.041666666666666664 + (im * (im * 0.001388888888888889)));
	tmp = 0.0;
	if (im <= 5e+63)
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * (t_0 * t_0)))) / (0.5 - (im * t_0)));
	else
		tmp = (1.0 + ((re * re) * -0.5)) * (im * (im * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.041666666666666664 + N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 5e+63], N[(1.0 + N[(N[(N[(im * im), $MachinePrecision] * N[(0.25 - N[(N[(im * im), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 5.00000000000000011e63

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   9. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 57.4%

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

   10. Simplified 57.4%

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

       1. associate-*r* N/A

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

       2. flip-+ N/A

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

       3. associate-*r/ N/A

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

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

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

   12. Applied egg-rr 47.0%

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

   if 5.00000000000000011e63 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. exp-neg N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-out N/A

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

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

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

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

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

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

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

   7. Simplified 93.8%

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

   8. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 77.5%

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

   10. Simplified 77.5%

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

   11. Taylor expanded in im around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 77.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   13. Simplified 77.5%

       \[\leadsto \left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\\ \mathbf{if}\;im \leq 400:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))
   (if (<= im 400.0)
     (+ 1.0 (* im (* im (+ 0.5 (* (* im im) t_0)))))
     (*
      (+ 1.0 (* (* re re) (+ -0.5 (* (* re re) 0.041666666666666664))))
      (+ 1.0 (* (* im im) (+ 0.5 (* im (* im t_0)))))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double tmp;
	if (im <= 400.0) {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * t_0))));
	} else {
		tmp = (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)))) * (1.0 + ((im * im) * (0.5 + (im * (im * t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)
    if (im <= 400.0d0) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * t_0))))
    else
        tmp = (1.0d0 + ((re * re) * ((-0.5d0) + ((re * re) * 0.041666666666666664d0)))) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	double tmp;
	if (im <= 400.0) {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * t_0))));
	} else {
		tmp = (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)))) * (1.0 + ((im * im) * (0.5 + (im * (im * t_0)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889)
	tmp = 0
	if im <= 400.0:
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * t_0))))
	else:
		tmp = (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)))) * (1.0 + ((im * im) * (0.5 + (im * (im * t_0)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))
	tmp = 0.0
	if (im <= 400.0)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * t_0)))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(Float64(re * re) * 0.041666666666666664)))) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.041666666666666664 + ((im * im) * 0.001388888888888889);
	tmp = 0.0;
	if (im <= 400.0)
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * t_0))));
	else
		tmp = (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)))) * (1.0 + ((im * im) * (0.5 + (im * (im * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 400.0], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 400

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 95.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 95.6%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   9. 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. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 61.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 61.8%

       \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)} \]

   if 400 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 73.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 73.4%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {re}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left({re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(re \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 69.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 69.6%

       \[\leadsto \color{blue}{\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 400:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 440:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 440.0)
   (+
    1.0
    (*
     im
     (*
      im
      (+
       0.5
       (*
        (* im im)
        (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))))))
   (*
    (+ 1.0 (* (* re re) (+ -0.5 (* (* re re) 0.041666666666666664))))
    (+ 1.0 (* (* im im) (+ 0.5 (* im (* im 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 440.0) {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	} else {
		tmp = (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)))) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 440.0d0) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0))))))
    else
        tmp = (1.0d0 + ((re * re) * ((-0.5d0) + ((re * re) * 0.041666666666666664d0)))) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 440.0) {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	} else {
		tmp = (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)))) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 440.0:
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))
	else:
		tmp = (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)))) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 440.0)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(Float64(re * re) * 0.041666666666666664)))) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 440.0)
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	else
		tmp = (1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)))) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 440.0], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 440

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 95.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 95.6%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   9. 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. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 61.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 61.8%

       \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)} \]

   if 440 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   7. Simplified 65.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {re}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left({re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(re \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 65.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 65.0%

       \[\leadsto \color{blue}{\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 440:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.4% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{if}\;im \leq 450:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (*
           im
           (*
            im
            (+
             0.5
             (*
              (* im im)
              (+
               0.041666666666666664
               (* (* im im) 0.001388888888888889)))))))))
   (if (<= im 450.0)
     t_0
     (if (<= im 7.2e+51)
       (+ 1.0 (* (* re re) (+ -0.5 (* (* re re) 0.041666666666666664))))
       t_0))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	double tmp;
	if (im <= 450.0) {
		tmp = t_0;
	} else if (im <= 7.2e+51) {
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0))))))
    if (im <= 450.0d0) then
        tmp = t_0
    else if (im <= 7.2d+51) then
        tmp = 1.0d0 + ((re * re) * ((-0.5d0) + ((re * re) * 0.041666666666666664d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	double tmp;
	if (im <= 450.0) {
		tmp = t_0;
	} else if (im <= 7.2e+51) {
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))
	tmp = 0
	if im <= 450.0:
		tmp = t_0
	elif im <= 7.2e+51:
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)))))))
	tmp = 0.0
	if (im <= 450.0)
		tmp = t_0;
	elseif (im <= 7.2e+51)
		tmp = Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(Float64(re * re) * 0.041666666666666664))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	tmp = 0.0;
	if (im <= 450.0)
		tmp = t_0;
	elseif (im <= 7.2e+51)
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 450.0], t$95$0, If[LessEqual[im, 7.2e+51], N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 450 or 7.20000000000000022e51 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 96.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 96.4%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   9. 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. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 64.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 64.6%

       \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)} \]

   if 450 < im < 7.20000000000000022e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   7. Simplified 3.1%

      \[\leadsto \color{blue}{\cos re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {re}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2}\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 46.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right) \]

   10. Simplified 46.4%

       \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 450:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+53}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 420.0)
   (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664)))))
   (if (<= im 1.1e+53)
     (+ 1.0 (* (* re re) (+ -0.5 (* (* re re) 0.041666666666666664))))
     (*
      (* (* im im) (* im im))
      (+ 0.041666666666666664 (* (* re re) -0.020833333333333332))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 420.0) {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	} else if (im <= 1.1e+53) {
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)));
	} 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 <= 420.0d0) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else if (im <= 1.1d+53) then
        tmp = 1.0d0 + ((re * re) * ((-0.5d0) + ((re * re) * 0.041666666666666664d0)))
    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 <= 420.0) {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	} else if (im <= 1.1e+53) {
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 420.0:
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	elif im <= 1.1e+53:
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)))
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 420.0)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	elseif (im <= 1.1e+53)
		tmp = Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(Float64(re * re) * 0.041666666666666664))));
	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 <= 420.0)
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	elseif (im <= 1.1e+53)
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)));
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 420.0], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+53], N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * 0.041666666666666664), $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 < 420

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   7. Simplified 94.4%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 61.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   10. Simplified 61.1%

       \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   if 420 < im < 1.09999999999999999e53

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   7. Simplified 3.1%

      \[\leadsto \color{blue}{\cos re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {re}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2}\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 46.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right) \]

   10. Simplified 46.4%

       \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)} \]

   if 1.09999999999999999e53 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   7. Simplified 90.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 74.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 74.4%

       \[\leadsto \color{blue}{\left(1 + \left(re \cdot re\right) \cdot -0.5\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]

   11. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       12. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} \cdot 1 + \color{blue}{\frac{1}{24} \cdot \left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + \color{blue}{\frac{1}{24}} \cdot \left(\frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{24} \cdot \left(\frac{-1}{2} \cdot {re}^{2}\right)\right)}\right)\right) \]

       15. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(\frac{1}{24} \cdot \frac{-1}{2}\right) \cdot \color{blue}{{re}^{2}}\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       20. metadata-eval 74.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{48}\right)\right)\right) \]

   13. Simplified 74.4%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+53}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.8% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot 0.041666666666666664\\ \mathbf{if}\;im \leq 370:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+53}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) 0.041666666666666664)))
   (if (<= im 370.0)
     (+ 1.0 (* im (* im (+ 0.5 t_0))))
     (if (<= im 5.5e+53)
       (+ 1.0 (* (* re re) (+ -0.5 (* (* re re) 0.041666666666666664))))
       (* im (* im t_0))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double tmp;
	if (im <= 370.0) {
		tmp = 1.0 + (im * (im * (0.5 + t_0)));
	} else if (im <= 5.5e+53) {
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)));
	} else {
		tmp = im * (im * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im * im) * 0.041666666666666664d0
    if (im <= 370.0d0) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + t_0)))
    else if (im <= 5.5d+53) then
        tmp = 1.0d0 + ((re * re) * ((-0.5d0) + ((re * re) * 0.041666666666666664d0)))
    else
        tmp = im * (im * t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double tmp;
	if (im <= 370.0) {
		tmp = 1.0 + (im * (im * (0.5 + t_0)));
	} else if (im <= 5.5e+53) {
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)));
	} else {
		tmp = im * (im * t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * 0.041666666666666664
	tmp = 0
	if im <= 370.0:
		tmp = 1.0 + (im * (im * (0.5 + t_0)))
	elif im <= 5.5e+53:
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)))
	else:
		tmp = im * (im * t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * 0.041666666666666664)
	tmp = 0.0
	if (im <= 370.0)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + t_0))));
	elseif (im <= 5.5e+53)
		tmp = Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(Float64(re * re) * 0.041666666666666664))));
	else
		tmp = Float64(im * Float64(im * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * 0.041666666666666664;
	tmp = 0.0;
	if (im <= 370.0)
		tmp = 1.0 + (im * (im * (0.5 + t_0)));
	elseif (im <= 5.5e+53)
		tmp = 1.0 + ((re * re) * (-0.5 + ((re * re) * 0.041666666666666664)));
	else
		tmp = im * (im * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]}, If[LessEqual[im, 370.0], N[(1.0 + N[(im * N[(im * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.5e+53], N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 370

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   7. Simplified 94.4%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 61.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   10. Simplified 61.1%

       \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   if 370 < im < 5.49999999999999975e53

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   7. Simplified 3.1%

      \[\leadsto \color{blue}{\cos re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {re}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2}\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 46.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right) \]

   10. Simplified 46.4%

       \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(re \cdot re\right)\right)} \]

   if 5.49999999999999975e53 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   7. Simplified 90.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 70.1%

       \[\leadsto \color{blue}{1} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]

   11. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   12. 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. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \left(im \cdot \color{blue}{im}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot im\right) \cdot \color{blue}{im} \]

       6. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot im\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot im\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right) \]

       13. *-lowering-*.f64 70.1%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right) \]

   13. Simplified 70.1%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 370:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+53}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.9% accurate, 19.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.2:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.2)
   (+ 1.0 (* 0.5 (* im im)))
   (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.2) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (im * im) * (0.5 + ((im * im) * 0.041666666666666664));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.2d0) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = (im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.2) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (im * im) * (0.5 + ((im * im) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.2:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = (im * im) * (0.5 + ((im * im) * 0.041666666666666664))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.2)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.2)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = (im * im) * (0.5 + ((im * im) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.2], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.2000000000000002

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   7. Simplified 95.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 61.7%

       \[\leadsto \color{blue}{1} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   12. 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 55.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   13. Simplified 55.2%

       \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   if 2.2000000000000002 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   7. Simplified 64.1%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 50.2%

       \[\leadsto \color{blue}{1} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]

   11. Taylor expanded in im around inf

       \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto {im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}} + \color{blue}{\frac{1}{24}}\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4} + \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]

       3. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) + \color{blue}{\frac{1}{24}} \cdot {im}^{4} \]

       4. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \left(\frac{1}{{im}^{2}} \cdot \frac{1}{2}\right) + \frac{1}{24} \cdot {im}^{4} \]

       5. associate-*r* N/A

          \[\leadsto \left({im}^{4} \cdot \frac{1}{{im}^{2}}\right) \cdot \frac{1}{2} + \color{blue}{\frac{1}{24}} \cdot {im}^{4} \]

       6. associate-*r/ N/A

          \[\leadsto \frac{{im}^{4} \cdot 1}{{im}^{2}} \cdot \frac{1}{2} + \frac{1}{24} \cdot {im}^{4} \]

       7. *-rgt-identity N/A

          \[\leadsto \frac{{im}^{4}}{{im}^{2}} \cdot \frac{1}{2} + \frac{1}{24} \cdot {im}^{4} \]

       8. metadata-eval N/A

          \[\leadsto \frac{{im}^{\left(2 \cdot 2\right)}}{{im}^{2}} \cdot \frac{1}{2} + \frac{1}{24} \cdot {im}^{4} \]

       9. pow-sqr N/A

          \[\leadsto \frac{{im}^{2} \cdot {im}^{2}}{{im}^{2}} \cdot \frac{1}{2} + \frac{1}{24} \cdot {im}^{4} \]

       10. associate-/l* N/A

           \[\leadsto \left({im}^{2} \cdot \frac{{im}^{2}}{{im}^{2}}\right) \cdot \frac{1}{2} + \frac{1}{24} \cdot {im}^{4} \]

       11. *-inverses N/A

           \[\leadsto \left({im}^{2} \cdot 1\right) \cdot \frac{1}{2} + \frac{1}{24} \cdot {im}^{4} \]

       12. *-rgt-identity N/A

           \[\leadsto {im}^{2} \cdot \frac{1}{2} + \frac{1}{24} \cdot {im}^{4} \]

       13. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot {im}^{2} + \color{blue}{\frac{1}{24}} \cdot {im}^{4} \]

       14. metadata-eval N/A

           \[\leadsto \frac{1}{2} \cdot {im}^{2} + \frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)} \]

       15. pow-sqr N/A

           \[\leadsto \frac{1}{2} \cdot {im}^{2} + \frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right) \]

       16. associate-*l* N/A

           \[\leadsto \frac{1}{2} \cdot {im}^{2} + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}} \]

       17. distribute-rgt-in N/A

           \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {im}^{2}\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {im}^{2}\right)\right) \]

       21. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right) \]

   13. Simplified 50.2%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.9% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7)
   (+ 1.0 (* 0.5 (* im im)))
   (* im (* im (* (* im im) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = im * (im * ((im * im) * 0.041666666666666664));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.7d0) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = im * (im * ((im * im) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = im * (im * ((im * im) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = im * (im * ((im * im) * 0.041666666666666664))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = im * (im * ((im * im) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.7000000000000002

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   7. Simplified 95.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 61.7%

       \[\leadsto \color{blue}{1} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   12. 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 55.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   13. Simplified 55.2%

       \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   if 3.7000000000000002 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   7. Simplified 64.1%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 50.2%

       \[\leadsto \color{blue}{1} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]

   11. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   12. 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. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \left(im \cdot \color{blue}{im}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot im\right) \cdot \color{blue}{im} \]

       6. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot im\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot im\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right) \]

       13. *-lowering-*.f64 50.2%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right) \]

   13. Simplified 50.2%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 55.9% accurate, 23.7× speedup

Math

\[\begin{array}{l} \\ 1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	return 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
end function

Java

public static double code(double re, double im) {
	return 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
}

Python

def code(re, im):
	return 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))

Julia

function code(re, im)
	return Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
end

Wolfram

code[re_, im_] := N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

   2. associate-*l* N/A

      \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

   10. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

   11. exp-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

   12. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

   15. exp-lowering-exp.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
6. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

   6. *-commutative N/A

      \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

   7. distribute-rgt-out N/A

      \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

   8. *-commutative N/A

      \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

   9. distribute-lft-in N/A

      \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

   11. distribute-lft-out N/A

       \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

   13. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

7. Simplified 87.3%

   \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

8. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 58.7%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

10. Simplified 58.7%

    \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
11. Add Preprocessing

Alternative 17: 38.7% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1.42) 1.0 (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.42) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.42d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.42) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.42:
		tmp = 1.0
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.42)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.42)
		tmp = 1.0;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.42], 1.0, N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.4199999999999999

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 73.6%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   7. Simplified 73.6%

      \[\leadsto \color{blue}{\cos re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 43.6%

       \[\leadsto \color{blue}{1} \]

   if 1.4199999999999999 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      2. associate-*l* N/A

         \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      15. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   7. Simplified 64.1%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 50.2%

       \[\leadsto \color{blue}{1} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   12. 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 31.5%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   13. Simplified 31.5%

       \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   14. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {im}^{2}} \]
   15. 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 31.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

   16. Simplified 31.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 47.5% 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. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

   2. associate-*l* N/A

      \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

   10. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

   11. exp-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

   12. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

   15. exp-lowering-exp.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
6. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \cos re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \cos re \cdot 1 + \left({im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos re}\right) \]

   6. *-commutative N/A

      \[\leadsto \cos re \cdot 1 + \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right) \]

   7. distribute-rgt-out N/A

      \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

   8. *-commutative N/A

      \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\frac{1}{2}}\right) \]

   9. distribute-lft-in N/A

      \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \cos re \cdot 1 + \cos re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right) \]

   11. distribute-lft-out N/A

       \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

   13. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

7. Simplified 87.3%

   \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

8. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 58.7%

    \[\leadsto \color{blue}{1} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]

11. Taylor expanded in im around 0

    \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
12. 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 49.1%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

13. Simplified 49.1%

    \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]
14. Add Preprocessing

Alternative 19: 29.6% 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. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

   2. associate-*l* N/A

      \[\leadsto \cos re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)}\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right)\right)\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(e^{im} \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot e^{im}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

   10. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right)\right) \]

   11. exp-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

   12. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

   15. exp-lowering-exp.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\cos re} \]
6. Step-by-step derivation

   1. cos-lowering-cos.f64 55.4%

      \[\leadsto \mathsf{cos.f64}\left(re\right) \]

7. Simplified 55.4%

   \[\leadsto \color{blue}{\cos re} \]

8. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 33.1%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))