math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.1s

Alternatives: 17

Speedup: 1.5×

• 50.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. 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: 95.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.983:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.983)
   (*
    (* (cos re) 0.5)
    (+
     2.0
     (*
      (* im im)
      (+
       1.0
       (*
        (* im im)
        (+ 0.08333333333333333 (* (* im im) 0.002777777777777778)))))))
   (cosh im)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.983) {
		tmp = (cos(re) * 0.5) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	} else {
		tmp = cosh(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= 0.983d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + ((im * im) * (1.0d0 + ((im * im) * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0))))))
    else
        tmp = cosh(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.983) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	} else {
		tmp = Math.cosh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.983:
		tmp = (math.cos(re) * 0.5) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))))
	else:
		tmp = math.cosh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.983)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(Float64(im * im) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))));
	else
		tmp = cosh(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.983)
		tmp = (cos(re) * 0.5) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	else
		tmp = cosh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.983], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(N[(im * im), $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cosh[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.982999999999999985

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 96.3%

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

   5. Simplified 96.3%

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

   if 0.982999999999999985 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. 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 99.9%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.983:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.983:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.983)
   (*
    (cos re)
    (+ 1.0 (* (* im im) (+ 0.5 (* im (* im 0.041666666666666664))))))
   (cosh im)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.983) {
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	} else {
		tmp = cosh(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= 0.983d0) then
        tmp = cos(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    else
        tmp = cosh(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.983) {
		tmp = Math.cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	} else {
		tmp = Math.cosh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.983:
		tmp = math.cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))))
	else:
		tmp = math.cosh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.983)
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	else
		tmp = cosh(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.983)
		tmp = cos(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	else
		tmp = cosh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.983], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cosh[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.982999999999999985

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Simplified 93.4%

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

   if 0.982999999999999985 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. 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 99.9%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 99.9%

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

   9. Applied egg-rr 99.9%

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

Alternative 4: 84.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{if}\;im \leq 0.00013:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+153}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ 2.0 (* im im)))))
   (if (<= im 0.00013) t_0 (if (<= im 1.5e+153) (cosh im) t_0))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (2.0 + (im * im));
	double tmp;
	if (im <= 0.00013) {
		tmp = t_0;
	} else if (im <= 1.5e+153) {
		tmp = cosh(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    if (im <= 0.00013d0) then
        tmp = t_0
    else if (im <= 1.5d+153) then
        tmp = cosh(im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	double tmp;
	if (im <= 0.00013) {
		tmp = t_0;
	} else if (im <= 1.5e+153) {
		tmp = Math.cosh(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.cos(re) * 0.5) * (2.0 + (im * im))
	tmp = 0
	if im <= 0.00013:
		tmp = t_0
	elif im <= 1.5e+153:
		tmp = math.cosh(im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)))
	tmp = 0.0
	if (im <= 0.00013)
		tmp = t_0;
	elseif (im <= 1.5e+153)
		tmp = cosh(im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (cos(re) * 0.5) * (2.0 + (im * im));
	tmp = 0.0;
	if (im <= 0.00013)
		tmp = t_0;
	elseif (im <= 1.5e+153)
		tmp = cosh(im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.00013], t$95$0, If[LessEqual[im, 1.5e+153], N[Cosh[im], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.29999999999999989e-4 or 1.50000000000000009e153 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 85.7%

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

   5. Simplified 85.7%

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

   if 1.29999999999999989e-4 < im < 1.50000000000000009e153

   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 70.2%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 70.2%

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

   9. Applied egg-rr 70.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00013:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+153}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.000115:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 0.000115) (cos re) (cosh im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.000115) {
		tmp = cos(re);
	} else {
		tmp = cosh(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.000115d0) then
        tmp = cos(re)
    else
        tmp = cosh(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.000115) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.cosh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.000115:
		tmp = math.cos(re)
	else:
		tmp = math.cosh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.000115)
		tmp = cos(re);
	else
		tmp = cosh(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.000115)
		tmp = cos(re);
	else
		tmp = cosh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.000115], N[Cos[re], $MachinePrecision], N[Cosh[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.15e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 65.2%

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

   5. Simplified 65.2%

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

   if 1.15e-4 < 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 77.5%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh im \]

      2. cosh-lowering-cosh.f64 77.5%

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

   9. Applied egg-rr 77.5%

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

Alternative 6: 64.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot \left(-0.25 + \left(re \cdot re\right) \cdot 0.020833333333333332\right)\right)\right) \cdot \left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7e-5)
   (cos re)
   (*
    (+ 0.5 (* re (* re (+ -0.25 (* (* re re) 0.020833333333333332)))))
    (+ 2.0 (* im (* im (+ 1.0 (* (* im im) 0.08333333333333333))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7e-5) {
		tmp = cos(re);
	} else {
		tmp = (0.5 + (re * (re * (-0.25 + ((re * re) * 0.020833333333333332))))) * (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7d-5) then
        tmp = cos(re)
    else
        tmp = (0.5d0 + (re * (re * ((-0.25d0) + ((re * re) * 0.020833333333333332d0))))) * (2.0d0 + (im * (im * (1.0d0 + ((im * im) * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7e-5) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 + (re * (re * (-0.25 + ((re * re) * 0.020833333333333332))))) * (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7e-5:
		tmp = math.cos(re)
	else:
		tmp = (0.5 + (re * (re * (-0.25 + ((re * re) * 0.020833333333333332))))) * (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7e-5)
		tmp = cos(re);
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * Float64(-0.25 + Float64(Float64(re * re) * 0.020833333333333332))))) * Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(Float64(im * im) * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7e-5)
		tmp = cos(re);
	else
		tmp = (0.5 + (re * (re * (-0.25 + ((re * re) * 0.020833333333333332))))) * (2.0 + (im * (im * (1.0 + ((im * im) * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7e-5], N[Cos[re], $MachinePrecision], N[(N[(0.5 + N[(re * N[(re * N[(-0.25 + N[(N[(re * re), $MachinePrecision] * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(im * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.9999999999999994e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 65.2%

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

   5. Simplified 65.2%

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

   if 6.9999999999999994e-5 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 82.8%

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

   5. Simplified 82.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \left(re \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{48} \cdot {re}^{2} + \frac{-1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{4} + \frac{1}{48} \cdot {re}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\frac{1}{48} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \left({re}^{2} \cdot \frac{1}{48}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{1}{48}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{48}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 65.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{48}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{12}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 65.9%

      \[\leadsto \color{blue}{\left(0.5 + re \cdot \left(re \cdot \left(-0.25 + \left(re \cdot re\right) \cdot 0.020833333333333332\right)\right)\right)} \cdot \left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 59.0% accurate, 16.2× speedup

Math

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

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(Float64(im * im) * Float64(0.041666666666666664 + Float64(Float64(im * 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[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $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 64.6%

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

8. Taylor expanded in im around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 59.4%

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

10. Simplified 59.4%

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

Alternative 8: 58.8% accurate, 18.1× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return 1.0 + ((im * im) * ((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) * ((im * im) * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $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 64.6%

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

8. Taylor expanded in im around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 59.4%

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

10. Simplified 59.4%

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

11. Taylor expanded in im around inf

    \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\left({im}^{4} \cdot \left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)\right)}\right)\right) \]
12. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{\left(3 + 1\right)} \cdot \left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

    2. pow-plus N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left({im}^{3} \cdot im\right) \cdot \left(\color{blue}{\frac{1}{720}} + \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

    3. metadata-eval N/A

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

    4. pow-plus N/A

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

    5. associate-*r* N/A

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

    6. unpow2 N/A

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

    7. associate-*r* N/A

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

    8. distribute-rgt-in N/A

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

    9. associate-*l* N/A

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

    10. lft-mult-inverse N/A

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

    11. metadata-eval N/A

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

    12. +-commutative N/A

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

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

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

    14. unpow2 N/A

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

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

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

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

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

    17. *-commutative N/A

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

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

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

    19. unpow2 N/A

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

    20. *-lowering-*.f64 59.0%

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

13. Simplified 59.0%

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

Alternative 9: 51.4% accurate, 19.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.0)
   (+ 1.0 (* 0.5 (* im im)))
   (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (im * im) * (0.5 + ((im * im) * 0.041666666666666664));
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[im, 2.0], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 83.7%

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

   5. Simplified 83.7%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 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%

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

   if 2 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Simplified 82.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 65.8%

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

   8. Simplified 65.8%

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

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

          \[\leadsto {im}^{\left(3 + 1\right)} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \]

       2. pow-plus N/A

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

       3. unpow3 N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-in N/A

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

       9. associate-*l* N/A

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

       10. lft-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. +-commutative N/A

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

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

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

       14. unpow2 N/A

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

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

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 65.8%

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

   11. Simplified 65.8%

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

Alternative 10: 38.1% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.66 \cdot 10^{+131}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.2e+32)
   1.0
   (if (<= im 1.66e+131) (* (* re re) -0.5) (* 0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.2e+32) {
		tmp = 1.0;
	} else if (im <= 1.66e+131) {
		tmp = (re * re) * -0.5;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.2d+32) then
        tmp = 1.0d0
    else if (im <= 1.66d+131) then
        tmp = (re * re) * (-0.5d0)
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.2e+32) {
		tmp = 1.0;
	} else if (im <= 1.66e+131) {
		tmp = (re * re) * -0.5;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.2e+32:
		tmp = 1.0
	elif im <= 1.66e+131:
		tmp = (re * re) * -0.5
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.2e+32)
		tmp = 1.0;
	elseif (im <= 1.66e+131)
		tmp = Float64(Float64(re * re) * -0.5);
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.2e+32)
		tmp = 1.0;
	elseif (im <= 1.66e+131)
		tmp = (re * re) * -0.5;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.2e+32], 1.0, If[LessEqual[im, 1.66e+131], N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision], N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.2000000000000001e32

   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 63.0%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 34.6%

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

   if 4.2000000000000001e32 < im < 1.65999999999999992e131

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 27.4%

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

   8. Simplified 27.4%

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

   9. Taylor expanded in re around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 26.6%

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

   11. Simplified 26.6%

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

   if 1.65999999999999992e131 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 87.2%

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

   8. Simplified 87.2%

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

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

          \[\leadsto {im}^{\left(3 + 1\right)} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \]

       2. pow-plus N/A

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

       3. unpow3 N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-in N/A

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

       9. associate-*l* N/A

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

       10. lft-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. +-commutative N/A

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

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

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

       14. unpow2 N/A

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

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

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 87.2%

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

   11. Simplified 87.2%

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

   12. Taylor expanded in im around 0

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

   14. Simplified 73.4%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.66 \cdot 10^{+131}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.8% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.001388888888888889 * N[(im * N[(im * N[(im * im), $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 64.6%

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

8. Taylor expanded in im around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 59.4%

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

10. Simplified 59.4%

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

11. Taylor expanded in im around inf

    \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\left(\frac{1}{720} \cdot {im}^{4}\right)}\right)\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\frac{1}{720}, \color{blue}{\left({im}^{4}\right)}\right)\right)\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\frac{1}{720}, \left({im}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right)\right) \]

    3. pow-plus N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\frac{1}{720}, \left({im}^{3} \cdot \color{blue}{im}\right)\right)\right)\right) \]

    4. cube-unmult N/A

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

    5. unpow2 N/A

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

    6. *-commutative N/A

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

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

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

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

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

    9. unpow2 N/A

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

    10. *-lowering-*.f64 59.0%

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

13. Simplified 59.0%

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

Alternative 12: 51.4% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.5:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.5)
   (+ 1.0 (* 0.5 (* im im)))
   (* 0.041666666666666664 (* im (* im (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.5) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 0.041666666666666664 * (im * (im * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.5d0) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = 0.041666666666666664d0 * (im * (im * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.5) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 0.041666666666666664 * (im * (im * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.5:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = 0.041666666666666664 * (im * (im * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.5)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(im * Float64(im * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.5)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = 0.041666666666666664 * (im * (im * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.5], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 83.7%

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

   5. Simplified 83.7%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 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%

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

   if 3.5 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Simplified 82.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 65.8%

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

   8. Simplified 65.8%

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

   9. Taylor expanded in im around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-plus N/A

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

       4. *-commutative N/A

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

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

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 65.8%

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

   11. Simplified 65.8%

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

Alternative 13: 55.8% 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. Taylor expanded in im around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-out N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. distribute-lft-in N/A

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

5. Simplified 91.9%

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

6. Taylor expanded in re around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 58.2%

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

8. Simplified 58.2%

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

Alternative 14: 55.7% accurate, 28.0× speedup

Math

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

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(im * Float64(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[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $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. Taylor expanded in im around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-out N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. distribute-lft-in N/A

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

5. Simplified 91.9%

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

6. Taylor expanded in re around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 58.2%

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

8. Simplified 58.2%

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

9. Taylor expanded in im around inf

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

    1. *-commutative N/A

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

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

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

    3. unpow2 N/A

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

    4. *-lowering-*.f64 57.9%

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

11. Simplified 57.9%

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

Alternative 15: 30.6% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.8 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.8e+32) 1.0 (* (* re re) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.8e+32) {
		tmp = 1.0;
	} else {
		tmp = (re * re) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.8d+32) then
        tmp = 1.0d0
    else
        tmp = (re * re) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.8e+32) {
		tmp = 1.0;
	} else {
		tmp = (re * re) * -0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.8e+32:
		tmp = 1.0
	else:
		tmp = (re * re) * -0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.8e+32)
		tmp = 1.0;
	else
		tmp = Float64(Float64(re * re) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.8e+32)
		tmp = 1.0;
	else
		tmp = (re * re) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.8e+32], 1.0, N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.79999999999999983e32

   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 63.0%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 34.6%

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

   if 4.79999999999999983e32 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 14.0%

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

   8. Simplified 14.0%

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

   9. Taylor expanded in re around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 13.2%

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

   11. Simplified 13.2%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.0% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 75.4%

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

5. Simplified 75.4%

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

6. Taylor expanded in re around 0

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

   1. distribute-lft-in N/A

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

   2. metadata-eval N/A

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

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

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 46.2%

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

8. Simplified 46.2%

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

Alternative 17: 27.9% 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 49.2%

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

5. Simplified 49.2%

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

6. Taylor expanded in re around 0

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

8. Simplified 27.2%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))