math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.2s

Alternatives: 25

Speedup: 1.5×

• 48.5% 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: 87.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.016:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.016)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 7.2e+51)
     (* (cosh im) (+ 1.0 (* (* re re) -0.5)))
     (*
      (cos re)
      (+
       1.0
       (*
        (* im im)
        (+ 0.5 (* im (* im (* (* im im) 0.001388888888888889))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.016) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 7.2e+51) {
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * ((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 <= 0.016d0) 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)))
    else
        tmp = cos(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * ((im * im) * 0.001388888888888889d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.016) {
		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));
	} else {
		tmp = Math.cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * ((im * im) * 0.001388888888888889))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.016:
		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))
	else:
		tmp = math.cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * ((im * im) * 0.001388888888888889))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.016)
		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) * -0.5)));
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(Float64(im * im) * 0.001388888888888889)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.016)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 7.2e+51)
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	else
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * ((im * im) * 0.001388888888888889))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.016], 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] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.016

   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 86.8%

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

   5. Simplified 86.8%

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

   if 0.016 < 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. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 88.4%

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

   9. Simplified 88.4%

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

   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. 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 im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in im around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{720} \cdot {im}^{3}\right)}\right)\right)\right)\right), \mathsf{cos.f64}\left(re\right)\right) \]
   11. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.016:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.32:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.32)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 2.6e+77)
     (* (cosh im) (+ 1.0 (* (* re re) -0.5)))
     (* (cos re) (* (* im im) (* im (* im 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.32) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 2.6e+77) {
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = cos(re) * ((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 <= 0.32d0) 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)))
    else
        tmp = cos(re) * ((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 <= 0.32) {
		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));
	} else {
		tmp = Math.cos(re) * ((im * im) * (im * (im * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.32:
		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))
	else:
		tmp = math.cos(re) * ((im * im) * (im * (im * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.32)
		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) * -0.5)));
	else
		tmp = Float64(cos(re) * Float64(Float64(im * im) * Float64(im * Float64(im * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.32)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 2.6e+77)
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	else
		tmp = cos(re) * ((im * im) * (im * (im * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.32], 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] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.320000000000000007

   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 86.8%

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

   5. Simplified 86.8%

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

   if 0.320000000000000007 < 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

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 79.3%

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

   9. Simplified 79.3%

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

   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. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.32:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.025:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.025)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 2.6e+77)
     (cosh im)
     (* (cos re) (* (* im im) (* im (* im 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.025) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 2.6e+77) {
		tmp = cosh(im);
	} else {
		tmp = cos(re) * ((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 <= 0.025d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 2.6d+77) then
        tmp = cosh(im)
    else
        tmp = cos(re) * ((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 <= 0.025) {
		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) * ((im * im) * (im * (im * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.025:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 2.6e+77:
		tmp = math.cosh(im)
	else:
		tmp = math.cos(re) * ((im * im) * (im * (im * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.025)
		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(Float64(im * im) * Float64(im * Float64(im * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.025)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 2.6e+77)
		tmp = cosh(im);
	else
		tmp = cos(re) * ((im * im) * (im * (im * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.025], 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[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.025000000000000001

   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 86.8%

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

   5. Simplified 86.8%

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

   if 0.025000000000000001 < 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 50.1%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{1} \]
   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 50.1%

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

   11. Applied egg-rr 50.1%

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

   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. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.025:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{if}\;im \leq 0.023:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ 2.0 (* im im)))))
   (if (<= im 0.023) t_0 (if (<= im 8e+153) (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 <= 0.023) {
		tmp = t_0;
	} else if (im <= 8e+153) {
		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 <= 0.023d0) then
        tmp = t_0
    else if (im <= 8d+153) 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 <= 0.023) {
		tmp = t_0;
	} else if (im <= 8e+153) {
		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 <= 0.023:
		tmp = t_0
	elif im <= 8e+153:
		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 <= 0.023)
		tmp = t_0;
	elseif (im <= 8e+153)
		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 <= 0.023)
		tmp = t_0;
	elseif (im <= 8e+153)
		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, 0.023], t$95$0, If[LessEqual[im, 8e+153], N[Cosh[im], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.023 or 8e153 < 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 87.8%

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

   5. Simplified 87.8%

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

   if 0.023 < im < 8e153

   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 64.1%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{1} \]
   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 64.1%

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

   11. Applied egg-rr 64.1%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.023:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00044:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00044)
   (cos re)
   (if (<= im 8e+153) (cosh im) (* (* (cos re) 0.5) (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00044) {
		tmp = cos(re);
	} else if (im <= 8e+153) {
		tmp = cosh(im);
	} else {
		tmp = (cos(re) * 0.5) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00044d0) then
        tmp = cos(re)
    else if (im <= 8d+153) then
        tmp = cosh(im)
    else
        tmp = (cos(re) * 0.5d0) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00044) {
		tmp = Math.cos(re);
	} else if (im <= 8e+153) {
		tmp = Math.cosh(im);
	} else {
		tmp = (Math.cos(re) * 0.5) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00044:
		tmp = math.cos(re)
	elif im <= 8e+153:
		tmp = math.cosh(im)
	else:
		tmp = (math.cos(re) * 0.5) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00044)
		tmp = cos(re);
	elseif (im <= 8e+153)
		tmp = cosh(im);
	else
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00044)
		tmp = cos(re);
	elseif (im <= 8e+153)
		tmp = cosh(im);
	else
		tmp = (cos(re) * 0.5) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00044], N[Cos[re], $MachinePrecision], If[LessEqual[im, 8e+153], N[Cosh[im], $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.40000000000000016e-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 64.0%

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

   7. Simplified 64.0%

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

   if 4.40000000000000016e-4 < im < 8e153

   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 64.1%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{1} \]
   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 64.1%

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

   11. Applied egg-rr 64.1%

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

   if 8e153 < 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 96.5%

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

   5. Simplified 96.5%

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

   6. Taylor expanded in im around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 96.5%

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

   8. Simplified 96.5%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00044:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00019:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 0.00019) (cos re) (cosh im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00019) {
		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 <= 0.00019d0) 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 <= 0.00019) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.cosh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00019:
		tmp = math.cos(re)
	else:
		tmp = math.cosh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00019)
		tmp = cos(re);
	else
		tmp = cosh(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00019)
		tmp = cos(re);
	else
		tmp = cosh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00019], N[Cos[re], $MachinePrecision], N[Cosh[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.9000000000000001e-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 64.0%

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

   7. Simplified 64.0%

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

   if 1.9000000000000001e-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 69.9%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{1} \]
   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 69.9%

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

   11. Applied egg-rr 69.9%

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

Alternative 8: 68.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot 0.08333333333333333\\ \mathbf{if}\;im \leq 1100:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(4 + \left(im \cdot \left(im \cdot \left(1 + t\_0\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(-1 - t\_0\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)}{2 - im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) 0.08333333333333333)))
   (if (<= im 1100.0)
     (cos re)
     (if (<= im 2e+64)
       (/
        (*
         (+ 4.0 (* (* im (* im (+ 1.0 t_0))) (* im (* im (- -1.0 t_0)))))
         (+ 0.5 (* (* re re) -0.25)))
        (- 2.0 (* im im)))
       (+
        1.0
        (*
         im
         (*
          im
          (+
           0.5
           (*
            (* im im)
            (+
             (* (* im im) 0.001388888888888889)
             0.041666666666666664))))))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * 0.08333333333333333;
	double tmp;
	if (im <= 1100.0) {
		tmp = cos(re);
	} else if (im <= 2e+64) {
		tmp = ((4.0 + ((im * (im * (1.0 + t_0))) * (im * (im * (-1.0 - t_0))))) * (0.5 + ((re * re) * -0.25))) / (2.0 - (im * im));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 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 * im) * 0.08333333333333333d0
    if (im <= 1100.0d0) then
        tmp = cos(re)
    else if (im <= 2d+64) then
        tmp = ((4.0d0 + ((im * (im * (1.0d0 + t_0))) * (im * (im * ((-1.0d0) - t_0))))) * (0.5d0 + ((re * re) * (-0.25d0)))) / (2.0d0 - (im * im))
    else
        tmp = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * (((im * im) * 0.001388888888888889d0) + 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * 0.08333333333333333;
	double tmp;
	if (im <= 1100.0) {
		tmp = Math.cos(re);
	} else if (im <= 2e+64) {
		tmp = ((4.0 + ((im * (im * (1.0 + t_0))) * (im * (im * (-1.0 - t_0))))) * (0.5 + ((re * re) * -0.25))) / (2.0 - (im * im));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * 0.08333333333333333
	tmp = 0
	if im <= 1100.0:
		tmp = math.cos(re)
	elif im <= 2e+64:
		tmp = ((4.0 + ((im * (im * (1.0 + t_0))) * (im * (im * (-1.0 - t_0))))) * (0.5 + ((re * re) * -0.25))) / (2.0 - (im * im))
	else:
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * 0.08333333333333333)
	tmp = 0.0
	if (im <= 1100.0)
		tmp = cos(re);
	elseif (im <= 2e+64)
		tmp = Float64(Float64(Float64(4.0 + Float64(Float64(im * Float64(im * Float64(1.0 + t_0))) * Float64(im * Float64(im * Float64(-1.0 - t_0))))) * Float64(0.5 + Float64(Float64(re * re) * -0.25))) / Float64(2.0 - Float64(im * im)));
	else
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * Float64(Float64(Float64(im * im) * 0.001388888888888889) + 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * 0.08333333333333333;
	tmp = 0.0;
	if (im <= 1100.0)
		tmp = cos(re);
	elseif (im <= 2e+64)
		tmp = ((4.0 + ((im * (im * (1.0 + t_0))) * (im * (im * (-1.0 - t_0))))) * (0.5 + ((re * re) * -0.25))) / (2.0 - (im * im));
	else
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]}, If[LessEqual[im, 1100.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2e+64], N[(N[(N[(4.0 + N[(N[(im * N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1100

   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 63.5%

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

   7. Simplified 63.5%

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

   if 1100 < im < 2.00000000000000004e64

   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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 4.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 4.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \color{blue}{\frac{-1}{4} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]

      3. distribute-rgt-out N/A

         \[\leadsto \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

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

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

   8. Simplified 33.0%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
   9. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{2 \cdot 2 - \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right)}{2 - \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)} \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{4} \cdot \left(re \cdot re\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left(2 \cdot 2 - \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(re \cdot re\right)\right)}{\color{blue}{2 - \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot 2 - \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(re \cdot re\right)\right)\right), \color{blue}{\left(2 - \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right)}\right) \]

   10. Applied egg-rr 51.4%

       \[\leadsto \color{blue}{\frac{\left(4 - \left(im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)}{2 - im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)}} \]

   11. Taylor expanded in im around 0

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

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right)\right), \mathsf{\_.f64}\left(2, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       2. *-lowering-*.f64 51.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   13. Simplified 51.6%

       \[\leadsto \frac{\left(4 - \left(im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)}{2 - \color{blue}{im \cdot im}} \]

   if 2.00000000000000004e64 < 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 im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in re around 0

       \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   11. 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 82.9%

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

   12. Simplified 82.9%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1100:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(4 + \left(im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(-1 - \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)}{2 - im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot 0.08333333333333333\\ \mathbf{if}\;im \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(4 + \left(im \cdot \left(im \cdot \left(1 + t\_0\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(-1 - t\_0\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)}{2 - im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) 0.08333333333333333)))
   (if (<= im 3.05e-61)
     1.0
     (if (<= im 2.15e+65)
       (/
        (*
         (+ 4.0 (* (* im (* im (+ 1.0 t_0))) (* im (* im (- -1.0 t_0)))))
         (+ 0.5 (* (* re re) -0.25)))
        (- 2.0 (* im im)))
       (+
        1.0
        (*
         im
         (*
          im
          (+
           0.5
           (*
            (* im im)
            (+
             (* (* im im) 0.001388888888888889)
             0.041666666666666664))))))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * 0.08333333333333333;
	double tmp;
	if (im <= 3.05e-61) {
		tmp = 1.0;
	} else if (im <= 2.15e+65) {
		tmp = ((4.0 + ((im * (im * (1.0 + t_0))) * (im * (im * (-1.0 - t_0))))) * (0.5 + ((re * re) * -0.25))) / (2.0 - (im * im));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 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 * im) * 0.08333333333333333d0
    if (im <= 3.05d-61) then
        tmp = 1.0d0
    else if (im <= 2.15d+65) then
        tmp = ((4.0d0 + ((im * (im * (1.0d0 + t_0))) * (im * (im * ((-1.0d0) - t_0))))) * (0.5d0 + ((re * re) * (-0.25d0)))) / (2.0d0 - (im * im))
    else
        tmp = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * (((im * im) * 0.001388888888888889d0) + 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * 0.08333333333333333;
	double tmp;
	if (im <= 3.05e-61) {
		tmp = 1.0;
	} else if (im <= 2.15e+65) {
		tmp = ((4.0 + ((im * (im * (1.0 + t_0))) * (im * (im * (-1.0 - t_0))))) * (0.5 + ((re * re) * -0.25))) / (2.0 - (im * im));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * 0.08333333333333333
	tmp = 0
	if im <= 3.05e-61:
		tmp = 1.0
	elif im <= 2.15e+65:
		tmp = ((4.0 + ((im * (im * (1.0 + t_0))) * (im * (im * (-1.0 - t_0))))) * (0.5 + ((re * re) * -0.25))) / (2.0 - (im * im))
	else:
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * 0.08333333333333333)
	tmp = 0.0
	if (im <= 3.05e-61)
		tmp = 1.0;
	elseif (im <= 2.15e+65)
		tmp = Float64(Float64(Float64(4.0 + Float64(Float64(im * Float64(im * Float64(1.0 + t_0))) * Float64(im * Float64(im * Float64(-1.0 - t_0))))) * Float64(0.5 + Float64(Float64(re * re) * -0.25))) / Float64(2.0 - Float64(im * im)));
	else
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * Float64(Float64(Float64(im * im) * 0.001388888888888889) + 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * 0.08333333333333333;
	tmp = 0.0;
	if (im <= 3.05e-61)
		tmp = 1.0;
	elseif (im <= 2.15e+65)
		tmp = ((4.0 + ((im * (im * (1.0 + t_0))) * (im * (im * (-1.0 - t_0))))) * (0.5 + ((re * re) * -0.25))) / (2.0 - (im * im));
	else
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]}, If[LessEqual[im, 3.05e-61], 1.0, If[LessEqual[im, 2.15e+65], N[(N[(N[(4.0 + N[(N[(im * N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.05e-61

   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 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 34.7%

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

   if 3.05e-61 < im < 2.15000000000000023e65

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 30.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 30.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \color{blue}{\frac{-1}{4} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]

      3. distribute-rgt-out N/A

         \[\leadsto \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

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

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

   8. Simplified 39.8%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
   9. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{2 \cdot 2 - \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right)}{2 - \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)} \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{4} \cdot \left(re \cdot re\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left(2 \cdot 2 - \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(re \cdot re\right)\right)}{\color{blue}{2 - \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot 2 - \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(re \cdot re\right)\right)\right), \color{blue}{\left(2 - \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \frac{1}{12}\right)\right)\right)}\right) \]

   10. Applied egg-rr 52.1%

       \[\leadsto \color{blue}{\frac{\left(4 - \left(im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)}{2 - im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)}} \]

   11. Taylor expanded in im around 0

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

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right)\right), \mathsf{\_.f64}\left(2, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       2. *-lowering-*.f64 52.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   13. Simplified 52.2%

       \[\leadsto \frac{\left(4 - \left(im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)}{2 - \color{blue}{im \cdot im}} \]

   if 2.15000000000000023e65 < 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 im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in re around 0

       \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   11. 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 82.9%

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

   12. Simplified 82.9%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(4 + \left(im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(-1 - \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)}{2 - im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\\ \mathbf{if}\;im \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+65}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (*
           im
           (*
            im
            (+
             0.5
             (*
              (* im im)
              (+
               (* (* im im) 0.001388888888888889)
               0.041666666666666664))))))))
   (if (<= im 3.05e-61)
     1.0
     (if (<= im 4.1e+65) (* (+ 1.0 (* (* re re) -0.5)) t_0) t_0))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	double tmp;
	if (im <= 3.05e-61) {
		tmp = 1.0;
	} else if (im <= 4.1e+65) {
		tmp = (1.0 + ((re * re) * -0.5)) * t_0;
	} 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) * (((im * im) * 0.001388888888888889d0) + 0.041666666666666664d0)))))
    if (im <= 3.05d-61) then
        tmp = 1.0d0
    else if (im <= 4.1d+65) then
        tmp = (1.0d0 + ((re * re) * (-0.5d0))) * t_0
    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) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	double tmp;
	if (im <= 3.05e-61) {
		tmp = 1.0;
	} else if (im <= 4.1e+65) {
		tmp = (1.0 + ((re * re) * -0.5)) * t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))
	tmp = 0
	if im <= 3.05e-61:
		tmp = 1.0
	elif im <= 4.1e+65:
		tmp = (1.0 + ((re * re) * -0.5)) * t_0
	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(Float64(Float64(im * im) * 0.001388888888888889) + 0.041666666666666664))))))
	tmp = 0.0
	if (im <= 3.05e-61)
		tmp = 1.0;
	elseif (im <= 4.1e+65)
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * -0.5)) * t_0);
	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) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	tmp = 0.0;
	if (im <= 3.05e-61)
		tmp = 1.0;
	elseif (im <= 4.1e+65)
		tmp = (1.0 + ((re * re) * -0.5)) * t_0;
	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[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3.05e-61], 1.0, If[LessEqual[im, 4.1e+65], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.05e-61

   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 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 34.7%

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

   if 3.05e-61 < im < 4.1000000000000001e65

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   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 im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 46.9%

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

   9. Simplified 46.9%

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

   10. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. metadata-eval N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt1-in N/A

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

   12. Simplified 46.2%

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

   if 4.1000000000000001e65 < 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 im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in re around 0

       \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   11. 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 82.9%

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

   12. Simplified 82.9%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+65}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot \left(-0.25 + \left(re \cdot re\right) \cdot \left(0.020833333333333332 + \left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.05e-61)
   1.0
   (if (<= im 3.4e+39)
     (*
      (+ 2.0 (* im im))
      (+
       0.5
       (*
        re
        (*
         re
         (+
          -0.25
          (*
           (* re re)
           (+ 0.020833333333333332 (* (* re re) -0.0006944444444444445))))))))
     (+
      1.0
      (*
       im
       (*
        im
        (+
         0.5
         (*
          (* im im)
          (+ (* (* im im) 0.001388888888888889) 0.041666666666666664)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.05e-61) {
		tmp = 1.0;
	} else if (im <= 3.4e+39) {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * (-0.25 + ((re * re) * (0.020833333333333332 + ((re * re) * -0.0006944444444444445)))))));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 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 <= 3.05d-61) then
        tmp = 1.0d0
    else if (im <= 3.4d+39) then
        tmp = (2.0d0 + (im * im)) * (0.5d0 + (re * (re * ((-0.25d0) + ((re * re) * (0.020833333333333332d0 + ((re * re) * (-0.0006944444444444445d0))))))))
    else
        tmp = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * (((im * im) * 0.001388888888888889d0) + 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.05e-61) {
		tmp = 1.0;
	} else if (im <= 3.4e+39) {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * (-0.25 + ((re * re) * (0.020833333333333332 + ((re * re) * -0.0006944444444444445)))))));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.05e-61:
		tmp = 1.0
	elif im <= 3.4e+39:
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * (-0.25 + ((re * re) * (0.020833333333333332 + ((re * re) * -0.0006944444444444445)))))))
	else:
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.05e-61)
		tmp = 1.0;
	elseif (im <= 3.4e+39)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(re * Float64(re * Float64(-0.25 + Float64(Float64(re * re) * Float64(0.020833333333333332 + Float64(Float64(re * re) * -0.0006944444444444445))))))));
	else
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * Float64(Float64(Float64(im * im) * 0.001388888888888889) + 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.05e-61)
		tmp = 1.0;
	elseif (im <= 3.4e+39)
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * (-0.25 + ((re * re) * (0.020833333333333332 + ((re * re) * -0.0006944444444444445)))))));
	else
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.05e-61], 1.0, If[LessEqual[im, 3.4e+39], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(re * N[(re * N[(-0.25 + N[(N[(re * re), $MachinePrecision] * N[(0.020833333333333332 + N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.05e-61

   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 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 34.7%

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

   if 3.05e-61 < im < 3.3999999999999999e39

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(2 + {im}^{2}\right)}\right) \]
   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 36.6%

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

   5. Simplified 36.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\left(re \cdot re\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \frac{-1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{4} + {re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{48}, \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{48}, \left({re}^{2} \cdot \frac{-1}{1440}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{1440}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{1440}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

      17. *-lowering-*.f64 48.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{1440}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, im\right)\right)\right) \]

   8. Simplified 48.2%

      \[\leadsto \color{blue}{\left(0.5 + re \cdot \left(re \cdot \left(-0.25 + \left(re \cdot re\right) \cdot \left(0.020833333333333332 + \left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right)\right)} \cdot \left(2 + im \cdot im\right) \]

   if 3.3999999999999999e39 < 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 im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 98.0%

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

   9. Simplified 98.0%

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

   10. Taylor expanded in re around 0

       \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   11. 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 76.8%

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

   12. Simplified 76.8%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot \left(-0.25 + \left(re \cdot re\right) \cdot \left(0.020833333333333332 + \left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.0% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\\ \mathbf{if}\;im \leq 26:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (*
           im
           (*
            im
            (+
             0.5
             (*
              (* im im)
              (+
               (* (* im im) 0.001388888888888889)
               0.041666666666666664))))))))
   (if (<= im 26.0)
     t_0
     (if (<= im 3.4e+39)
       (+
        1.0
        (*
         (* re re)
         (+
          -0.5
          (*
           re
           (*
            re
            (+ 0.041666666666666664 (* (* re re) -0.001388888888888889)))))))
       t_0))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	double tmp;
	if (im <= 26.0) {
		tmp = t_0;
	} else if (im <= 3.4e+39) {
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))));
	} 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) * (((im * im) * 0.001388888888888889d0) + 0.041666666666666664d0)))))
    if (im <= 26.0d0) then
        tmp = t_0
    else if (im <= 3.4d+39) then
        tmp = 1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * (0.041666666666666664d0 + ((re * re) * (-0.001388888888888889d0)))))))
    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) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	double tmp;
	if (im <= 26.0) {
		tmp = t_0;
	} else if (im <= 3.4e+39) {
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))
	tmp = 0
	if im <= 26.0:
		tmp = t_0
	elif im <= 3.4e+39:
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))))
	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(Float64(Float64(im * im) * 0.001388888888888889) + 0.041666666666666664))))))
	tmp = 0.0
	if (im <= 26.0)
		tmp = t_0;
	elseif (im <= 3.4e+39)
		tmp = Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.001388888888888889)))))));
	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) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
	tmp = 0.0;
	if (im <= 26.0)
		tmp = t_0;
	elseif (im <= 3.4e+39)
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))));
	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[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 26.0], t$95$0, If[LessEqual[im, 3.4e+39], N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(re * N[(re * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 26 or 3.3999999999999999e39 < 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 im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 96.1%

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

   9. Simplified 96.1%

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

   10. Taylor expanded in re around 0

       \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   11. 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 62.7%

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

   12. Simplified 62.7%

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

   if 26 < im < 3.3999999999999999e39

   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 3.6%

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

   7. Simplified 3.6%

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

   8. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 41.8%

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

   10. Simplified 41.8%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 26:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.1% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+65}:\\ \;\;\;\;\left(im \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00012)
   (* 0.5 (+ 2.0 (* im im)))
   (if (<= im 4.2e+65)
     (*
      (* im (+ 0.5 (* re (* re -0.25))))
      (* im (+ 1.0 (* im (* im 0.08333333333333333)))))
     (* 0.5 (* im (* im (* (* im im) 0.08333333333333333)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 4.2e+65) {
		tmp = (im * (0.5 + (re * (re * -0.25)))) * (im * (1.0 + (im * (im * 0.08333333333333333))));
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00012d0) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else if (im <= 4.2d+65) then
        tmp = (im * (0.5d0 + (re * (re * (-0.25d0))))) * (im * (1.0d0 + (im * (im * 0.08333333333333333d0))))
    else
        tmp = 0.5d0 * (im * (im * ((im * im) * 0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 4.2e+65) {
		tmp = (im * (0.5 + (re * (re * -0.25)))) * (im * (1.0 + (im * (im * 0.08333333333333333))));
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00012:
		tmp = 0.5 * (2.0 + (im * im))
	elif im <= 4.2e+65:
		tmp = (im * (0.5 + (re * (re * -0.25)))) * (im * (1.0 + (im * (im * 0.08333333333333333))))
	else:
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00012)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 4.2e+65)
		tmp = Float64(Float64(im * Float64(0.5 + Float64(re * Float64(re * -0.25)))) * Float64(im * Float64(1.0 + Float64(im * Float64(im * 0.08333333333333333)))));
	else
		tmp = Float64(0.5 * Float64(im * Float64(im * Float64(Float64(im * im) * 0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00012)
		tmp = 0.5 * (2.0 + (im * im));
	elseif (im <= 4.2e+65)
		tmp = (im * (0.5 + (re * (re * -0.25)))) * (im * (1.0 + (im * (im * 0.08333333333333333))));
	else
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00012], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e+65], N[(N[(im * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + N[(im * N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.20000000000000003e-4

   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 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 50.7%

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

   8. Simplified 50.7%

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

   if 1.20000000000000003e-4 < im < 4.19999999999999983e65

   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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 5.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 5.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \color{blue}{\frac{-1}{4} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]

      3. distribute-rgt-out N/A

         \[\leadsto \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

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

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

   8. Simplified 31.0%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   9. Taylor expanded in im around inf

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

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

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

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(im \cdot im\right) \cdot {im}^{2}\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot \left(im \cdot {im}^{2}\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \left(im \cdot {im}^{2}\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2}\right)\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot im\right)\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \left(\frac{1}{{im}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{/.f64}\left(1, \left({im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{/.f64}\left(1, \left(im \cdot im\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       13. *-lowering-*.f64 31.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

   11. Simplified 31.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.08333333333333333 + \frac{1}{im \cdot im}\right)\right)} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\frac{1}{12} + \frac{1}{im \cdot im}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot \left(\left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right) \cdot \left(\frac{1}{12} + \frac{1}{im \cdot im}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot \left(im \cdot \color{blue}{\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{12} + \frac{1}{im \cdot im}\right)\right)}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \color{blue}{\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{12} + \frac{1}{im \cdot im}\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{im \cdot im} + \color{blue}{\frac{1}{12}}\right)\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{im \cdot im} + \color{blue}{\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{12}}\right) \]

       7. cube-unmult N/A

          \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left({im}^{3} \cdot \frac{1}{im \cdot im} + \left(\color{blue}{im} \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{12}\right) \]

       8. inv-pow N/A

          \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left({im}^{3} \cdot {\left(im \cdot im\right)}^{-1} + \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \frac{1}{12}\right) \]

       9. pow2 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left({im}^{3} \cdot {\left({im}^{2}\right)}^{-1} + \left(im \cdot \left(\color{blue}{im} \cdot im\right)\right) \cdot \frac{1}{12}\right) \]

       10. pow-pow N/A

           \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left({im}^{3} \cdot {im}^{\left(2 \cdot -1\right)} + \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \frac{1}{12}\right) \]

       11. pow-prod-up N/A

           \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left({im}^{\left(3 + 2 \cdot -1\right)} + \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot \frac{1}{12}\right) \]

       12. metadata-eval N/A

           \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left({im}^{\left(3 + -2\right)} + \left(im \cdot \left(im \cdot \color{blue}{im}\right)\right) \cdot \frac{1}{12}\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left({im}^{1} + \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \frac{1}{12}\right) \]

       14. unpow1 N/A

           \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left(im + \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot \frac{1}{12}\right) \]

       15. *-rgt-identity N/A

           \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left(im \cdot 1 + \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot \frac{1}{12}\right) \]

       16. associate-*r* N/A

           \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left(im \cdot 1 + im \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \frac{1}{12}\right)}\right) \]

       17. distribute-lft-in N/A

           \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot im\right) \cdot \left(im \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \frac{1}{12}\right)}\right) \]

   13. Applied egg-rr 31.0%

       \[\leadsto \color{blue}{\left(\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot im\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   if 4.19999999999999983e65 < 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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 97.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 97.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

   8. Simplified 80.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \]

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{12} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{12}} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{12}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 80.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right) \]

   11. Simplified 80.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+65}:\\ \;\;\;\;\left(im \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.1% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+65}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00012)
   (* 0.5 (+ 2.0 (* im im)))
   (if (<= im 4.2e+65)
     (*
      (* (* im im) (* im im))
      (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))
     (* 0.5 (* im (* im (* (* im im) 0.08333333333333333)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 4.2e+65) {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00012d0) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else if (im <= 4.2d+65) then
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    else
        tmp = 0.5d0 * (im * (im * ((im * im) * 0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 4.2e+65) {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00012:
		tmp = 0.5 * (2.0 + (im * im))
	elif im <= 4.2e+65:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	else:
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00012)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 4.2e+65)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	else
		tmp = Float64(0.5 * Float64(im * Float64(im * Float64(Float64(im * im) * 0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00012)
		tmp = 0.5 * (2.0 + (im * im));
	elseif (im <= 4.2e+65)
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	else
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00012], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e+65], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.20000000000000003e-4

   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 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 50.7%

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

   8. Simplified 50.7%

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

   if 1.20000000000000003e-4 < im < 4.19999999999999983e65

   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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 5.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 5.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \color{blue}{\frac{-1}{4} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]

      3. distribute-rgt-out N/A

         \[\leadsto \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

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

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

   8. Simplified 31.0%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   9. Taylor expanded in im around inf

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

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

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

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(im \cdot im\right) \cdot {im}^{2}\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot \left(im \cdot {im}^{2}\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \left(im \cdot {im}^{2}\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2}\right)\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot im\right)\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \left(\frac{1}{{im}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{/.f64}\left(1, \left({im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{/.f64}\left(1, \left(im \cdot im\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       13. *-lowering-*.f64 31.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

   11. Simplified 31.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.08333333333333333 + \frac{1}{im \cdot im}\right)\right)} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]

   12. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{12} \cdot {im}^{4}\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{4} \cdot \frac{1}{12}\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{4} \cdot {re}^{2}\right) \]

       3. associate-*l* N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{12} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right)} \]

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

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{12} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

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

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

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

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{12} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{12} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{12} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{12} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right)\right) \]

       12. distribute-lft-in N/A

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

       13. metadata-eval N/A

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

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

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

       15. *-commutative N/A

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

       16. *-commutative N/A

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

       17. associate-*l* N/A

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

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

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

       21. unpow2 N/A

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

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

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

       23. metadata-eval 30.9%

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

   14. Simplified 30.9%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]

   if 4.19999999999999983e65 < 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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 97.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 97.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

   8. Simplified 80.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \]

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{12} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{12}} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{12}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 80.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right) \]

   11. Simplified 80.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.1% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+65}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + re \cdot \left(re \cdot -0.020833333333333332\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00012)
   (* 0.5 (+ 2.0 (* im im)))
   (if (<= im 4.2e+65)
     (*
      (* im im)
      (*
       (* im im)
       (+ 0.041666666666666664 (* re (* re -0.020833333333333332)))))
     (* 0.5 (* im (* im (* (* im im) 0.08333333333333333)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 4.2e+65) {
		tmp = (im * im) * ((im * im) * (0.041666666666666664 + (re * (re * -0.020833333333333332))));
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00012d0) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else if (im <= 4.2d+65) then
        tmp = (im * im) * ((im * im) * (0.041666666666666664d0 + (re * (re * (-0.020833333333333332d0)))))
    else
        tmp = 0.5d0 * (im * (im * ((im * im) * 0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 4.2e+65) {
		tmp = (im * im) * ((im * im) * (0.041666666666666664 + (re * (re * -0.020833333333333332))));
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00012:
		tmp = 0.5 * (2.0 + (im * im))
	elif im <= 4.2e+65:
		tmp = (im * im) * ((im * im) * (0.041666666666666664 + (re * (re * -0.020833333333333332))))
	else:
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00012)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 4.2e+65)
		tmp = Float64(Float64(im * im) * Float64(Float64(im * im) * Float64(0.041666666666666664 + Float64(re * Float64(re * -0.020833333333333332)))));
	else
		tmp = Float64(0.5 * Float64(im * Float64(im * Float64(Float64(im * im) * 0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00012)
		tmp = 0.5 * (2.0 + (im * im));
	elseif (im <= 4.2e+65)
		tmp = (im * im) * ((im * im) * (0.041666666666666664 + (re * (re * -0.020833333333333332))));
	else
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00012], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e+65], N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 + N[(re * N[(re * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.20000000000000003e-4

   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 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 50.7%

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

   8. Simplified 50.7%

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

   if 1.20000000000000003e-4 < im < 4.19999999999999983e65

   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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 5.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 5.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \color{blue}{\frac{-1}{4} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]

      3. distribute-rgt-out N/A

         \[\leadsto \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

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

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

   8. Simplified 31.0%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{12} \cdot {im}^{4}\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{12} \cdot {im}^{\left(2 \cdot 2\right)}\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{12} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(\frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{4} \cdot {re}^{2}\right) \]

       5. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{4} \cdot {re}^{2}\right) \]

       6. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{12} \cdot {im}^{2}\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right)} \]

       7. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \left(\frac{1}{12} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right)}\right) \]

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

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

       9. unpow2 N/A

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

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

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. *-commutative N/A

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

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

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

       15. unpow2 N/A

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

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

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

       17. distribute-rgt-in N/A

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

       18. metadata-eval N/A

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

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

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

       20. *-commutative N/A

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

       21. associate-*r* N/A

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

       22. *-commutative N/A

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

       23. unpow2 N/A

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

       24. associate-*l* N/A

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

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

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

   11. Simplified 30.9%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + re \cdot \left(re \cdot -0.020833333333333332\right)\right)\right)} \]

   if 4.19999999999999983e65 < 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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 97.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 97.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

   8. Simplified 80.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \]

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{12} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{12}} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{12}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 80.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right) \]

   11. Simplified 80.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 43.3% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+65}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.05e-61)
   1.0
   (if (<= im 4.2e+65)
     (* (+ 2.0 (* im im)) (+ 0.5 (* (* re re) -0.25)))
     (* 0.5 (* im (* im (* (* im im) 0.08333333333333333)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.05e-61) {
		tmp = 1.0;
	} else if (im <= 4.2e+65) {
		tmp = (2.0 + (im * im)) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.05d-61) then
        tmp = 1.0d0
    else if (im <= 4.2d+65) then
        tmp = (2.0d0 + (im * im)) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = 0.5d0 * (im * (im * ((im * im) * 0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.05e-61) {
		tmp = 1.0;
	} else if (im <= 4.2e+65) {
		tmp = (2.0 + (im * im)) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.05e-61:
		tmp = 1.0
	elif im <= 4.2e+65:
		tmp = (2.0 + (im * im)) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.05e-61)
		tmp = 1.0;
	elseif (im <= 4.2e+65)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = Float64(0.5 * Float64(im * Float64(im * Float64(Float64(im * im) * 0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.05e-61)
		tmp = 1.0;
	elseif (im <= 4.2e+65)
		tmp = (2.0 + (im * im)) * (0.5 + ((re * re) * -0.25));
	else
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.05e-61], 1.0, If[LessEqual[im, 4.2e+65], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.05e-61

   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 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 34.7%

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

   if 3.05e-61 < im < 4.19999999999999983e65

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(2 + {im}^{2}\right)}\right) \]
   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 30.1%

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

   5. Simplified 30.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 39.3%

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

   8. Simplified 39.3%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   if 4.19999999999999983e65 < 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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 97.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 97.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

   8. Simplified 80.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \]

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{12} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{12}} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{12}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 80.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right) \]

   11. Simplified 80.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+65}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.0% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+65}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00012)
   (* 0.5 (+ 2.0 (* im im)))
   (if (<= im 4.2e+65)
     (* (* im im) (+ 0.5 (* re (* re -0.25))))
     (* 0.5 (* im (* im (* (* im im) 0.08333333333333333)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 4.2e+65) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00012d0) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else if (im <= 4.2d+65) then
        tmp = (im * im) * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = 0.5d0 * (im * (im * ((im * im) * 0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 4.2e+65) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00012:
		tmp = 0.5 * (2.0 + (im * im))
	elif im <= 4.2e+65:
		tmp = (im * im) * (0.5 + (re * (re * -0.25)))
	else:
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00012)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 4.2e+65)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = Float64(0.5 * Float64(im * Float64(im * Float64(Float64(im * im) * 0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00012)
		tmp = 0.5 * (2.0 + (im * im));
	elseif (im <= 4.2e+65)
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	else
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00012], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e+65], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.20000000000000003e-4

   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 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 50.7%

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

   8. Simplified 50.7%

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

   if 1.20000000000000003e-4 < im < 4.19999999999999983e65

   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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 5.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 5.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \color{blue}{\frac{-1}{4} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]

      3. distribute-rgt-out N/A

         \[\leadsto \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

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

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

   8. Simplified 31.0%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   9. Taylor expanded in im around inf

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

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

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

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(im \cdot im\right) \cdot {im}^{2}\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot \left(im \cdot {im}^{2}\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \left(im \cdot {im}^{2}\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2}\right)\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot im\right)\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(\frac{1}{12} + \frac{1}{{im}^{2}}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \left(\frac{1}{{im}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{/.f64}\left(1, \left({im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{/.f64}\left(1, \left(im \cdot im\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

       13. *-lowering-*.f64 31.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right) \]

   11. Simplified 31.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.08333333333333333 + \frac{1}{im \cdot im}\right)\right)} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]

   12. Taylor expanded in im around 0

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

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

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

       2. unpow2 N/A

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

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

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

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

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

       9. *-lowering-*.f64 30.4%

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

   14. Simplified 30.4%

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

   if 4.19999999999999983e65 < 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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 97.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 97.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

   8. Simplified 80.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \]

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{12} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{12}} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{12}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 80.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right) \]

   11. Simplified 80.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 51.9% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+65}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00012)
   (* 0.5 (+ 2.0 (* im im)))
   (if (<= im 1.95e+65)
     (+ 1.0 (* (* re re) -0.5))
     (* 0.5 (* im (* im (* (* im im) 0.08333333333333333)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 1.95e+65) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00012d0) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else if (im <= 1.95d+65) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    else
        tmp = 0.5d0 * (im * (im * ((im * im) * 0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 1.95e+65) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00012:
		tmp = 0.5 * (2.0 + (im * im))
	elif im <= 1.95e+65:
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00012)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.95e+65)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	else
		tmp = Float64(0.5 * Float64(im * Float64(im * Float64(Float64(im * im) * 0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00012)
		tmp = 0.5 * (2.0 + (im * im));
	elseif (im <= 1.95e+65)
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = 0.5 * (im * (im * ((im * im) * 0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00012], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.95e+65], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.20000000000000003e-4

   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 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 50.7%

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

   8. Simplified 50.7%

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

   if 1.20000000000000003e-4 < im < 1.9499999999999999e65

   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 4.0%

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

   7. Simplified 4.0%

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

   8. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 17.7%

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

   10. Simplified 17.7%

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

   if 1.9499999999999999e65 < 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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 97.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 97.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

   8. Simplified 80.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \]

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{12} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{12} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{12}} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{12}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 80.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right) \]

   11. Simplified 80.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 51.9% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+65}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00012)
   (* 0.5 (+ 2.0 (* im im)))
   (if (<= im 3.6e+65)
     (+ 1.0 (* (* re re) -0.5))
     (* im (* im (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 3.6e+65) {
		tmp = 1.0 + ((re * re) * -0.5);
	} 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 <= 0.00012d0) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else if (im <= 3.6d+65) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    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 <= 0.00012) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 3.6e+65) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = im * (im * ((im * im) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00012:
		tmp = 0.5 * (2.0 + (im * im))
	elif im <= 3.6e+65:
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = im * (im * ((im * im) * 0.041666666666666664))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00012)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 3.6e+65)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	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 <= 0.00012)
		tmp = 0.5 * (2.0 + (im * im));
	elseif (im <= 3.6e+65)
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = im * (im * ((im * im) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00012], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.6e+65], 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]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.20000000000000003e-4

   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 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 50.7%

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

   8. Simplified 50.7%

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

   if 1.20000000000000003e-4 < im < 3.59999999999999978e65

   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 4.0%

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

   7. Simplified 4.0%

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

   8. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 17.7%

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

   10. Simplified 17.7%

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

   if 3.59999999999999978e65 < 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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\frac{1}{12} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{12} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 97.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 97.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

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

   8. Simplified 80.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 78.7%

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

   11. Simplified 78.7%

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

Alternative 20: 58.4% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ 1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (+
  1.0
  (*
   im
   (*
    im
    (+
     0.5
     (*
      (* im im)
      (+ (* (* im im) 0.001388888888888889) 0.041666666666666664)))))))

C

double code(double re, double im) {
	return 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 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) * (((im * im) * 0.001388888888888889d0) + 0.041666666666666664d0)))))
end function

Java

public static double code(double re, double im) {
	return 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
}

Python

def code(re, im):
	return 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))

Julia

function code(re, im)
	return Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * Float64(Float64(Float64(im * im) * 0.001388888888888889) + 0.041666666666666664))))))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + (im * (im * (0.5 + ((im * im) * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))));
end

Wolfram

code[re_, im_] := N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. 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 im around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 90.8%

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

9. Simplified 90.8%

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

10. Taylor expanded in re around 0

    \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
11. 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 59.1%

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

12. Simplified 59.1%

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

13. Final simplification 59.1%

    \[\leadsto 1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right) \]
14. Add Preprocessing

Alternative 21: 48.0% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.3 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.3e+85)
   (* 0.5 (+ 2.0 (* im im)))
   (* 0.041666666666666664 (* re (* re (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.3e+85) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = 0.041666666666666664 * (re * (re * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.3d+85) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else
        tmp = 0.041666666666666664d0 * (re * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.3e+85) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = 0.041666666666666664 * (re * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.3e+85:
		tmp = 0.5 * (2.0 + (im * im))
	else:
		tmp = 0.041666666666666664 * (re * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.3e+85)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(re * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.3e+85)
		tmp = 0.5 * (2.0 + (im * im));
	else
		tmp = 0.041666666666666664 * (re * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.3e+85], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(re * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.30000000000000005e85

   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 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 53.3%

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

   8. Simplified 53.3%

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

   if 1.30000000000000005e85 < re

   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 48.4%

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

   7. Simplified 48.4%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 28.0%

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

   10. Simplified 28.0%

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

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{24} \cdot {re}^{4}} \]
   12. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left(re \cdot {re}^{\color{blue}{3}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{3}\right)}\right)\right) \]

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 28.0%

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

   13. Simplified 28.0%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 48.3% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.25 \cdot 10^{+186}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.25e+186) (* 0.5 (+ 2.0 (* im im))) (+ 1.0 (* (* re re) -0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.25e+186) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 4.25d+186) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.25e+186) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.25e+186:
		tmp = 0.5 * (2.0 + (im * im))
	else:
		tmp = 1.0 + ((re * re) * -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.25e+186)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.25e+186)
		tmp = 0.5 * (2.0 + (im * im));
	else
		tmp = 1.0 + ((re * re) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.25e+186], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.2499999999999999e186

   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 75.3%

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

   5. Simplified 75.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 50.4%

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

   8. Simplified 50.4%

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

   if 4.2499999999999999e186 < re

   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 56.9%

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

   7. Simplified 56.9%

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

   8. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 27.0%

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

   10. Simplified 27.0%

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

Alternative 23: 38.5% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 0.00012) 1.0 (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		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 <= 0.00012d0) 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 <= 0.00012) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00012:
		tmp = 1.0
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00012)
		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 <= 0.00012)
		tmp = 1.0;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00012], 1.0, N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.20000000000000003e-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 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 35.8%

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

   if 1.20000000000000003e-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 69.9%

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

   10. Taylor expanded in im around 0

       \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   11. 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 32.6%

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

   12. Simplified 32.6%

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

   13. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {im}^{2}} \]
   14. 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 32.6%

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

   15. Simplified 32.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 47.3% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. 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 75.3%

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

5. Simplified 75.3%

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

6. Taylor expanded in re around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(2 + {im}^{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(2, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

   4. *-lowering-*.f64 46.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

8. Simplified 46.2%

   \[\leadsto \color{blue}{0.5 \cdot \left(2 + im \cdot im\right)} \]
9. Add Preprocessing

Alternative 25: 29.0% 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 49.1%

      \[\leadsto \mathsf{cos.f64}\left(re\right) \]

7. Simplified 49.1%

   \[\leadsto \color{blue}{\cos re} \]

8. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 27.6%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))