math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.1s

Alternatives: 18

Speedup: 1.5×

• 50.2% 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. metadata-eval N/A

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

   4. div-inv N/A

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

   5. +-commutative N/A

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

   6. cosh-def N/A

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

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

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

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

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

   9. cos-lowering-cos.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 72.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.75e-12)
   (cos re)
   (if (<= im 2.6e+77)
     (* (cosh im) (+ 1.0 (* re (* re -0.5))))
     (*
      (cos re)
      (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = cos(re);
	} else if (im <= 2.6e+77) {
		tmp = cosh(im) * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.75d-12) then
        tmp = cos(re)
    else if (im <= 2.6d+77) then
        tmp = cosh(im) * (1.0d0 + (re * (re * (-0.5d0))))
    else
        tmp = cos(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = Math.cos(re);
	} else if (im <= 2.6e+77) {
		tmp = Math.cosh(im) * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = Math.cos(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.75e-12:
		tmp = math.cos(re)
	elif im <= 2.6e+77:
		tmp = math.cosh(im) * (1.0 + (re * (re * -0.5)))
	else:
		tmp = math.cos(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.75e-12)
		tmp = cos(re);
	elseif (im <= 2.6e+77)
		tmp = Float64(cosh(im) * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	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 (im <= 1.75e-12)
		tmp = cos(re);
	elseif (im <= 2.6e+77)
		tmp = cosh(im) * (1.0 + (re * (re * -0.5)));
	else
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.75e-12], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(N[Cosh[im], $MachinePrecision] * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if im < 1.75e-12

   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 69.4%

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

   5. Simplified 69.4%

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

   if 1.75e-12 < im < 2.6000000000000002e77

   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. metadata-eval N/A

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

      4. div-inv N/A

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

      5. +-commutative N/A

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

      6. cosh-def N/A

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

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

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

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

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

      9. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 95.2%

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

   7. Simplified 95.2%

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

   if 2.6000000000000002e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

   5. Simplified 100.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(im \cdot im\right) \cdot 0.5\\ \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+52}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(1 + re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(t\_0 + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* im im) 0.5))))
   (if (<= im 1.75e-12)
     (cos re)
     (if (<= im 1.15e+52)
       (cosh im)
       (if (<= im 1.35e+154)
         (*
          (+ 1.0 (* re (* re -0.5)))
          (+
           t_0
           (*
            (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))
            (* im (* im (* im im))))))
         (* (cos re) t_0))))))

C

double code(double re, double im) {
	double t_0 = 1.0 + ((im * im) * 0.5);
	double tmp;
	if (im <= 1.75e-12) {
		tmp = cos(re);
	} else if (im <= 1.15e+52) {
		tmp = cosh(im);
	} else if (im <= 1.35e+154) {
		tmp = (1.0 + (re * (re * -0.5))) * (t_0 + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	} else {
		tmp = cos(re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((im * im) * 0.5d0)
    if (im <= 1.75d-12) then
        tmp = cos(re)
    else if (im <= 1.15d+52) then
        tmp = cosh(im)
    else if (im <= 1.35d+154) then
        tmp = (1.0d0 + (re * (re * (-0.5d0)))) * (t_0 + ((0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)) * (im * (im * (im * im)))))
    else
        tmp = cos(re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + ((im * im) * 0.5);
	double tmp;
	if (im <= 1.75e-12) {
		tmp = Math.cos(re);
	} else if (im <= 1.15e+52) {
		tmp = Math.cosh(im);
	} else if (im <= 1.35e+154) {
		tmp = (1.0 + (re * (re * -0.5))) * (t_0 + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	} else {
		tmp = Math.cos(re) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + ((im * im) * 0.5)
	tmp = 0
	if im <= 1.75e-12:
		tmp = math.cos(re)
	elif im <= 1.15e+52:
		tmp = math.cosh(im)
	elif im <= 1.35e+154:
		tmp = (1.0 + (re * (re * -0.5))) * (t_0 + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))))
	else:
		tmp = math.cos(re) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(Float64(im * im) * 0.5))
	tmp = 0.0
	if (im <= 1.75e-12)
		tmp = cos(re);
	elseif (im <= 1.15e+52)
		tmp = cosh(im);
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(1.0 + Float64(re * Float64(re * -0.5))) * Float64(t_0 + Float64(Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)) * Float64(im * Float64(im * Float64(im * im))))));
	else
		tmp = Float64(cos(re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + ((im * im) * 0.5);
	tmp = 0.0;
	if (im <= 1.75e-12)
		tmp = cos(re);
	elseif (im <= 1.15e+52)
		tmp = cosh(im);
	elseif (im <= 1.35e+154)
		tmp = (1.0 + (re * (re * -0.5))) * (t_0 + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	else
		tmp = cos(re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.75e-12], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.15e+52], N[Cosh[im], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[(N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.75e-12

   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 69.4%

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

   5. Simplified 69.4%

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

   if 1.75e-12 < im < 1.15e52

   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. metadata-eval N/A

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

      4. div-inv N/A

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

      5. +-commutative N/A

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

      6. cosh-def N/A

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

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

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

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

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

      9. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 76.5%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 76.5%

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

   9. Applied egg-rr 76.5%

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

   if 1.15e52 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 73.9%

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

   8. Simplified 73.9%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+52}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(1 + re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(\left(1 + \left(im \cdot im\right) \cdot 0.5\right) + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.75e-12)
   (cos re)
   (if (<= im 1.35e+154)
     (* (cosh im) (+ 1.0 (* re (* re -0.5))))
     (* (cos re) (+ 1.0 (* (* im im) 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = cosh(im) * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = cos(re) * (1.0 + ((im * im) * 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 <= 1.75d-12) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = cosh(im) * (1.0d0 + (re * (re * (-0.5d0))))
    else
        tmp = cos(re) * (1.0d0 + ((im * im) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = Math.cosh(im) * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = Math.cos(re) * (1.0 + ((im * im) * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.75e-12:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = math.cosh(im) * (1.0 + (re * (re * -0.5)))
	else:
		tmp = math.cos(re) * (1.0 + ((im * im) * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.75e-12)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(cosh(im) * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.75e-12)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = cosh(im) * (1.0 + (re * (re * -0.5)));
	else
		tmp = cos(re) * (1.0 + ((im * im) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.75e-12], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[Cosh[im], $MachinePrecision] * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.75e-12

   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 69.4%

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

   5. Simplified 69.4%

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

   if 1.75e-12 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. +-commutative N/A

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

      6. cosh-def N/A

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

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

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

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

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

      9. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 82.5%

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

   7. Simplified 82.5%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(\left(1 + \left(im \cdot im\right) \cdot 0.5\right) + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.75e-12)
   (cos re)
   (if (<= im 3.5e+51)
     (cosh im)
     (if (<= im 5.2e+134)
       (*
        (+ 1.0 (* re (* re -0.5)))
        (+
         (+ 1.0 (* (* im im) 0.5))
         (*
          (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))
          (* im (* im (* im im))))))
       (+
        1.0
        (*
         (* im im)
         (+
          0.5
          (*
           im
           (*
            im
            (+
             0.041666666666666664
             (* im (* im 0.001388888888888889))))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = cos(re);
	} else if (im <= 3.5e+51) {
		tmp = cosh(im);
	} else if (im <= 5.2e+134) {
		tmp = (1.0 + (re * (re * -0.5))) * ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	} else {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.75d-12) then
        tmp = cos(re)
    else if (im <= 3.5d+51) then
        tmp = cosh(im)
    else if (im <= 5.2d+134) then
        tmp = (1.0d0 + (re * (re * (-0.5d0)))) * ((1.0d0 + ((im * im) * 0.5d0)) + ((0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)) * (im * (im * (im * im)))))
    else
        tmp = 1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = Math.cos(re);
	} else if (im <= 3.5e+51) {
		tmp = Math.cosh(im);
	} else if (im <= 5.2e+134) {
		tmp = (1.0 + (re * (re * -0.5))) * ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	} else {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.75e-12:
		tmp = math.cos(re)
	elif im <= 3.5e+51:
		tmp = math.cosh(im)
	elif im <= 5.2e+134:
		tmp = (1.0 + (re * (re * -0.5))) * ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))))
	else:
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.75e-12)
		tmp = cos(re);
	elseif (im <= 3.5e+51)
		tmp = cosh(im);
	elseif (im <= 5.2e+134)
		tmp = Float64(Float64(1.0 + Float64(re * Float64(re * -0.5))) * Float64(Float64(1.0 + Float64(Float64(im * im) * 0.5)) + Float64(Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)) * Float64(im * Float64(im * Float64(im * im))))));
	else
		tmp = Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.75e-12)
		tmp = cos(re);
	elseif (im <= 3.5e+51)
		tmp = cosh(im);
	elseif (im <= 5.2e+134)
		tmp = (1.0 + (re * (re * -0.5))) * ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	else
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.75e-12], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.5e+51], N[Cosh[im], $MachinePrecision], If[LessEqual[im, 5.2e+134], N[(N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.75e-12

   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 69.4%

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

   5. Simplified 69.4%

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

   if 1.75e-12 < im < 3.5e51

   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. metadata-eval N/A

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

      4. div-inv N/A

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

      5. +-commutative N/A

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

      6. cosh-def N/A

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

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

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

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

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

      9. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 76.5%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 76.5%

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

   9. Applied egg-rr 76.5%

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

   if 3.5e51 < im < 5.2000000000000003e134

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 78.9%

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

   8. Simplified 78.9%

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

   if 5.2000000000000003e134 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. 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. metadata-eval N/A

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

      4. div-inv N/A

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

      5. +-commutative N/A

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

      6. cosh-def N/A

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

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

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

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

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

      9. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 84.8%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 84.8%

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

   9. Applied egg-rr 84.8%

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

   10. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-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 84.8%

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

   12. Simplified 84.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(\left(1 + \left(im \cdot im\right) \cdot 0.5\right) + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\left(\left(1 + \left(im \cdot im\right) \cdot 0.5\right) + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.75e-12)
   (cos re)
   (if (<= im 2e+134)
     (*
      (+
       (+ 1.0 (* (* im im) 0.5))
       (*
        (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))
        (* im (* im (* im im)))))
      (+
       1.0
       (*
        (* re re)
        (+
         -0.5
         (*
          re
          (*
           re
           (+ 0.041666666666666664 (* (* re re) -0.001388888888888889))))))))
     (+
      1.0
      (*
       (* im im)
       (+
        0.5
        (*
         im
         (*
          im
          (+ 0.041666666666666664 (* im (* im 0.001388888888888889)))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = cos(re);
	} else if (im <= 2e+134) {
		tmp = ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))));
	} else {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.75d-12) then
        tmp = cos(re)
    else if (im <= 2d+134) then
        tmp = ((1.0d0 + ((im * im) * 0.5d0)) + ((0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)) * (im * (im * (im * im))))) * (1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * (0.041666666666666664d0 + ((re * re) * (-0.001388888888888889d0))))))))
    else
        tmp = 1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = Math.cos(re);
	} else if (im <= 2e+134) {
		tmp = ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))));
	} else {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.75e-12:
		tmp = math.cos(re)
	elif im <= 2e+134:
		tmp = ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))))
	else:
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.75e-12)
		tmp = cos(re);
	elseif (im <= 2e+134)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(im * im) * 0.5)) + Float64(Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)) * Float64(im * Float64(im * Float64(im * im))))) * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.001388888888888889))))))));
	else
		tmp = Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.75e-12)
		tmp = cos(re);
	elseif (im <= 2e+134)
		tmp = ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))));
	else
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.75e-12], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2e+134], N[(N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.75e-12

   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 69.4%

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

   5. Simplified 69.4%

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

   if 1.75e-12 < im < 1.99999999999999984e134

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Simplified 63.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 57.3%

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

   8. Simplified 57.3%

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

   if 1.99999999999999984e134 < im

   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. metadata-eval N/A

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

      4. div-inv N/A

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

      5. +-commutative N/A

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

      6. cosh-def N/A

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

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

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

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

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

      9. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 84.8%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 84.8%

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

   9. Applied egg-rr 84.8%

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

   10. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-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 84.8%

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

   12. Simplified 84.8%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\left(\left(1 + \left(im \cdot im\right) \cdot 0.5\right) + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{if}\;im \leq 330:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+134}:\\ \;\;\;\;\left(\left(1 + \left(im \cdot im\right) \cdot 0.5\right) + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (*
           (* im im)
           (+
            0.5
            (*
             im
             (*
              im
              (+
               0.041666666666666664
               (* im (* im 0.001388888888888889))))))))))
   (if (<= im 330.0)
     t_0
     (if (<= im 3.1e+134)
       (*
        (+
         (+ 1.0 (* (* im im) 0.5))
         (*
          (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))
          (* im (* im (* im im)))))
        (+
         1.0
         (*
          (* re re)
          (+
           -0.5
           (*
            re
            (*
             re
             (+ 0.041666666666666664 (* (* re re) -0.001388888888888889))))))))
       t_0))))

C

double code(double re, double im) {
	double t_0 = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	double tmp;
	if (im <= 330.0) {
		tmp = t_0;
	} else if (im <= 3.1e+134) {
		tmp = ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))))))
    if (im <= 330.0d0) then
        tmp = t_0
    else if (im <= 3.1d+134) then
        tmp = ((1.0d0 + ((im * im) * 0.5d0)) + ((0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)) * (im * (im * (im * im))))) * (1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * (0.041666666666666664d0 + ((re * re) * (-0.001388888888888889d0))))))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	double tmp;
	if (im <= 330.0) {
		tmp = t_0;
	} else if (im <= 3.1e+134) {
		tmp = ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))))
	tmp = 0
	if im <= 330.0:
		tmp = t_0
	elif im <= 3.1e+134:
		tmp = ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889))))))))
	tmp = 0.0
	if (im <= 330.0)
		tmp = t_0;
	elseif (im <= 3.1e+134)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(im * im) * 0.5)) + Float64(Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)) * Float64(im * Float64(im * Float64(im * im))))) * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.001388888888888889))))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	tmp = 0.0;
	if (im <= 330.0)
		tmp = t_0;
	elseif (im <= 3.1e+134)
		tmp = ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im))))) * (1.0 + ((re * re) * (-0.5 + (re * (re * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(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]}, If[LessEqual[im, 330.0], t$95$0, If[LessEqual[im, 3.1e+134], N[(N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 330 or 3.09999999999999982e134 < im

   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. metadata-eval N/A

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

      4. div-inv N/A

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

      5. +-commutative N/A

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

      6. cosh-def N/A

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

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

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

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

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

      9. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 61.5%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 61.5%

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

   9. Applied egg-rr 61.5%

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

   10. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-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 58.1%

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

   12. Simplified 58.1%

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

   if 330 < im < 3.09999999999999982e134

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Simplified 61.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 54.8%

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

   8. Simplified 54.8%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 330:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+134}:\\ \;\;\;\;\left(\left(1 + \left(im \cdot im\right) \cdot 0.5\right) + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{if}\;im \leq 380:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(\left(1 + \left(im \cdot im\right) \cdot 0.5\right) + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (*
           (* im im)
           (+
            0.5
            (*
             im
             (*
              im
              (+
               0.041666666666666664
               (* im (* im 0.001388888888888889))))))))))
   (if (<= im 380.0)
     t_0
     (if (<= im 2e+134)
       (*
        (+ 1.0 (* re (* re -0.5)))
        (+
         (+ 1.0 (* (* im im) 0.5))
         (*
          (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))
          (* im (* im (* im im))))))
       t_0))))

C

double code(double re, double im) {
	double t_0 = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	double tmp;
	if (im <= 380.0) {
		tmp = t_0;
	} else if (im <= 2e+134) {
		tmp = (1.0 + (re * (re * -0.5))) * ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))))))
    if (im <= 380.0d0) then
        tmp = t_0
    else if (im <= 2d+134) then
        tmp = (1.0d0 + (re * (re * (-0.5d0)))) * ((1.0d0 + ((im * im) * 0.5d0)) + ((0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)) * (im * (im * (im * im)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	double tmp;
	if (im <= 380.0) {
		tmp = t_0;
	} else if (im <= 2e+134) {
		tmp = (1.0 + (re * (re * -0.5))) * ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))))
	tmp = 0
	if im <= 380.0:
		tmp = t_0
	elif im <= 2e+134:
		tmp = (1.0 + (re * (re * -0.5))) * ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889))))))))
	tmp = 0.0
	if (im <= 380.0)
		tmp = t_0;
	elseif (im <= 2e+134)
		tmp = Float64(Float64(1.0 + Float64(re * Float64(re * -0.5))) * Float64(Float64(1.0 + Float64(Float64(im * im) * 0.5)) + Float64(Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)) * Float64(im * Float64(im * Float64(im * im))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	tmp = 0.0;
	if (im <= 380.0)
		tmp = t_0;
	elseif (im <= 2e+134)
		tmp = (1.0 + (re * (re * -0.5))) * ((1.0 + ((im * im) * 0.5)) + ((0.041666666666666664 + ((im * im) * 0.001388888888888889)) * (im * (im * (im * im)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = 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]}, If[LessEqual[im, 380.0], t$95$0, If[LessEqual[im, 2e+134], N[(N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 380 or 1.99999999999999984e134 < im

   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. metadata-eval N/A

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

      4. div-inv N/A

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

      5. +-commutative N/A

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

      6. cosh-def N/A

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

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

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

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

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

      9. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 61.5%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 61.5%

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

   9. Applied egg-rr 61.5%

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

   10. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-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 58.1%

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

   12. Simplified 58.1%

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

   if 380 < im < 1.99999999999999984e134

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Simplified 61.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 51.8%

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

   8. Simplified 51.8%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 380:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(\left(1 + \left(im \cdot im\right) \cdot 0.5\right) + \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9.6 \cdot 10^{+91}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot 0.5\right) \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9.6e+91)
   (+
    1.0
    (*
     (* im im)
     (+
      0.5
      (*
       im
       (* im (+ 0.041666666666666664 (* im (* im 0.001388888888888889))))))))
   (*
    (+ 1.0 (* (* im im) 0.5))
    (+
     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 <= 9.6e+91) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + ((im * im) * 0.5)) * (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 <= 9.6d+91) then
        tmp = 1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))))))
    else
        tmp = (1.0d0 + ((im * im) * 0.5d0)) * (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 <= 9.6e+91) {
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + ((im * im) * 0.5)) * (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 <= 9.6e+91:
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))))
	else:
		tmp = (1.0 + ((im * im) * 0.5)) * (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 <= 9.6e+91)
		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(1.0 + Float64(Float64(im * im) * 0.5)) * 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 <= 9.6e+91)
		tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
	else
		tmp = (1.0 + ((im * im) * 0.5)) * (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, 9.6e+91], 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[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.59999999999999932e91

   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. metadata-eval N/A

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

      4. div-inv N/A

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

      5. +-commutative N/A

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

      6. cosh-def N/A

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

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

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

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

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

      9. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 71.0%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 71.0%

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

   9. Applied egg-rr 71.0%

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

   10. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-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 63.9%

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

   12. Simplified 63.9%

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

   if 9.59999999999999932e91 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 76.1%

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

   5. Simplified 76.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 30.9%

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

   8. Simplified 30.9%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9.6 \cdot 10^{+91}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot 0.5\right) \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.8% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9.6 \cdot 10^{+91}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9.6e+91)
   (+
    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 <= 9.6e+91) {
		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 <= 9.6d+91) 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 <= 9.6e+91) {
		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 <= 9.6e+91:
		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 <= 9.6e+91)
		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 <= 9.6e+91)
		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, 9.6e+91], 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 < 9.59999999999999932e91

   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. metadata-eval N/A

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

      4. div-inv N/A

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

      5. +-commutative N/A

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

      6. cosh-def N/A

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

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

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

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

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

      9. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 71.0%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 71.0%

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

   9. Applied egg-rr 71.0%

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

   10. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-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 63.9%

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

   12. Simplified 63.9%

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

   if 9.59999999999999932e91 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 61.2%

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

   5. Simplified 61.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)} \]
   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 30.9%

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

   8. Simplified 30.9%

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

Alternative 11: 49.4% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(im \cdot im\right) \cdot 0.5\\ \mathbf{if}\;im \leq 680:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+134}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(-0.5 + im \cdot \left(im \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* im im) 0.5))))
   (if (<= im 680.0)
     t_0
     (if (<= im 6.2e+134) (* (* re re) (+ -0.5 (* im (* im -0.25)))) t_0))))

C

double code(double re, double im) {
	double t_0 = 1.0 + ((im * im) * 0.5);
	double tmp;
	if (im <= 680.0) {
		tmp = t_0;
	} else if (im <= 6.2e+134) {
		tmp = (re * re) * (-0.5 + (im * (im * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((im * im) * 0.5d0)
    if (im <= 680.0d0) then
        tmp = t_0
    else if (im <= 6.2d+134) then
        tmp = (re * re) * ((-0.5d0) + (im * (im * (-0.25d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + ((im * im) * 0.5);
	double tmp;
	if (im <= 680.0) {
		tmp = t_0;
	} else if (im <= 6.2e+134) {
		tmp = (re * re) * (-0.5 + (im * (im * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + ((im * im) * 0.5)
	tmp = 0
	if im <= 680.0:
		tmp = t_0
	elif im <= 6.2e+134:
		tmp = (re * re) * (-0.5 + (im * (im * -0.25)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(Float64(im * im) * 0.5))
	tmp = 0.0
	if (im <= 680.0)
		tmp = t_0;
	elseif (im <= 6.2e+134)
		tmp = Float64(Float64(re * re) * Float64(-0.5 + Float64(im * Float64(im * -0.25))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + ((im * im) * 0.5);
	tmp = 0.0;
	if (im <= 680.0)
		tmp = t_0;
	elseif (im <= 6.2e+134)
		tmp = (re * re) * (-0.5 + (im * (im * -0.25)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 680.0], t$95$0, If[LessEqual[im, 6.2e+134], N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(im * N[(im * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 680 or 6.19999999999999963e134 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 86.1%

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

   5. Simplified 86.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 53.0%

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

   8. Simplified 53.0%

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

   if 680 < im < 6.19999999999999963e134

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

   5. Simplified 49.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

   8. Simplified 45.7%

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

   9. Taylor expanded in re around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 19.6%

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

   11. Simplified 19.6%

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

   12. Taylor expanded in im around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. associate-*r* N/A

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

       6. distribute-lft-in N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. distribute-lft-in N/A

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

       11. metadata-eval N/A

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

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

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

       13. associate-*r* N/A

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

       14. metadata-eval N/A

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

       15. unpow2 N/A

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

       16. associate-*r* N/A

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

       17. *-commutative N/A

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

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

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

       19. *-commutative N/A

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

       20. *-lowering-*.f64 19.9%

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

   14. Simplified 19.9%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot 0.5\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+134}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(-0.5 + im \cdot \left(im \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ 1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (+
  1.0
  (*
   (* im im)
   (+
    0.5
    (*
     im
     (* im (+ 0.041666666666666664 (* im (* im 0.001388888888888889)))))))))

C

double code(double re, double im) {
	return 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0)))))))
end function

Java

public static double code(double re, double im) {
	return 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
}

Python

def code(re, im):
	return 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))))

Julia

function code(re, im)
	return Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889))))))))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889)))))));
end

Wolfram

code[re_, im_] := 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]

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. metadata-eval N/A

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

   4. div-inv N/A

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

   5. +-commutative N/A

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

   6. cosh-def N/A

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

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

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

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

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

   9. cos-lowering-cos.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in re around 0

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

7. Simplified 62.8%

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

   1. *-rgt-identity N/A

      \[\leadsto \cosh im \]

   2. cosh-lowering-cosh.f64 62.8%

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

9. Applied egg-rr 62.8%

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

10. Taylor expanded in im around 0

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

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

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

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

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

    3. unpow2 N/A

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

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

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

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

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

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. *-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 56.5%

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

12. Simplified 56.5%

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

Alternative 13: 39.1% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+127}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 650.0)
   1.0
   (if (<= im 2.1e+127) (* -0.5 (* re re)) (* (* im im) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 650.0) {
		tmp = 1.0;
	} else if (im <= 2.1e+127) {
		tmp = -0.5 * (re * re);
	} else {
		tmp = (im * im) * 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 <= 650.0d0) then
        tmp = 1.0d0
    else if (im <= 2.1d+127) then
        tmp = (-0.5d0) * (re * re)
    else
        tmp = (im * im) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 650.0) {
		tmp = 1.0;
	} else if (im <= 2.1e+127) {
		tmp = -0.5 * (re * re);
	} else {
		tmp = (im * im) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 650.0:
		tmp = 1.0
	elif im <= 2.1e+127:
		tmp = -0.5 * (re * re)
	else:
		tmp = (im * im) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 650.0)
		tmp = 1.0;
	elseif (im <= 2.1e+127)
		tmp = Float64(-0.5 * Float64(re * re));
	else
		tmp = Float64(Float64(im * im) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 650.0)
		tmp = 1.0;
	elseif (im <= 2.1e+127)
		tmp = -0.5 * (re * re);
	else
		tmp = (im * im) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 650.0], 1.0, If[LessEqual[im, 2.1e+127], N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 650

   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 69.4%

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

   5. Simplified 69.4%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 35.9%

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

   if 650 < im < 2.09999999999999992e127

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

   5. Simplified 47.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

   8. Simplified 44.0%

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

   9. Taylor expanded in re around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 20.2%

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

   11. Simplified 20.2%

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

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   13. 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 14.6%

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

   14. Simplified 14.6%

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

   if 2.09999999999999992e127 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 86.9%

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

   5. Simplified 86.9%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 72.2%

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

   8. Simplified 72.2%

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

   9. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 72.2%

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

   11. Simplified 72.2%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+127}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.7% accurate, 23.7× speedup

Math

\[\begin{array}{l} \\ 1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	return 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
end function

Java

public static double code(double re, double im) {
	return 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
}

Python

def code(re, im):
	return 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))

Julia

function code(re, im)
	return Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
end

Wolfram

code[re_, im_] := N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. 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. metadata-eval N/A

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

   4. div-inv N/A

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

   5. +-commutative N/A

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

   6. cosh-def N/A

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

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

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

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

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

   9. cos-lowering-cos.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in re around 0

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

7. Simplified 62.8%

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

8. Taylor expanded in im around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 55.0%

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

10. Simplified 55.0%

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

Alternative 15: 56.5% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ 1 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (+ 1.0 (* (* im im) (* (* im im) 0.041666666666666664))))

C

double code(double re, double im) {
	return 1.0 + ((im * im) * ((im * im) * 0.041666666666666664));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + ((im * im) * ((im * im) * 0.041666666666666664d0))
end function

Java

public static double code(double re, double im) {
	return 1.0 + ((im * im) * ((im * im) * 0.041666666666666664));
}

Python

def code(re, im):
	return 1.0 + ((im * im) * ((im * im) * 0.041666666666666664))

Julia

function code(re, im)
	return Float64(1.0 + Float64(Float64(im * im) * Float64(Float64(im * im) * 0.041666666666666664)))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + ((im * im) * ((im * im) * 0.041666666666666664));
end

Wolfram

code[re_, im_] := N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. distribute-lft-in N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt1-in N/A

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

   6. associate-*r* N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. associate-+r+ N/A

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

   12. +-commutative N/A

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

5. Simplified 87.4%

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

6. Taylor expanded in re around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 55.0%

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

8. Simplified 55.0%

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

9. Taylor expanded in im around inf

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

    1. *-commutative N/A

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

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

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

    3. unpow2 N/A

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

    4. *-lowering-*.f64 54.8%

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

11. Simplified 54.8%

    \[\leadsto 1 + \left(im \cdot im\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)} \]
12. Add Preprocessing

Alternative 16: 31.3% accurate, 30.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 36000.0)
		tmp = 1.0;
	else
		tmp = Float64(-0.5 * Float64(re * re));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[im, 36000.0], 1.0, N[(-0.5 * N[(re * re), $MachinePrecision]), $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 69.4%

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

   5. Simplified 69.4%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 35.9%

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

   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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

   5. Simplified 75.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

         \[\leadsto \left(\frac{-1}{2} \cdot {re}^{2} + 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \cdot \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{2} \cdot {re}^{2}\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{2}\right)\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \frac{-1}{2}\right)\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \frac{-1}{2}\right)\right)\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \frac{-1}{2}\right)\right)\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right)\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

   8. Simplified 59.4%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 17.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 17.3%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   13. 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 12.2%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right) \]

   14. Simplified 12.2%

       \[\leadsto \color{blue}{-0.5 \cdot \left(re \cdot re\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 47.7% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ 1 + \left(im \cdot im\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ 1.0 (* (* im im) 0.5)))

C

double code(double re, double im) {
	return 1.0 + ((im * im) * 0.5);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + ((im * im) * 0.5d0)
end function

Java

public static double code(double re, double im) {
	return 1.0 + ((im * im) * 0.5);
}

Python

def code(re, im):
	return 1.0 + ((im * im) * 0.5)

Julia

function code(re, im)
	return Float64(1.0 + Float64(Float64(im * im) * 0.5))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + ((im * im) * 0.5);
end

Wolfram

code[re_, im_] := N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\cos re + \frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \cos re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\cos re} \]

   2. distribute-rgt1-in N/A

      \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

   3. unpow2 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

   4. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

   5. *-commutative N/A

      \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   6. *-commutative N/A

      \[\leadsto \cos re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)}\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(1 + \color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{\color{blue}{2}}\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

   16. *-lowering-*.f64 75.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

5. Simplified 75.9%

   \[\leadsto \color{blue}{\cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

   4. *-lowering-*.f64 46.8%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

8. Simplified 46.8%

   \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

9. Final simplification 46.8%

   \[\leadsto 1 + \left(im \cdot im\right) \cdot 0.5 \]
10. Add Preprocessing

Alternative 18: 28.6% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\cos re} \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 52.5%

      \[\leadsto \mathsf{cos.f64}\left(re\right) \]

5. Simplified 52.5%

   \[\leadsto \color{blue}{\cos re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 27.4%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))