math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.4s

Alternatives: 17

Speedup: 1.5×

• 49.9% 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

\[\begin{array}{l} \\ \frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* 0.5 (cos re)) (/ 1.0 (* 2.0 (cosh im)))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) / (1.0 / (2.0 * cosh(im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) / (1.0d0 / (2.0d0 * cosh(im)))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) / (1.0 / (2.0 * Math.cosh(im)));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) / (1.0 / (2.0 * math.cosh(im)))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) / Float64(1.0 / Float64(2.0 * cosh(im))))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) / (1.0 / (2.0 * cosh(im)));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. flip3-+ N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}{\color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}} \]

   2. clear-num N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{1}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

   3. un-div-inv N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \cos re}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \cos re\right), \color{blue}{\left(\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)}\right) \]

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

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

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

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

   7. clear-num N/A

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

   8. flip3-+ N/A

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}}} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 1.0)
   (*
    (cos re)
    (+
     1.0
     (*
      im
      (*
       im
       (+
        0.5
        (*
         im
         (*
          im
          (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))))))))
   (cosh im)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 1.0) {
		tmp = cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))));
	} 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 (cos(re) <= 1.0d0) then
        tmp = cos(re) * (1.0d0 + (im * (im * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0))))))))
    else
        tmp = cosh(im)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 1

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}, \mathsf{cos.f64}\left(re\right)\right) \]
   6. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 95.3%

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

   7. Simplified 95.3%

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

   if 1 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. flip3-+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}{\color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}} \]

      2. clear-num N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{1}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \cos re}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \cos re\right), \color{blue}{\left(\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)}\right) \]

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

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

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

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

      7. clear-num N/A

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

      8. flip3-+ N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. /-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\cos re \cdot 1\right) \cdot \cosh im \]

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 59.6%

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

       1. *-rgt-identity N/A

          \[\leadsto \cosh im \]

       2. cosh-lowering-cosh.f64 59.6%

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

   11. Applied egg-rr 59.6%

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

4. Final simplification 95.3%

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

Alternative 3: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. flip3-+ N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}{\color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}} \]

   2. clear-num N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{1}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

   3. un-div-inv N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \cos re}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \cos re\right), \color{blue}{\left(\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)}\right) \]

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

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

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

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

   7. clear-num N/A

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

   8. flip3-+ N/A

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}}} \]
5. Step-by-step derivation

   1. associate-/r/ N/A

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

   2. /-rgt-identity N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

      \[\leadsto \left(\cos re \cdot 1\right) \cdot \cosh im \]

   7. *-rgt-identity N/A

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

   8. *-commutative N/A

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

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

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

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

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

   11. cos-lowering-cos.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 4: 92.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 27.5:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.041666666666666664\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 27.5)
   (*
    (cos re)
    (+ 1.0 (* (* im im) (+ 0.5 (* 0.041666666666666664 (* im im))))))
   (if (<= im 1.15e+77)
     (* (cosh im) (+ 1.0 (* (* re re) -0.5)))
     (* (* (cos re) 0.041666666666666664) (* (* im im) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 27.5) {
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))));
	} else if (im <= 1.15e+77) {
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = (cos(re) * 0.041666666666666664) * ((im * im) * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 27.5d0) then
        tmp = cos(re) * (1.0d0 + ((im * im) * (0.5d0 + (0.041666666666666664d0 * (im * im)))))
    else if (im <= 1.15d+77) then
        tmp = cosh(im) * (1.0d0 + ((re * re) * (-0.5d0)))
    else
        tmp = (cos(re) * 0.041666666666666664d0) * ((im * im) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 27.5) {
		tmp = Math.cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))));
	} else if (im <= 1.15e+77) {
		tmp = Math.cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = (Math.cos(re) * 0.041666666666666664) * ((im * im) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 27.5:
		tmp = math.cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))))
	elif im <= 1.15e+77:
		tmp = math.cosh(im) * (1.0 + ((re * re) * -0.5))
	else:
		tmp = (math.cos(re) * 0.041666666666666664) * ((im * im) * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 27.5)
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))));
	elseif (im <= 1.15e+77)
		tmp = Float64(cosh(im) * Float64(1.0 + Float64(Float64(re * re) * -0.5)));
	else
		tmp = Float64(Float64(cos(re) * 0.041666666666666664) * Float64(Float64(im * im) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 27.5)
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (0.041666666666666664 * (im * im)))));
	elseif (im <= 1.15e+77)
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	else
		tmp = (cos(re) * 0.041666666666666664) * ((im * im) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 27.5], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(N[Cosh[im], $MachinePrecision] * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 27.5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 94.6%

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

   if 27.5 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]
   6. 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 70.0%

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

   7. Simplified 70.0%

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

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 27.5:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.041666666666666664\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00325:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.041666666666666664\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00325)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 1.15e+77)
     (* (cosh im) (+ 1.0 (* (* re re) -0.5)))
     (* (* (cos re) 0.041666666666666664) (* (* im im) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00325) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = (cos(re) * 0.041666666666666664) * ((im * im) * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00325d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else if (im <= 1.15d+77) then
        tmp = cosh(im) * (1.0d0 + ((re * re) * (-0.5d0)))
    else
        tmp = (cos(re) * 0.041666666666666664d0) * ((im * im) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00325) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = Math.cosh(im) * (1.0 + ((re * re) * -0.5));
	} else {
		tmp = (Math.cos(re) * 0.041666666666666664) * ((im * im) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00325:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	elif im <= 1.15e+77:
		tmp = math.cosh(im) * (1.0 + ((re * re) * -0.5))
	else:
		tmp = (math.cos(re) * 0.041666666666666664) * ((im * im) * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00325)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.15e+77)
		tmp = Float64(cosh(im) * Float64(1.0 + Float64(Float64(re * re) * -0.5)));
	else
		tmp = Float64(Float64(cos(re) * 0.041666666666666664) * Float64(Float64(im * im) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00325)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 1.15e+77)
		tmp = cosh(im) * (1.0 + ((re * re) * -0.5));
	else
		tmp = (cos(re) * 0.041666666666666664) * ((im * im) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00325], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(N[Cosh[im], $MachinePrecision] * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00324999999999999985

   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 83.3%

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

   5. Simplified 83.3%

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

   if 0.00324999999999999985 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]
   6. 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 67.2%

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

   7. Simplified 67.2%

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

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00325:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.041666666666666664\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00325 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.00325) (not (<= im 1.35e+154)))
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (cosh im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.00325) || !(im <= 1.35e+154)) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} 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 <= 0.00325d0) .or. (.not. (im <= 1.35d+154))) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else
        tmp = cosh(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 0.00325) || !(im <= 1.35e+154)) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else {
		tmp = Math.cosh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.00325) or not (im <= 1.35e+154):
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	else:
		tmp = math.cosh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.00325) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = cosh(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 0.00325) || ~((im <= 1.35e+154)))
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	else
		tmp = cosh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.00325], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cosh[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.00324999999999999985 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 86.1%

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

   5. Simplified 86.1%

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

   if 0.00324999999999999985 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. flip3-+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}{\color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}} \]

      2. clear-num N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{1}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \cos re}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \cos re\right), \color{blue}{\left(\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)}\right) \]

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

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

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

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

      7. clear-num N/A

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

      8. flip3-+ N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. /-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\cos re \cdot 1\right) \cdot \cosh im \]

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 82.8%

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

       1. *-rgt-identity N/A

          \[\leadsto \cosh im \]

       2. cosh-lowering-cosh.f64 82.8%

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

   11. Applied egg-rr 82.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00325 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0018:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+132}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0018)
   (cos re)
   (if (<= im 3e+132)
     (cosh im)
     (*
      (* (* im im) (* im im))
      (+ 0.041666666666666664 (* (* re re) -0.020833333333333332))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0018) {
		tmp = cos(re);
	} else if (im <= 3e+132) {
		tmp = cosh(im);
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0018d0) then
        tmp = cos(re)
    else if (im <= 3d+132) then
        tmp = cosh(im)
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0018) {
		tmp = Math.cos(re);
	} else if (im <= 3e+132) {
		tmp = Math.cosh(im);
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0018:
		tmp = math.cos(re)
	elif im <= 3e+132:
		tmp = math.cosh(im)
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0018)
		tmp = cos(re);
	elseif (im <= 3e+132)
		tmp = cosh(im);
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0018)
		tmp = cos(re);
	elseif (im <= 3e+132)
		tmp = cosh(im);
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0018], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3e+132], N[Cosh[im], $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0018

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 67.1%

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

   5. Simplified 67.1%

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

   if 0.0018 < im < 2.9999999999999998e132

   1. Initial program 100.0%

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

      1. flip3-+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}{\color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}} \]

      2. clear-num N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{1}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \cos re}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \cos re\right), \color{blue}{\left(\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)}\right) \]

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

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

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

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

      7. clear-num N/A

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

      8. flip3-+ N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. /-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\cos re \cdot 1\right) \cdot \cosh im \]

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 84.7%

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

       1. *-rgt-identity N/A

          \[\leadsto \cosh im \]

       2. cosh-lowering-cosh.f64 84.7%

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

   11. Applied egg-rr 84.7%

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

   if 2.9999999999999998e132 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

   8. Simplified 75.6%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. associate-*r* N/A

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

      7. cube-unmult N/A

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

      8. *-commutative N/A

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

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

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

      10. cube-unmult N/A

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

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

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

      12. *-lowering-*.f64 75.6%

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

   10. Applied egg-rr 75.6%

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

   11. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

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

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

       12. distribute-rgt-in N/A

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

       13. metadata-eval N/A

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

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

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. metadata-eval N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 75.6%

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

   13. Simplified 75.6%

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

Alternative 8: 66.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00325:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(1 + \frac{\left(im \cdot im\right) \cdot \left(0.125 + 7.233796296296296 \cdot 10^{-5} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)}{0.25}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00325)
   (cos re)
   (*
    (+ 1.0 (* (* re re) -0.5))
    (+
     1.0
     (/
      (*
       (* im im)
       (+
        0.125
        (* 7.233796296296296e-5 (* (* im im) (* (* im im) (* im im))))))
      0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00325) {
		tmp = cos(re);
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (1.0 + (((im * im) * (0.125 + (7.233796296296296e-5 * ((im * im) * ((im * im) * (im * im)))))) / 0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00325d0) then
        tmp = cos(re)
    else
        tmp = (1.0d0 + ((re * re) * (-0.5d0))) * (1.0d0 + (((im * im) * (0.125d0 + (7.233796296296296d-5 * ((im * im) * ((im * im) * (im * im)))))) / 0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00325) {
		tmp = Math.cos(re);
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (1.0 + (((im * im) * (0.125 + (7.233796296296296e-5 * ((im * im) * ((im * im) * (im * im)))))) / 0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00325:
		tmp = math.cos(re)
	else:
		tmp = (1.0 + ((re * re) * -0.5)) * (1.0 + (((im * im) * (0.125 + (7.233796296296296e-5 * ((im * im) * ((im * im) * (im * im)))))) / 0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00325)
		tmp = cos(re);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * -0.5)) * Float64(1.0 + Float64(Float64(Float64(im * im) * Float64(0.125 + Float64(7.233796296296296e-5 * Float64(Float64(im * im) * Float64(Float64(im * im) * Float64(im * im)))))) / 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00325)
		tmp = cos(re);
	else
		tmp = (1.0 + ((re * re) * -0.5)) * (1.0 + (((im * im) * (0.125 + (7.233796296296296e-5 * ((im * im) * ((im * im) * (im * im)))))) / 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00325], N[Cos[re], $MachinePrecision], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(im * im), $MachinePrecision] * N[(0.125 + N[(7.233796296296296e-5 * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.00324999999999999985

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 67.1%

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

   5. Simplified 67.1%

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

   if 0.00324999999999999985 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 84.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

   8. Simplified 63.2%

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

      1. associate-*r* N/A

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

      2. flip3-+ N/A

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

      3. associate-*r/ N/A

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

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

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

   10. Applied egg-rr 7.7%

       \[\leadsto \left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(1 + \color{blue}{\frac{\left(im \cdot im\right) \cdot \left(0.125 + 7.233796296296296 \cdot 10^{-5} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)}{0.25 + \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right) - 0.5\right)}}\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{1}{13824}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right), \color{blue}{\frac{1}{4}}\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 65.7%

       \[\leadsto \left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(1 + \frac{\left(im \cdot im\right) \cdot \left(0.125 + 7.233796296296296 \cdot 10^{-5} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)}{\color{blue}{0.25}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 58.8% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4 \cdot 10^{+132}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4e+132)
   (+
    1.0
    (*
     (* im im)
     (+
      0.5
      (*
       im
       (* im (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))))))
   (*
    (* (* im im) (* im im))
    (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4e+132) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4d+132) then
        tmp = 1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0))))))
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4e+132) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4e+132:
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4e+132)
		tmp = Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)))))));
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4e+132)
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4e+132], N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.99999999999999996e132

   1. Initial program 100.0%

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

      1. flip3-+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}{\color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}} \]

      2. clear-num N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{1}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \cos re}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \cos re\right), \color{blue}{\left(\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)}\right) \]

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

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

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

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

      7. clear-num N/A

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

      8. flip3-+ N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. /-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\cos re \cdot 1\right) \cdot \cosh im \]

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 57.5%

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

   10. Taylor expanded in im around 0

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

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

       3. unpow2 N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 54.0%

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

   12. Simplified 54.0%

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

   if 3.99999999999999996e132 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

   8. Simplified 75.6%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. associate-*r* N/A

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

      7. cube-unmult N/A

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

      8. *-commutative N/A

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

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

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

      10. cube-unmult N/A

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

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

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

      12. *-lowering-*.f64 75.6%

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

   10. Applied egg-rr 75.6%

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

   11. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

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

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

       12. distribute-rgt-in N/A

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

       13. metadata-eval N/A

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

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

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. metadata-eval N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 75.6%

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

   13. Simplified 75.6%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 56.0% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.65 \cdot 10^{+132}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.65e+132)
   (+ 1.0 (* im (* im (+ 0.5 (* 0.041666666666666664 (* im im))))))
   (*
    (* (* im im) (* im im))
    (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.65e+132) {
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.65d+132) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (0.041666666666666664d0 * (im * im)))))
    else
        tmp = ((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.65e+132) {
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	} else {
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.65e+132:
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))))
	else:
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.65e+132)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))));
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.65e+132)
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	else
		tmp = ((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.65e+132], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.65e132

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 90.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 51.7%

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

   8. Simplified 51.7%

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

   if 2.65e132 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

   8. Simplified 75.6%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. associate-*r* N/A

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

      7. cube-unmult N/A

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

      8. *-commutative N/A

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

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

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

      10. cube-unmult N/A

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

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

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

      12. *-lowering-*.f64 75.6%

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

   10. Applied egg-rr 75.6%

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

   11. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

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

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

       12. distribute-rgt-in N/A

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

       13. metadata-eval N/A

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

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

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. metadata-eval N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 75.6%

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

   13. Simplified 75.6%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.65 \cdot 10^{+132}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.9% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+142}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+142)
   (+ 1.0 (* im (* im (+ 0.5 (* 0.041666666666666664 (* im im))))))
   (*
    (* im im)
    (*
     (* im im)
     (+ 0.041666666666666664 (* (* re re) -0.020833333333333332))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+142) {
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	} else {
		tmp = (im * im) * ((im * im) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.1d+142) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (0.041666666666666664d0 * (im * im)))))
    else
        tmp = (im * im) * ((im * im) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+142) {
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	} else {
		tmp = (im * im) * ((im * im) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+142:
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))))
	else:
		tmp = (im * im) * ((im * im) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+142)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))));
	else
		tmp = Float64(Float64(im * im) * Float64(Float64(im * im) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+142)
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	else
		tmp = (im * im) * ((im * im) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+142], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.1e142

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 90.9%

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

   6. Taylor expanded in re around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 52.2%

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

   8. Simplified 52.2%

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

   if 2.1e142 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

   8. Simplified 76.9%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

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

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

       15. unpow2 N/A

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

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

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

       17. distribute-lft-in N/A

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

       18. metadata-eval N/A

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

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

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

       20. *-commutative N/A

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

       21. *-commutative N/A

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

       22. associate-*l* N/A

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

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

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

   11. Simplified 76.9%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+142}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.5% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9.8 \cdot 10^{+204}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9.8e+204)
   (+ 1.0 (* im (* im (+ 0.5 (* 0.041666666666666664 (* im im))))))
   (* (* re re) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9.8e+204) {
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	} else {
		tmp = (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 <= 9.8d+204) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (0.041666666666666664d0 * (im * im)))))
    else
        tmp = (re * re) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9.8e+204) {
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	} else {
		tmp = (re * re) * -0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9.8e+204:
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))))
	else:
		tmp = (re * re) * -0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9.8e+204)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(0.041666666666666664 * Float64(im * im))))));
	else
		tmp = Float64(Float64(re * re) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9.8e+204)
		tmp = 1.0 + (im * (im * (0.5 + (0.041666666666666664 * (im * im)))));
	else
		tmp = (re * re) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9.8e+204], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.7999999999999995e204

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 91.6%

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

   6. Taylor expanded in re around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 57.8%

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

   8. Simplified 57.8%

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

   if 9.7999999999999995e204 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 45.0%

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

   5. Simplified 45.0%

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

   6. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. *-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 37.3%

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

   8. Simplified 37.3%

      \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot -0.5} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\frac{-1}{2}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right) \]

       4. *-lowering-*.f64 37.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right) \]

   11. Simplified 37.3%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9.8 \cdot 10^{+204}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.3% accurate, 19.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9.8 \cdot 10^{+204}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9.8e+204)
   (+ 1.0 (* im (* im (* 0.041666666666666664 (* im im)))))
   (* (* re re) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9.8e+204) {
		tmp = 1.0 + (im * (im * (0.041666666666666664 * (im * im))));
	} else {
		tmp = (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 <= 9.8d+204) then
        tmp = 1.0d0 + (im * (im * (0.041666666666666664d0 * (im * im))))
    else
        tmp = (re * re) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9.8e+204) {
		tmp = 1.0 + (im * (im * (0.041666666666666664 * (im * im))));
	} else {
		tmp = (re * re) * -0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9.8e+204:
		tmp = 1.0 + (im * (im * (0.041666666666666664 * (im * im))))
	else:
		tmp = (re * re) * -0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9.8e+204)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.041666666666666664 * Float64(im * im)))));
	else
		tmp = Float64(Float64(re * re) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9.8e+204)
		tmp = 1.0 + (im * (im * (0.041666666666666664 * (im * im))));
	else
		tmp = (re * re) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9.8e+204], N[(1.0 + N[(im * N[(im * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.7999999999999995e204

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 91.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}, 1\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)}\right), 1\right)\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right), 1\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right), 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right), 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\left(im \cdot im\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right), 1\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(im \cdot \left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right), 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right), 1\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right), 1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right), 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right), 1\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right), 1\right)\right) \]

      12. *-lowering-*.f64 90.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right), 1\right)\right) \]

   8. Simplified 90.7%

      \[\leadsto \cos re \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} + 1\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 57.4%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   11. Simplified 57.4%

       \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   if 9.7999999999999995e204 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 45.0%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 45.0%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      3. *-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 37.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right) \]

   8. Simplified 37.3%

      \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot -0.5} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\frac{-1}{2}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right) \]

       4. *-lowering-*.f64 37.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right) \]

   11. Simplified 37.3%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9.8 \cdot 10^{+204}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.5% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+173}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.6e+173) (+ 1.0 (* 0.5 (* im im))) (+ 1.0 (* (* re re) -0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.6e+173) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.6d+173) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.6e+173) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.6e+173:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = 1.0 + ((re * re) * -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.6e+173)
		tmp = Float64(1.0 + Float64(0.5 * 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 <= 1.6e+173)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = 1.0 + ((re * re) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.6e+173], N[(1.0 + N[(0.5 * 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 < 1.6000000000000001e173

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}{\color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}} \]

      2. clear-num N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{1}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \cos re}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \cos re\right), \color{blue}{\left(\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \cos re\right), \left(\frac{\color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \left(\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \left(\frac{1}{\color{blue}{\frac{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}}}\right)\right) \]

      8. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \left(\frac{1}{e^{\mathsf{neg}\left(im\right)} + \color{blue}{e^{im}}}\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \cos re}{1} \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

      2. /-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{2} \cdot \cosh im\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot 2\right) \cdot \color{blue}{\cosh im} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\cos re \cdot \frac{1}{2}\right) \cdot 2\right) \cdot \cosh im \]

      5. associate-*l* N/A

         \[\leadsto \left(\cos re \cdot \left(\frac{1}{2} \cdot 2\right)\right) \cdot \cosh \color{blue}{im} \]

      6. metadata-eval N/A

         \[\leadsto \left(\cos re \cdot 1\right) \cdot \cosh im \]

      7. *-rgt-identity N/A

         \[\leadsto \cos re \cdot \cosh \color{blue}{im} \]

      8. *-commutative N/A

         \[\leadsto \cosh im \cdot \color{blue}{\cos re} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cosh im, \color{blue}{\cos re}\right) \]

      10. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \cos \color{blue}{re}\right) \]

      11. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{cos.f64}\left(re\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \color{blue}{1}\right) \]
   8. Step-by-step derivation

   9. Simplified 64.2%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   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 44.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   12. Simplified 44.7%

       \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   if 1.6000000000000001e173 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 42.3%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 42.3%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      3. *-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 43.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right) \]

   8. Simplified 43.2%

      \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 48.5% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+173}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.6e+173) (+ 1.0 (* 0.5 (* im im))) (* (* re re) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.6e+173) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (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 <= 1.6d+173) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = (re * re) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.6e+173) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (re * re) * -0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.6e+173:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = (re * re) * -0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.6e+173)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(Float64(re * re) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.6e+173)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = (re * re) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.6e+173], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.6000000000000001e173

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}{\color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}} \]

      2. clear-num N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \frac{1}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \cos re}{\color{blue}{\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \cos re\right), \color{blue}{\left(\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \cos re\right), \left(\frac{\color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \left(\frac{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}}{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \left(\frac{1}{\color{blue}{\frac{{\left(e^{\mathsf{neg}\left(im\right)}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{\mathsf{neg}\left(im\right)} \cdot e^{\mathsf{neg}\left(im\right)} + \left(e^{im} \cdot e^{im} - e^{\mathsf{neg}\left(im\right)} \cdot e^{im}\right)}}}\right)\right) \]

      8. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \left(\frac{1}{e^{\mathsf{neg}\left(im\right)} + \color{blue}{e^{im}}}\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \cos re}{1} \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

      2. /-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{2} \cdot \cosh im\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot 2\right) \cdot \color{blue}{\cosh im} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\cos re \cdot \frac{1}{2}\right) \cdot 2\right) \cdot \cosh im \]

      5. associate-*l* N/A

         \[\leadsto \left(\cos re \cdot \left(\frac{1}{2} \cdot 2\right)\right) \cdot \cosh \color{blue}{im} \]

      6. metadata-eval N/A

         \[\leadsto \left(\cos re \cdot 1\right) \cdot \cosh im \]

      7. *-rgt-identity N/A

         \[\leadsto \cos re \cdot \cosh \color{blue}{im} \]

      8. *-commutative N/A

         \[\leadsto \cosh im \cdot \color{blue}{\cos re} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cosh im, \color{blue}{\cos re}\right) \]

      10. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \cos \color{blue}{re}\right) \]

      11. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \mathsf{cos.f64}\left(re\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(im\right), \color{blue}{1}\right) \]
   8. Step-by-step derivation

   9. Simplified 64.2%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   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 44.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   12. Simplified 44.7%

       \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   if 1.6000000000000001e173 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 42.3%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 42.3%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      3. *-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 43.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right) \]

   8. Simplified 43.2%

      \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot -0.5} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\frac{-1}{2}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right) \]

       4. *-lowering-*.f64 43.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right) \]

   11. Simplified 43.2%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 31.3% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 7e+19) 1.0 (* (* re re) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7e+19) {
		tmp = 1.0;
	} else {
		tmp = (re * re) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7d+19) then
        tmp = 1.0d0
    else
        tmp = (re * re) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7e+19) {
		tmp = 1.0;
	} else {
		tmp = (re * re) * -0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7e+19:
		tmp = 1.0
	else:
		tmp = (re * re) * -0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7e+19)
		tmp = 1.0;
	else
		tmp = Float64(Float64(re * re) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7e+19)
		tmp = 1.0;
	else
		tmp = (re * re) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7e+19], 1.0, N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7e19

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 65.8%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 65.8%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 31.4%

      \[\leadsto \color{blue}{1} \]

   if 7e19 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      3. *-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 16.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right) \]

   8. Simplified 16.8%

      \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot -0.5} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\frac{-1}{2}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right) \]

       4. *-lowering-*.f64 15.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right) \]

   11. Simplified 15.9%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 28.4% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\cos re} \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 50.6%

      \[\leadsto \mathsf{cos.f64}\left(re\right) \]

5. Simplified 50.6%

   \[\leadsto \color{blue}{\cos re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 24.4%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))