math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 13.3s

Alternatives: 17

Speedup: 1.5×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. cosh-undef N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

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

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

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

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

   11. cos-lowering-cos.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.99998:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.99998)
   (*
    (* (cos re) 0.5)
    (+
     2.0
     (*
      (* im im)
      (+
       1.0
       (*
        im
        (* im (+ 0.08333333333333333 (* (* im im) 0.002777777777777778))))))))
   (cosh im)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.99998) {
		tmp = (cos(re) * 0.5) * (2.0 + ((im * im) * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))));
	} 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) <= 0.99998d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + ((im * im) * (1.0d0 + (im * (im * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0)))))))
    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) <= 0.99998) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + ((im * im) * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))));
	} else {
		tmp = Math.cosh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.99998:
		tmp = (math.cos(re) * 0.5) * (2.0 + ((im * im) * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))
	else:
		tmp = math.cosh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.99998)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(Float64(im * im) * Float64(1.0 + Float64(im * Float64(im * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778))))))));
	else
		tmp = cosh(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.99998)
		tmp = (cos(re) * 0.5) * (2.0 + ((im * im) * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))));
	else
		tmp = cosh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.99998], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(N[(im * im), $MachinePrecision] * N[(1.0 + N[(im * N[(im * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cosh[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.99997999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot im\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{12}, \left({im}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 92.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 92.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)} \]

   if 0.99997999999999998 < (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. *-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

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

   7. Simplified 100.0%

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

      1. *-rgt-identity N/A

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

      2. *-lft-identity N/A

         \[\leadsto \cosh im \]

      3. cosh-lowering-cosh.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.99998:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+79}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+146}:\\ \;\;\;\;\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.5e-13)
   (cos re)
   (if (<= im 2e+79)
     (cosh im)
     (if (<= im 2.1e+146)
       (*
        (+ 2.0 (* im (* im (+ 1.0 (* (* im im) 0.08333333333333333)))))
        (+ 0.5 (* -0.25 (* re re))))
       (* (* (cos re) 0.5) (+ 2.0 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = cos(re);
	} else if (im <= 2e+79) {
		tmp = cosh(im);
	} else if (im <= 2.1e+146) {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = (cos(re) * 0.5) * (2.0 + (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 <= 9.5d-13) then
        tmp = cos(re)
    else if (im <= 2d+79) then
        tmp = cosh(im)
    else if (im <= 2.1d+146) then
        tmp = (2.0d0 + (im * (im * (1.0d0 + ((im * im) * 0.08333333333333333d0))))) * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = Math.cos(re);
	} else if (im <= 2e+79) {
		tmp = Math.cosh(im);
	} else if (im <= 2.1e+146) {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.5e-13:
		tmp = math.cos(re)
	elif im <= 2e+79:
		tmp = math.cosh(im)
	elif im <= 2.1e+146:
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.5e-13)
		tmp = cos(re);
	elseif (im <= 2e+79)
		tmp = cosh(im);
	elseif (im <= 2.1e+146)
		tmp = Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(Float64(im * im) * 0.08333333333333333))))) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.5e-13)
		tmp = cos(re);
	elseif (im <= 2e+79)
		tmp = cosh(im);
	elseif (im <= 2.1e+146)
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	else
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.5e-13], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2e+79], N[Cosh[im], $MachinePrecision], If[LessEqual[im, 2.1e+146], N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 9.49999999999999991e-13

   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 69.8%

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

   5. Simplified 69.8%

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

   if 9.49999999999999991e-13 < im < 1.99999999999999993e79

   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

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

   7. Simplified 78.9%

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

      1. *-rgt-identity N/A

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

      2. *-lft-identity N/A

         \[\leadsto \cosh im \]

      3. cosh-lowering-cosh.f64 78.9%

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

   9. Applied egg-rr 78.9%

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

   if 1.99999999999999993e79 < im < 2.1000000000000001e146

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

   8. Simplified 77.8%

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

   if 2.1000000000000001e146 < 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 97.1%

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

   5. Simplified 97.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+79}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+146}:\\ \;\;\;\;\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{-49}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e-49)
   (cosh im)
   (*
    (cos re)
    (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1e-49) {
		tmp = cosh(im);
	} else {
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1d-49) then
        tmp = cosh(im)
    else
        tmp = cos(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1e-49) {
		tmp = Math.cosh(im);
	} else {
		tmp = Math.cos(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1e-49:
		tmp = math.cosh(im)
	else:
		tmp = math.cos(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1e-49)
		tmp = cosh(im);
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1e-49)
		tmp = cosh(im);
	else
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1e-49], N[Cosh[im], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.99999999999999936e-50

   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

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

   7. Simplified 74.1%

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

      1. *-rgt-identity N/A

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

      2. *-lft-identity N/A

         \[\leadsto \cosh im \]

      3. cosh-lowering-cosh.f64 74.1%

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

   9. Applied egg-rr 74.1%

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

   if 9.99999999999999936e-50 < 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

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 93.0%

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

Alternative 5: 68.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+79}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.5e-13)
   (cos re)
   (if (<= im 3e+79)
     (cosh im)
     (*
      (+ 2.0 (* im (* im (+ 1.0 (* (* im im) 0.08333333333333333)))))
      (+ 0.5 (* -0.25 (* re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = cos(re);
	} else if (im <= 3e+79) {
		tmp = cosh(im);
	} else {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 9.5d-13) then
        tmp = cos(re)
    else if (im <= 3d+79) then
        tmp = cosh(im)
    else
        tmp = (2.0d0 + (im * (im * (1.0d0 + ((im * im) * 0.08333333333333333d0))))) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = Math.cos(re);
	} else if (im <= 3e+79) {
		tmp = Math.cosh(im);
	} else {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.5e-13:
		tmp = math.cos(re)
	elif im <= 3e+79:
		tmp = math.cosh(im)
	else:
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.5e-13)
		tmp = cos(re);
	elseif (im <= 3e+79)
		tmp = cosh(im);
	else
		tmp = Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(Float64(im * im) * 0.08333333333333333))))) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.5e-13)
		tmp = cos(re);
	elseif (im <= 3e+79)
		tmp = cosh(im);
	else
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.5e-13], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3e+79], N[Cosh[im], $MachinePrecision], N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 9.49999999999999991e-13

   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 69.8%

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

   5. Simplified 69.8%

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

   if 9.49999999999999991e-13 < im < 2.99999999999999974e79

   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

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

   7. Simplified 78.9%

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

      1. *-rgt-identity N/A

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

      2. *-lft-identity N/A

         \[\leadsto \cosh im \]

      3. cosh-lowering-cosh.f64 78.9%

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

   9. Applied egg-rr 78.9%

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

   if 2.99999999999999974e79 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

   8. Simplified 74.4%

      \[\leadsto \color{blue}{\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot im\right)\\ t_1 := im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\\ \mathbf{if}\;im \leq 27000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+32}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+51}:\\ \;\;\;\;1 + im \cdot \frac{0.125 \cdot t\_0 + t\_0 \cdot \left(t\_0 \cdot \left(t\_0 \cdot 7.233796296296296 \cdot 10^{-5}\right)\right)}{\left(im \cdot im\right) \cdot 0.25 + t\_1 \cdot \left(t\_1 - im \cdot 0.5\right)}\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+82}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im im))) (t_1 (* im (* (* im im) 0.041666666666666664))))
   (if (<= im 27000.0)
     (cos re)
     (if (<= im 7e+32)
       (+
        1.0
        (*
         (* re re)
         (+
          -0.5
          (*
           re
           (*
            re
            (+ 0.041666666666666664 (* (* re re) -0.001388888888888889)))))))
       (if (<= im 6.8e+51)
         (+
          1.0
          (*
           im
           (/
            (+ (* 0.125 t_0) (* t_0 (* t_0 (* t_0 7.233796296296296e-5))))
            (+ (* (* im im) 0.25) (* t_1 (- t_1 (* im 0.5)))))))
         (if (<= im 2e+82)
           (+
            1.0
            (*
             (* im im)
             (+
              0.5
              (*
               im
               (*
                im
                (+
                 0.041666666666666664
                 (* im (* im 0.001388888888888889))))))))
           (*
            (+ 2.0 (* im (* im (+ 1.0 (* (* im im) 0.08333333333333333)))))
            (+ 0.5 (* -0.25 (* re re))))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * im);
	double t_1 = im * ((im * im) * 0.041666666666666664);
	double tmp;
	if (im <= 27000.0) {
		tmp = cos(re);
	} else if (im <= 7e+32) {
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))));
	} else if (im <= 6.8e+51) {
		tmp = 1.0 + (im * (((0.125 * t_0) + (t_0 * (t_0 * (t_0 * 7.233796296296296e-5)))) / (((im * im) * 0.25) + (t_1 * (t_1 - (im * 0.5))))));
	} else if (im <= 2e+82) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	} else {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (im * im)
    t_1 = im * ((im * im) * 0.041666666666666664d0)
    if (im <= 27000.0d0) then
        tmp = cos(re)
    else if (im <= 7d+32) then
        tmp = 1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * (0.041666666666666664d0 + ((re * re) * (-0.001388888888888889d0)))))))
    else if (im <= 6.8d+51) then
        tmp = 1.0d0 + (im * (((0.125d0 * t_0) + (t_0 * (t_0 * (t_0 * 7.233796296296296d-5)))) / (((im * im) * 0.25d0) + (t_1 * (t_1 - (im * 0.5d0))))))
    else if (im <= 2d+82) then
        tmp = 1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))))))
    else
        tmp = (2.0d0 + (im * (im * (1.0d0 + ((im * im) * 0.08333333333333333d0))))) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * im);
	double t_1 = im * ((im * im) * 0.041666666666666664);
	double tmp;
	if (im <= 27000.0) {
		tmp = Math.cos(re);
	} else if (im <= 7e+32) {
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))));
	} else if (im <= 6.8e+51) {
		tmp = 1.0 + (im * (((0.125 * t_0) + (t_0 * (t_0 * (t_0 * 7.233796296296296e-5)))) / (((im * im) * 0.25) + (t_1 * (t_1 - (im * 0.5))))));
	} else if (im <= 2e+82) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	} else {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * im)
	t_1 = im * ((im * im) * 0.041666666666666664)
	tmp = 0
	if im <= 27000.0:
		tmp = math.cos(re)
	elif im <= 7e+32:
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))))
	elif im <= 6.8e+51:
		tmp = 1.0 + (im * (((0.125 * t_0) + (t_0 * (t_0 * (t_0 * 7.233796296296296e-5)))) / (((im * im) * 0.25) + (t_1 * (t_1 - (im * 0.5))))))
	elif im <= 2e+82:
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))))
	else:
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * im))
	t_1 = Float64(im * Float64(Float64(im * im) * 0.041666666666666664))
	tmp = 0.0
	if (im <= 27000.0)
		tmp = cos(re);
	elseif (im <= 7e+32)
		tmp = Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.001388888888888889)))))));
	elseif (im <= 6.8e+51)
		tmp = Float64(1.0 + Float64(im * Float64(Float64(Float64(0.125 * t_0) + Float64(t_0 * Float64(t_0 * Float64(t_0 * 7.233796296296296e-5)))) / Float64(Float64(Float64(im * im) * 0.25) + Float64(t_1 * Float64(t_1 - Float64(im * 0.5)))))));
	elseif (im <= 2e+82)
		tmp = Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889))))))));
	else
		tmp = Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(Float64(im * im) * 0.08333333333333333))))) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * im);
	t_1 = im * ((im * im) * 0.041666666666666664);
	tmp = 0.0;
	if (im <= 27000.0)
		tmp = cos(re);
	elseif (im <= 7e+32)
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))));
	elseif (im <= 6.8e+51)
		tmp = 1.0 + (im * (((0.125 * t_0) + (t_0 * (t_0 * (t_0 * 7.233796296296296e-5)))) / (((im * im) * 0.25) + (t_1 * (t_1 - (im * 0.5))))));
	elseif (im <= 2e+82)
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	else
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 27000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 7e+32], N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(re * N[(re * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.8e+51], N[(1.0 + N[(im * N[(N[(N[(0.125 * t$95$0), $MachinePrecision] + N[(t$95$0 * N[(t$95$0 * N[(t$95$0 * 7.233796296296296e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(im * im), $MachinePrecision] * 0.25), $MachinePrecision] + N[(t$95$1 * N[(t$95$1 - N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2e+82], N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 27000

   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 69.5%

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

   5. Simplified 69.5%

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

   if 27000 < im < 7.0000000000000002e32

   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

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 51.6%

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

   8. Simplified 51.6%

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

   if 7.0000000000000002e32 < im < 6.79999999999999969e51

   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. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 5.3%

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

   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 5.3%

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

   8. Simplified 5.3%

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

      1. distribute-rgt-in N/A

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

      2. flip3-+ N/A

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

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

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

   10. Applied egg-rr 100.0%

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

   if 6.79999999999999969e51 < im < 1.9999999999999999e82

   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

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0

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

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

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

      2. *-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. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

   if 1.9999999999999999e82 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

   8. Simplified 74.4%

      \[\leadsto \color{blue}{\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 27000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+32}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+51}:\\ \;\;\;\;1 + im \cdot \frac{0.125 \cdot \left(im \cdot \left(im \cdot im\right)\right) + \left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right)}{\left(im \cdot im\right) \cdot 0.25 + \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right) - im \cdot 0.5\right)}\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+82}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\\ t_1 := im \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 7.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + t\_0 \cdot \left(t\_0 \cdot t\_0\right)}{1 + t\_0 \cdot \left(t\_0 + -1\right)}\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+77}:\\ \;\;\;\;1 + \frac{\left(im \cdot im\right) \cdot \left(0.125 + t\_1 \cdot \left(t\_1 \cdot 7.233796296296296 \cdot 10^{-5}\right)\right)}{0.25 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001736111111111111 - 0.020833333333333332\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664))))
        (t_1 (* im (* im im))))
   (if (<= im 7.5e+38)
     (/ (+ 1.0 (* t_0 (* t_0 t_0))) (+ 1.0 (* t_0 (+ t_0 -1.0))))
     (if (<= im 5e+77)
       (+
        1.0
        (/
         (* (* im im) (+ 0.125 (* t_1 (* t_1 7.233796296296296e-5))))
         (+
          0.25
          (*
           (* im im)
           (- (* (* im im) 0.001736111111111111) 0.020833333333333332)))))
       (*
        (+ 2.0 (* im (* im (+ 1.0 (* (* im im) 0.08333333333333333)))))
        (+ 0.5 (* -0.25 (* re re))))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (0.5 + ((im * im) * 0.041666666666666664));
	double t_1 = im * (im * im);
	double tmp;
	if (im <= 7.5e+38) {
		tmp = (1.0 + (t_0 * (t_0 * t_0))) / (1.0 + (t_0 * (t_0 + -1.0)));
	} else if (im <= 5e+77) {
		tmp = 1.0 + (((im * im) * (0.125 + (t_1 * (t_1 * 7.233796296296296e-5)))) / (0.25 + ((im * im) * (((im * im) * 0.001736111111111111) - 0.020833333333333332))));
	} else {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))
    t_1 = im * (im * im)
    if (im <= 7.5d+38) then
        tmp = (1.0d0 + (t_0 * (t_0 * t_0))) / (1.0d0 + (t_0 * (t_0 + (-1.0d0))))
    else if (im <= 5d+77) then
        tmp = 1.0d0 + (((im * im) * (0.125d0 + (t_1 * (t_1 * 7.233796296296296d-5)))) / (0.25d0 + ((im * im) * (((im * im) * 0.001736111111111111d0) - 0.020833333333333332d0))))
    else
        tmp = (2.0d0 + (im * (im * (1.0d0 + ((im * im) * 0.08333333333333333d0))))) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (0.5 + ((im * im) * 0.041666666666666664));
	double t_1 = im * (im * im);
	double tmp;
	if (im <= 7.5e+38) {
		tmp = (1.0 + (t_0 * (t_0 * t_0))) / (1.0 + (t_0 * (t_0 + -1.0)));
	} else if (im <= 5e+77) {
		tmp = 1.0 + (((im * im) * (0.125 + (t_1 * (t_1 * 7.233796296296296e-5)))) / (0.25 + ((im * im) * (((im * im) * 0.001736111111111111) - 0.020833333333333332))));
	} else {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (0.5 + ((im * im) * 0.041666666666666664))
	t_1 = im * (im * im)
	tmp = 0
	if im <= 7.5e+38:
		tmp = (1.0 + (t_0 * (t_0 * t_0))) / (1.0 + (t_0 * (t_0 + -1.0)))
	elif im <= 5e+77:
		tmp = 1.0 + (((im * im) * (0.125 + (t_1 * (t_1 * 7.233796296296296e-5)))) / (0.25 + ((im * im) * (((im * im) * 0.001736111111111111) - 0.020833333333333332))))
	else:
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))
	t_1 = Float64(im * Float64(im * im))
	tmp = 0.0
	if (im <= 7.5e+38)
		tmp = Float64(Float64(1.0 + Float64(t_0 * Float64(t_0 * t_0))) / Float64(1.0 + Float64(t_0 * Float64(t_0 + -1.0))));
	elseif (im <= 5e+77)
		tmp = Float64(1.0 + Float64(Float64(Float64(im * im) * Float64(0.125 + Float64(t_1 * Float64(t_1 * 7.233796296296296e-5)))) / Float64(0.25 + Float64(Float64(im * im) * Float64(Float64(Float64(im * im) * 0.001736111111111111) - 0.020833333333333332)))));
	else
		tmp = Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(Float64(im * im) * 0.08333333333333333))))) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (0.5 + ((im * im) * 0.041666666666666664));
	t_1 = im * (im * im);
	tmp = 0.0;
	if (im <= 7.5e+38)
		tmp = (1.0 + (t_0 * (t_0 * t_0))) / (1.0 + (t_0 * (t_0 + -1.0)));
	elseif (im <= 5e+77)
		tmp = 1.0 + (((im * im) * (0.125 + (t_1 * (t_1 * 7.233796296296296e-5)))) / (0.25 + ((im * im) * (((im * im) * 0.001736111111111111) - 0.020833333333333332))));
	else
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 7.5e+38], N[(N[(1.0 + N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e+77], N[(1.0 + N[(N[(N[(im * im), $MachinePrecision] * N[(0.125 + N[(t$95$1 * N[(t$95$1 * 7.233796296296296e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.001736111111111111), $MachinePrecision] - 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 7.4999999999999999e38

   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. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 88.9%

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

   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 56.3%

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

   8. Simplified 56.3%

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

   10. Applied egg-rr 41.1%

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

   if 7.4999999999999999e38 < im < 5.00000000000000004e77

   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. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 6.5%

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

   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 6.5%

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

   8. Simplified 6.5%

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

      1. associate-*r* N/A

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

      2. flip3-+ N/A

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

      3. associate-*r/ N/A

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

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

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

   10. Applied egg-rr 100.0%

       \[\leadsto 1 + \color{blue}{\frac{\left(im \cdot im\right) \cdot \left(0.125 + \left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right)}{0.25 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001736111111111111 - 0.020833333333333332\right)}} \]

   if 5.00000000000000004e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

   8. Simplified 75.0%

      \[\leadsto \color{blue}{\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 46.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 2 \cdot 10^{+77}:\\ \;\;\;\;1 + \frac{\left(im \cdot im\right) \cdot \left(0.125 + t\_0 \cdot \left(t\_0 \cdot 7.233796296296296 \cdot 10^{-5}\right)\right)}{0.25 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001736111111111111 - 0.020833333333333332\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im im))))
   (if (<= im 2e+77)
     (+
      1.0
      (/
       (* (* im im) (+ 0.125 (* t_0 (* t_0 7.233796296296296e-5))))
       (+
        0.25
        (*
         (* im im)
         (- (* (* im im) 0.001736111111111111) 0.020833333333333332)))))
     (*
      (+ 2.0 (* im (* im (+ 1.0 (* (* im im) 0.08333333333333333)))))
      (+ 0.5 (* -0.25 (* re re)))))))

C

double code(double re, double im) {
	double t_0 = im * (im * im);
	double tmp;
	if (im <= 2e+77) {
		tmp = 1.0 + (((im * im) * (0.125 + (t_0 * (t_0 * 7.233796296296296e-5)))) / (0.25 + ((im * im) * (((im * im) * 0.001736111111111111) - 0.020833333333333332))));
	} else {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (im * im)
    if (im <= 2d+77) then
        tmp = 1.0d0 + (((im * im) * (0.125d0 + (t_0 * (t_0 * 7.233796296296296d-5)))) / (0.25d0 + ((im * im) * (((im * im) * 0.001736111111111111d0) - 0.020833333333333332d0))))
    else
        tmp = (2.0d0 + (im * (im * (1.0d0 + ((im * im) * 0.08333333333333333d0))))) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * im);
	double tmp;
	if (im <= 2e+77) {
		tmp = 1.0 + (((im * im) * (0.125 + (t_0 * (t_0 * 7.233796296296296e-5)))) / (0.25 + ((im * im) * (((im * im) * 0.001736111111111111) - 0.020833333333333332))));
	} else {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * im)
	tmp = 0
	if im <= 2e+77:
		tmp = 1.0 + (((im * im) * (0.125 + (t_0 * (t_0 * 7.233796296296296e-5)))) / (0.25 + ((im * im) * (((im * im) * 0.001736111111111111) - 0.020833333333333332))))
	else:
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * im))
	tmp = 0.0
	if (im <= 2e+77)
		tmp = Float64(1.0 + Float64(Float64(Float64(im * im) * Float64(0.125 + Float64(t_0 * Float64(t_0 * 7.233796296296296e-5)))) / Float64(0.25 + Float64(Float64(im * im) * Float64(Float64(Float64(im * im) * 0.001736111111111111) - 0.020833333333333332)))));
	else
		tmp = Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(Float64(im * im) * 0.08333333333333333))))) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * im);
	tmp = 0.0;
	if (im <= 2e+77)
		tmp = 1.0 + (((im * im) * (0.125 + (t_0 * (t_0 * 7.233796296296296e-5)))) / (0.25 + ((im * im) * (((im * im) * 0.001736111111111111) - 0.020833333333333332))));
	else
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 2e+77], N[(1.0 + N[(N[(N[(im * im), $MachinePrecision] * N[(0.125 + N[(t$95$0 * N[(t$95$0 * 7.233796296296296e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.001736111111111111), $MachinePrecision] - 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.99999999999999997e77

   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. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 85.9%

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

   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 54.4%

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

   8. Simplified 54.4%

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

      1. associate-*r* N/A

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

      2. flip3-+ N/A

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

      3. associate-*r/ N/A

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

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

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

   10. Applied egg-rr 43.8%

       \[\leadsto 1 + \color{blue}{\frac{\left(im \cdot im\right) \cdot \left(0.125 + \left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right)}{0.25 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001736111111111111 - 0.020833333333333332\right)}} \]

   if 1.99999999999999997e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

   8. Simplified 75.0%

      \[\leadsto \color{blue}{\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 58.7% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 10^{+81}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1e+81)
   (+
    1.0
    (*
     (* im im)
     (+
      0.5
      (*
       im
       (* im (+ 0.041666666666666664 (* im (* im 0.001388888888888889))))))))
   (*
    (+ 2.0 (* im (* im (+ 1.0 (* (* im im) 0.08333333333333333)))))
    (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1e+81) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	} else {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1d+81) then
        tmp = 1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))))))
    else
        tmp = (2.0d0 + (im * (im * (1.0d0 + ((im * im) * 0.08333333333333333d0))))) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1e+81) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	} else {
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1e+81:
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))))
	else:
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1e+81)
		tmp = Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889))))))));
	else
		tmp = Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(Float64(im * im) * 0.08333333333333333))))) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1e+81)
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	else
		tmp = (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333))))) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1e+81], N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.99999999999999921e80

   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

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

   7. Simplified 66.0%

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

   8. Taylor expanded in im around 0

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

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

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

      2. *-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. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 57.5%

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

   10. Simplified 57.5%

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

   if 9.99999999999999921e80 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

   8. Simplified 74.4%

      \[\leadsto \color{blue}{\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+48}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5e+48)
   (+
    1.0
    (*
     (* im im)
     (+
      0.5
      (*
       im
       (* im (+ 0.041666666666666664 (* im (* im 0.001388888888888889))))))))
   (+
    1.0
    (*
     (* re re)
     (+
      -0.5
      (*
       re
       (*
        re
        (+ 0.041666666666666664 (* (* re re) -0.001388888888888889)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5e+48) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	} else {
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5d+48) then
        tmp = 1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))))))
    else
        tmp = 1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * (0.041666666666666664d0 + ((re * re) * (-0.001388888888888889d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5e+48) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	} else {
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5e+48:
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))))
	else:
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5e+48)
		tmp = Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889))))))));
	else
		tmp = Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.001388888888888889)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5e+48)
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	else
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5e+48], N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(re * N[(re * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.99999999999999973e48

   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

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

   7. Simplified 74.1%

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

   8. Taylor expanded in im around 0

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

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

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

      2. *-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. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 65.7%

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

   10. Simplified 65.7%

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

   if 4.99999999999999973e48 < 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 54.8%

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

   5. Simplified 54.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 29.9%

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

   8. Simplified 29.9%

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

Alternative 11: 58.6% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7 \cdot 10^{+84}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7e+84)
   (+
    1.0
    (*
     (* im im)
     (+
      0.5
      (*
       im
       (* im (+ 0.041666666666666664 (* im (* im 0.001388888888888889))))))))
   (* (+ 0.5 (* -0.25 (* re re))) (+ 2.0 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7e+84) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (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 (re <= 7d+84) then
        tmp = 1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))))))
    else
        tmp = (0.5d0 + ((-0.25d0) * (re * re))) * (2.0d0 + (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 7e+84) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7e+84:
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))))
	else:
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7e+84)
		tmp = Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889))))))));
	else
		tmp = Float64(Float64(0.5 + Float64(-0.25 * Float64(re * re))) * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 7e+84)
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	else
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7e+84], N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.9999999999999998e84

   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

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

   7. Simplified 73.6%

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

   8. Taylor expanded in im around 0

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

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

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

      2. *-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. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 65.3%

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

   10. Simplified 65.3%

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

   if 6.9999999999999998e84 < 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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(2 + {im}^{2}\right)}\right) \]
   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 79.7%

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

   5. Simplified 79.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 29.7%

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

   8. Simplified 29.7%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7 \cdot 10^{+84}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.1% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 36000:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+32}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.041666666666666664 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 36000.0)
   (+ 1.0 (* 0.5 (* im im)))
   (if (<= im 8.5e+32)
     (+ 1.0 (* (* re re) -0.5))
     (* im (* 0.041666666666666664 (* im (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 36000.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if (im <= 8.5e+32) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = im * (0.041666666666666664 * (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 <= 36000.0d0) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else if (im <= 8.5d+32) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    else
        tmp = im * (0.041666666666666664d0 * (im * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 36000.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if (im <= 8.5e+32) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = im * (0.041666666666666664 * (im * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 36000.0:
		tmp = 1.0 + (0.5 * (im * im))
	elif im <= 8.5e+32:
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = im * (0.041666666666666664 * (im * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 36000.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	elseif (im <= 8.5e+32)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	else
		tmp = Float64(im * Float64(0.041666666666666664 * Float64(im * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 36000.0)
		tmp = 1.0 + (0.5 * (im * im));
	elseif (im <= 8.5e+32)
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = im * (0.041666666666666664 * (im * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 36000.0], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.5e+32], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.041666666666666664 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 36000

   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 84.7%

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

   5. Simplified 84.7%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

         \[\leadsto \frac{1}{2} \cdot 2 + \color{blue}{\frac{1}{2} \cdot {im}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto 1 + \color{blue}{\frac{1}{2}} \cdot {im}^{2} \]

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 54.1%

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

   8. Simplified 54.1%

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

   if 36000 < im < 8.4999999999999998e32

   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 44.9%

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

   8. Simplified 44.9%

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

   if 8.4999999999999998e32 < 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. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 82.8%

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

   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%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{\left(3 + \color{blue}{1}\right)} \]

       2. pow-plus N/A

          \[\leadsto \frac{1}{24} \cdot \left({im}^{3} \cdot \color{blue}{im}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{3}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{3}\right)} \]

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

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

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

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 52.2%

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

   11. Simplified 52.2%

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

Alternative 13: 56.0% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7 \cdot 10^{+84}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7e+84)
   (+ 1.0 (* im (* im (+ 0.5 (* im (* im 0.041666666666666664))))))
   (* (+ 0.5 (* -0.25 (* re re))) (+ 2.0 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7e+84) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (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 (re <= 7d+84) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    else
        tmp = (0.5d0 + ((-0.25d0) * (re * re))) * (2.0d0 + (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 7e+84) {
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7e+84:
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))))
	else:
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7e+84)
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	else
		tmp = Float64(Float64(0.5 + Float64(-0.25 * Float64(re * re))) * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 7e+84)
		tmp = 1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664)))));
	else
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7e+84], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.9999999999999998e84

   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. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 87.5%

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

   6. Taylor expanded in re around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 62.9%

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

   8. Simplified 62.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 62.9%

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

   10. Applied egg-rr 62.9%

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

   if 6.9999999999999998e84 < 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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(2 + {im}^{2}\right)}\right) \]
   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 79.7%

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

   5. Simplified 79.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 29.7%

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

   8. Simplified 29.7%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7 \cdot 10^{+84}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{-22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9e-22) 1.0 (* (+ 0.5 (* -0.25 (* re re))) (+ 2.0 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9e-22) {
		tmp = 1.0;
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (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 <= 9d-22) then
        tmp = 1.0d0
    else
        tmp = (0.5d0 + ((-0.25d0) * (re * re))) * (2.0d0 + (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9e-22) {
		tmp = 1.0;
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9e-22:
		tmp = 1.0
	else:
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9e-22)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.5 + Float64(-0.25 * Float64(re * re))) * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9e-22)
		tmp = 1.0;
	else
		tmp = (0.5 + (-0.25 * (re * re))) * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9e-22], 1.0, N[(N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.99999999999999973e-22

   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 69.8%

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

   5. Simplified 69.8%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 41.1%

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

   if 8.99999999999999973e-22 < 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 52.5%

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

   5. Simplified 52.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 52.9%

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

   8. Simplified 52.9%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{-22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.5% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.65 \cdot 10^{+154}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.65e+154) (+ 1.0 (* 0.5 (* im im))) (* (* re re) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.65e+154) {
		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 <= 2.65d+154) 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 <= 2.65e+154) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (re * re) * -0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.65e+154:
		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 <= 2.65e+154)
		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 <= 2.65e+154)
		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, 2.65e+154], 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 < 2.65000000000000012e154

   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 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

         \[\leadsto \frac{1}{2} \cdot 2 + \color{blue}{\frac{1}{2} \cdot {im}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto 1 + \color{blue}{\frac{1}{2}} \cdot {im}^{2} \]

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 53.9%

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

   8. Simplified 53.9%

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

   if 2.65000000000000012e154 < 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 51.6%

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

   5. Simplified 51.6%

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

   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 32.0%

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

   8. Simplified 32.0%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 32.0%

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

   11. Simplified 32.0%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.65 \cdot 10^{+154}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 31.5% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 36000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 36000.0) 1.0 (* (* re re) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 36000.0) {
		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 <= 36000.0d0) 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 <= 36000.0) {
		tmp = 1.0;
	} else {
		tmp = (re * re) * -0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 36000.0:
		tmp = 1.0
	else:
		tmp = (re * re) * -0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 36000.0)
		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 <= 36000.0)
		tmp = 1.0;
	else
		tmp = (re * re) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 36000.0], 1.0, N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 36000

   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 69.2%

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

   5. Simplified 69.2%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 40.7%

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

   if 36000 < 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 22.0%

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

   8. Simplified 22.0%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 21.2%

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

   11. Simplified 21.2%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 36000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 28.8% 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 54.7%

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

5. Simplified 54.7%

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

6. Taylor expanded in re around 0

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

8. Simplified 32.3%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024186 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))