math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.7s

Alternatives: 24

Speedup: 1.5×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (+ (/ (* 0.5 (sin re)) (exp im)) (* 0.5 (* (sin re) (exp im)))))

C

double code(double re, double im) {
	return ((0.5 * sin(re)) / exp(im)) + (0.5 * (sin(re) * exp(im)));
}

Fortran

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

Java

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

Python

def code(re, im):
	return ((0.5 * math.sin(re)) / math.exp(im)) + (0.5 * (math.sin(re) * math.exp(im)))

Julia

function code(re, im)
	return Float64(Float64(Float64(0.5 * sin(re)) / exp(im)) + Float64(0.5 * Float64(sin(re) * exp(im))))
end

MATLAB

function tmp = code(re, im)
	tmp = ((0.5 * sin(re)) / exp(im)) + (0.5 * (sin(re) * exp(im)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto e^{0 - im} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \color{blue}{e^{im} \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right), \color{blue}{\left(e^{im} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot e^{0 - im}\right), \left(\color{blue}{e^{im}} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)\right) \]

   4. sub0-neg N/A

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

   5. exp-neg N/A

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

   6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

   15. *-commutative N/A

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

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

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

   17. sin-lowering-sin.f64 N/A

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

   18. exp-lowering-exp.f64 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sin re}{e^{im}} + 0.5 \cdot \left(\sin re \cdot e^{im}\right)} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

   6. sub0-neg N/A

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

   7. cosh-undef N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. exp-0 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

   12. exp-0 N/A

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

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

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

   14. sin-lowering-sin.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 95.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{if}\;im \leq 0.52:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (sin re)
          (+
           1.0
           (*
            (* im im)
            (+
             0.5
             (*
              im
              (*
               im
               (+
                0.041666666666666664
                (* (* im im) 0.001388888888888889))))))))))
   (if (<= im 0.52) t_0 (if (<= im 7.2e+51) (* re (cosh im)) t_0))))

C

double code(double re, double im) {
	double t_0 = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	double tmp;
	if (im <= 0.52) {
		tmp = t_0;
	} else if (im <= 7.2e+51) {
		tmp = re * 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 = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))
    if (im <= 0.52d0) then
        tmp = t_0
    else if (im <= 7.2d+51) then
        tmp = re * 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.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	double tmp;
	if (im <= 0.52) {
		tmp = t_0;
	} else if (im <= 7.2e+51) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))
	tmp = 0
	if im <= 0.52:
		tmp = t_0
	elif im <= 7.2e+51:
		tmp = re * math.cosh(im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))))))
	tmp = 0.0
	if (im <= 0.52)
		tmp = t_0;
	elseif (im <= 7.2e+51)
		tmp = Float64(re * cosh(im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	tmp = 0.0;
	if (im <= 0.52)
		tmp = t_0;
	elseif (im <= 7.2e+51)
		tmp = re * cosh(im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.52], t$95$0, If[LessEqual[im, 7.2e+51], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.52000000000000002 or 7.20000000000000022e51 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 96.9%

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

   7. Simplified 96.9%

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

   if 0.52000000000000002 < im < 7.20000000000000022e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 72.7%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.52:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq 0.3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (sin re)
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))
   (if (<= im 0.3)
     t_0
     (if (<= im 5.8e+60)
       (* re (cosh im))
       (if (<= im 1.12e+77)
         (* re (+ 1.0 (* (* re re) -0.16666666666666666)))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= 0.3) {
		tmp = t_0;
	} else if (im <= 5.8e+60) {
		tmp = re * cosh(im);
	} else if (im <= 1.12e+77) {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	} 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 = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    if (im <= 0.3d0) then
        tmp = t_0
    else if (im <= 5.8d+60) then
        tmp = re * cosh(im)
    else if (im <= 1.12d+77) then
        tmp = re * (1.0d0 + ((re * re) * (-0.16666666666666666d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= 0.3) {
		tmp = t_0;
	} else if (im <= 5.8e+60) {
		tmp = re * Math.cosh(im);
	} else if (im <= 1.12e+77) {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	tmp = 0
	if im <= 0.3:
		tmp = t_0
	elif im <= 5.8e+60:
		tmp = re * math.cosh(im)
	elif im <= 1.12e+77:
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	tmp = 0.0
	if (im <= 0.3)
		tmp = t_0;
	elseif (im <= 5.8e+60)
		tmp = Float64(re * cosh(im));
	elseif (im <= 1.12e+77)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	tmp = 0.0;
	if (im <= 0.3)
		tmp = t_0;
	elseif (im <= 5.8e+60)
		tmp = re * cosh(im);
	elseif (im <= 1.12e+77)
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.3], t$95$0, If[LessEqual[im, 5.8e+60], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.12e+77], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.299999999999999989 or 1.1199999999999999e77 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 95.1%

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

   if 0.299999999999999989 < im < 5.79999999999999999e60

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 76.9%

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

   if 5.79999999999999999e60 < im < 1.1199999999999999e77

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.3:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{if}\;im \leq 0.057:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (* im im) 2.0))))
   (if (<= im 0.057)
     t_0
     (if (<= im 5.8e+60)
       (* re (cosh im))
       (if (<= im 1.35e+154)
         (*
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664))))
          (* re (+ 1.0 (* (* re re) -0.16666666666666666))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * ((im * im) + 2.0);
	double tmp;
	if (im <= 0.057) {
		tmp = t_0;
	} else if (im <= 5.8e+60) {
		tmp = re * cosh(im);
	} else if (im <= 1.35e+154) {
		tmp = (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664)))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	} 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 = (0.5d0 * sin(re)) * ((im * im) + 2.0d0)
    if (im <= 0.057d0) then
        tmp = t_0
    else if (im <= 5.8d+60) then
        tmp = re * cosh(im)
    else if (im <= 1.35d+154) then
        tmp = (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0)))) * (re * (1.0d0 + ((re * re) * (-0.16666666666666666d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * ((im * im) + 2.0);
	double tmp;
	if (im <= 0.057) {
		tmp = t_0;
	} else if (im <= 5.8e+60) {
		tmp = re * Math.cosh(im);
	} else if (im <= 1.35e+154) {
		tmp = (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664)))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * ((im * im) + 2.0)
	tmp = 0
	if im <= 0.057:
		tmp = t_0
	elif im <= 5.8e+60:
		tmp = re * math.cosh(im)
	elif im <= 1.35e+154:
		tmp = (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664)))) * (re * (1.0 + ((re * re) * -0.16666666666666666)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(Float64(im * im) + 2.0))
	tmp = 0.0
	if (im <= 0.057)
		tmp = t_0;
	elseif (im <= 5.8e+60)
		tmp = Float64(re * cosh(im));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))) * Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * ((im * im) + 2.0);
	tmp = 0.0;
	if (im <= 0.057)
		tmp = t_0;
	elseif (im <= 5.8e+60)
		tmp = re * cosh(im);
	elseif (im <= 1.35e+154)
		tmp = (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664)))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.057], t$95$0, If[LessEqual[im, 5.8e+60], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0570000000000000021 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 87.1%

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

   5. Simplified 87.1%

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

   if 0.0570000000000000021 < im < 5.79999999999999999e60

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 76.9%

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

   if 5.79999999999999999e60 < im < 1.35000000000000003e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 85.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 84.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 84.0%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.057:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot 0.041666666666666664\\ \mathbf{if}\;im \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + t\_0\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) 0.041666666666666664)))
   (if (<= im 1.18e-7)
     (sin re)
     (if (<= im 5.8e+60)
       (* re (cosh im))
       (if (<= im 5.6e+141)
         (*
          (+ 1.0 (* (* im im) (+ 0.5 t_0)))
          (* re (+ 1.0 (* (* re re) -0.16666666666666666))))
         (* im (* im (* re t_0))))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double tmp;
	if (im <= 1.18e-7) {
		tmp = sin(re);
	} else if (im <= 5.8e+60) {
		tmp = re * cosh(im);
	} else if (im <= 5.6e+141) {
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	} else {
		tmp = im * (im * (re * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im * im) * 0.041666666666666664d0
    if (im <= 1.18d-7) then
        tmp = sin(re)
    else if (im <= 5.8d+60) then
        tmp = re * cosh(im)
    else if (im <= 5.6d+141) then
        tmp = (1.0d0 + ((im * im) * (0.5d0 + t_0))) * (re * (1.0d0 + ((re * re) * (-0.16666666666666666d0))))
    else
        tmp = im * (im * (re * t_0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double tmp;
	if (im <= 1.18e-7) {
		tmp = Math.sin(re);
	} else if (im <= 5.8e+60) {
		tmp = re * Math.cosh(im);
	} else if (im <= 5.6e+141) {
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	} else {
		tmp = im * (im * (re * t_0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * 0.041666666666666664
	tmp = 0
	if im <= 1.18e-7:
		tmp = math.sin(re)
	elif im <= 5.8e+60:
		tmp = re * math.cosh(im)
	elif im <= 5.6e+141:
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)))
	else:
		tmp = im * (im * (re * t_0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * 0.041666666666666664)
	tmp = 0.0
	if (im <= 1.18e-7)
		tmp = sin(re);
	elseif (im <= 5.8e+60)
		tmp = Float64(re * cosh(im));
	elseif (im <= 5.6e+141)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + t_0))) * Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666))));
	else
		tmp = Float64(im * Float64(im * Float64(re * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * 0.041666666666666664;
	tmp = 0.0;
	if (im <= 1.18e-7)
		tmp = sin(re);
	elseif (im <= 5.8e+60)
		tmp = re * cosh(im);
	elseif (im <= 5.6e+141)
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	else
		tmp = im * (im * (re * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]}, If[LessEqual[im, 1.18e-7], N[Sin[re], $MachinePrecision], If[LessEqual[im, 5.8e+60], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6e+141], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.18e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 63.5%

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

   5. Simplified 63.5%

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

   if 1.18e-7 < im < 5.79999999999999999e60

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 75.3%

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

   if 5.79999999999999999e60 < im < 5.59999999999999982e141

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 84.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 83.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 83.3%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   if 5.59999999999999982e141 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 87.5%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

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

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 87.5%

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

   11. Simplified 87.5%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+141}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(-0.08333333333333333 + \frac{0.5}{re \cdot re}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.18e-7)
   (sin re)
   (if (<= im 1.75e+141)
     (*
      re
      (*
       re
       (*
        re
        (*
         (+
          2.0
          (*
           im
           (*
            im
            (+
             1.0
             (*
              im
              (*
               im
               (+ 0.08333333333333333 (* (* im im) 0.002777777777777778))))))))
         (+ -0.08333333333333333 (/ 0.5 (* re re)))))))
     (* im (* im (* re (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.18e-7) {
		tmp = sin(re);
	} else if (im <= 1.75e+141) {
		tmp = re * (re * (re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (-0.08333333333333333 + (0.5 / (re * re))))));
	} else {
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.18d-7) then
        tmp = sin(re)
    else if (im <= 1.75d+141) then
        tmp = re * (re * (re * ((2.0d0 + (im * (im * (1.0d0 + (im * (im * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0)))))))) * ((-0.08333333333333333d0) + (0.5d0 / (re * re))))))
    else
        tmp = im * (im * (re * ((im * im) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.18e-7) {
		tmp = Math.sin(re);
	} else if (im <= 1.75e+141) {
		tmp = re * (re * (re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (-0.08333333333333333 + (0.5 / (re * re))))));
	} else {
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.18e-7:
		tmp = math.sin(re)
	elif im <= 1.75e+141:
		tmp = re * (re * (re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (-0.08333333333333333 + (0.5 / (re * re))))))
	else:
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.18e-7)
		tmp = sin(re);
	elseif (im <= 1.75e+141)
		tmp = Float64(re * Float64(re * Float64(re * Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(im * Float64(im * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))))) * Float64(-0.08333333333333333 + Float64(0.5 / Float64(re * re)))))));
	else
		tmp = Float64(im * Float64(im * Float64(re * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.18e-7)
		tmp = sin(re);
	elseif (im <= 1.75e+141)
		tmp = re * (re * (re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (-0.08333333333333333 + (0.5 / (re * re))))));
	else
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.18e-7], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.75e+141], N[(re * N[(re * N[(re * N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(im * N[(im * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.08333333333333333 + N[(0.5 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(re * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.18e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 63.5%

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

   5. Simplified 63.5%

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

   if 1.18e-7 < im < 1.75e141

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 73.8%

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

   5. Simplified 73.8%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. fma-undefine N/A

         \[\leadsto re \cdot \left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2}\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2} + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 67.7%

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

   9. Taylor expanded in re around inf

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

   11. Simplified 55.0%

       \[\leadsto re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(-0.08333333333333333 + \frac{0.5}{re \cdot re}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

   13. Applied egg-rr 77.5%

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

   if 1.75e141 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 87.5%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

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

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 87.5%

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

   11. Simplified 87.5%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+141}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(-0.08333333333333333 + \frac{0.5}{re \cdot re}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\\ t_1 := im \cdot \left(-1 - t\_0\right)\\ \mathbf{if}\;im \leq 5.4 \cdot 10^{+51}:\\ \;\;\;\;re \cdot \frac{0.5 \cdot \left(4 + \left(im \cdot im\right) \cdot \left(\left(im \cdot \left(1 + t\_0\right)\right) \cdot t\_1\right)\right)}{2 + im \cdot t\_1}\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+142}:\\ \;\;\;\;re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (* im im)
          (+ 0.08333333333333333 (* im (* im 0.002777777777777778)))))
        (t_1 (* im (- -1.0 t_0))))
   (if (<= im 5.4e+51)
     (*
      re
      (/
       (* 0.5 (+ 4.0 (* (* im im) (* (* im (+ 1.0 t_0)) t_1))))
       (+ 2.0 (* im t_1))))
     (if (<= im 4.4e+142)
       (*
        re
        (*
         (+
          2.0
          (*
           im
           (*
            im
            (+
             1.0
             (*
              im
              (*
               im
               (+ 0.08333333333333333 (* (* im im) 0.002777777777777778))))))))
         (+ 0.5 (* (* re re) -0.08333333333333333))))
       (* im (* im (* re (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (0.08333333333333333 + (im * (im * 0.002777777777777778)));
	double t_1 = im * (-1.0 - t_0);
	double tmp;
	if (im <= 5.4e+51) {
		tmp = re * ((0.5 * (4.0 + ((im * im) * ((im * (1.0 + t_0)) * t_1)))) / (2.0 + (im * t_1)));
	} else if (im <= 4.4e+142) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)));
	} else {
		tmp = im * (im * (re * ((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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (im * im) * (0.08333333333333333d0 + (im * (im * 0.002777777777777778d0)))
    t_1 = im * ((-1.0d0) - t_0)
    if (im <= 5.4d+51) then
        tmp = re * ((0.5d0 * (4.0d0 + ((im * im) * ((im * (1.0d0 + t_0)) * t_1)))) / (2.0d0 + (im * t_1)))
    else if (im <= 4.4d+142) then
        tmp = re * ((2.0d0 + (im * (im * (1.0d0 + (im * (im * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0)))))))) * (0.5d0 + ((re * re) * (-0.08333333333333333d0))))
    else
        tmp = im * (im * (re * ((im * im) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (0.08333333333333333 + (im * (im * 0.002777777777777778)));
	double t_1 = im * (-1.0 - t_0);
	double tmp;
	if (im <= 5.4e+51) {
		tmp = re * ((0.5 * (4.0 + ((im * im) * ((im * (1.0 + t_0)) * t_1)))) / (2.0 + (im * t_1)));
	} else if (im <= 4.4e+142) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)));
	} else {
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (0.08333333333333333 + (im * (im * 0.002777777777777778)))
	t_1 = im * (-1.0 - t_0)
	tmp = 0
	if im <= 5.4e+51:
		tmp = re * ((0.5 * (4.0 + ((im * im) * ((im * (1.0 + t_0)) * t_1)))) / (2.0 + (im * t_1)))
	elif im <= 4.4e+142:
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)))
	else:
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(0.08333333333333333 + Float64(im * Float64(im * 0.002777777777777778))))
	t_1 = Float64(im * Float64(-1.0 - t_0))
	tmp = 0.0
	if (im <= 5.4e+51)
		tmp = Float64(re * Float64(Float64(0.5 * Float64(4.0 + Float64(Float64(im * im) * Float64(Float64(im * Float64(1.0 + t_0)) * t_1)))) / Float64(2.0 + Float64(im * t_1))));
	elseif (im <= 4.4e+142)
		tmp = Float64(re * Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(im * Float64(im * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))))) * Float64(0.5 + Float64(Float64(re * re) * -0.08333333333333333))));
	else
		tmp = Float64(im * Float64(im * Float64(re * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (0.08333333333333333 + (im * (im * 0.002777777777777778)));
	t_1 = im * (-1.0 - t_0);
	tmp = 0.0;
	if (im <= 5.4e+51)
		tmp = re * ((0.5 * (4.0 + ((im * im) * ((im * (1.0 + t_0)) * t_1)))) / (2.0 + (im * t_1)));
	elseif (im <= 4.4e+142)
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)));
	else
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(0.08333333333333333 + N[(im * N[(im * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 5.4e+51], N[(re * N[(N[(0.5 * N[(4.0 + N[(N[(im * im), $MachinePrecision] * N[(N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(im * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.4e+142], N[(re * N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(im * N[(im * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(re * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.39999999999999983e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in re around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{2} + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

   8. Simplified 58.4%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 37.5%

       \[\leadsto re \cdot \color{blue}{\frac{\left(4 - \left(im \cdot im\right) \cdot \left(\left(im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right) \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)\right)\right) \cdot 0.5}{2 - im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)}} \]

   if 5.39999999999999983e51 < im < 4.39999999999999974e142

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. fma-undefine N/A

         \[\leadsto re \cdot \left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2}\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2} + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 84.6%

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

   if 4.39999999999999974e142 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 87.5%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

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

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 87.5%

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

   11. Simplified 87.5%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.4 \cdot 10^{+51}:\\ \;\;\;\;re \cdot \frac{0.5 \cdot \left(4 + \left(im \cdot im\right) \cdot \left(\left(im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right) \cdot \left(im \cdot \left(-1 - \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)\right)\right)}{2 + im \cdot \left(im \cdot \left(-1 - \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)}\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+142}:\\ \;\;\;\;re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 132000:\\ \;\;\;\;re + \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.06 \cdot 10^{+142}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(-0.08333333333333333 + \frac{0.5}{re \cdot re}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 132000.0)
   (+
    re
    (*
     (+
      1.0
      (* (* im im) (+ 0.08333333333333333 (* im (* im 0.002777777777777778)))))
     (* re (* im (* 0.5 im)))))
   (if (<= im 2.06e+142)
     (*
      re
      (*
       re
       (*
        re
        (*
         (+
          2.0
          (*
           im
           (*
            im
            (+
             1.0
             (*
              im
              (*
               im
               (+ 0.08333333333333333 (* (* im im) 0.002777777777777778))))))))
         (+ -0.08333333333333333 (/ 0.5 (* re re)))))))
     (* im (* im (* re (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 132000.0) {
		tmp = re + ((1.0 + ((im * im) * (0.08333333333333333 + (im * (im * 0.002777777777777778))))) * (re * (im * (0.5 * im))));
	} else if (im <= 2.06e+142) {
		tmp = re * (re * (re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (-0.08333333333333333 + (0.5 / (re * re))))));
	} else {
		tmp = im * (im * (re * ((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 <= 132000.0d0) then
        tmp = re + ((1.0d0 + ((im * im) * (0.08333333333333333d0 + (im * (im * 0.002777777777777778d0))))) * (re * (im * (0.5d0 * im))))
    else if (im <= 2.06d+142) then
        tmp = re * (re * (re * ((2.0d0 + (im * (im * (1.0d0 + (im * (im * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0)))))))) * ((-0.08333333333333333d0) + (0.5d0 / (re * re))))))
    else
        tmp = im * (im * (re * ((im * im) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 132000.0) {
		tmp = re + ((1.0 + ((im * im) * (0.08333333333333333 + (im * (im * 0.002777777777777778))))) * (re * (im * (0.5 * im))));
	} else if (im <= 2.06e+142) {
		tmp = re * (re * (re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (-0.08333333333333333 + (0.5 / (re * re))))));
	} else {
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 132000.0:
		tmp = re + ((1.0 + ((im * im) * (0.08333333333333333 + (im * (im * 0.002777777777777778))))) * (re * (im * (0.5 * im))))
	elif im <= 2.06e+142:
		tmp = re * (re * (re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (-0.08333333333333333 + (0.5 / (re * re))))))
	else:
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 132000.0)
		tmp = Float64(re + Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.08333333333333333 + Float64(im * Float64(im * 0.002777777777777778))))) * Float64(re * Float64(im * Float64(0.5 * im)))));
	elseif (im <= 2.06e+142)
		tmp = Float64(re * Float64(re * Float64(re * Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(im * Float64(im * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))))) * Float64(-0.08333333333333333 + Float64(0.5 / Float64(re * re)))))));
	else
		tmp = Float64(im * Float64(im * Float64(re * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 132000.0)
		tmp = re + ((1.0 + ((im * im) * (0.08333333333333333 + (im * (im * 0.002777777777777778))))) * (re * (im * (0.5 * im))));
	elseif (im <= 2.06e+142)
		tmp = re * (re * (re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (-0.08333333333333333 + (0.5 / (re * re))))));
	else
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 132000.0], N[(re + N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.08333333333333333 + N[(im * N[(im * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.06e+142], N[(re * N[(re * N[(re * N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(im * N[(im * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.08333333333333333 + N[(0.5 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(re * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 132000

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in re around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{2} + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

   8. Simplified 60.7%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \left(\left(\frac{1}{2} \cdot \left(im \cdot im\right)\right) \cdot \left(1 + im \cdot \left(im \cdot \left(\frac{1}{12} + \left(im \cdot im\right) \cdot \frac{1}{360}\right)\right)\right) + \color{blue}{1}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\left(\frac{1}{2} \cdot \left(im \cdot im\right)\right) \cdot \left(1 + im \cdot \left(im \cdot \left(\frac{1}{12} + \left(im \cdot im\right) \cdot \frac{1}{360}\right)\right)\right)\right) + \color{blue}{re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\left(\frac{1}{2} \cdot \left(im \cdot im\right)\right) \cdot \left(1 + im \cdot \left(im \cdot \left(\frac{1}{12} + \left(im \cdot im\right) \cdot \frac{1}{360}\right)\right)\right)\right) + re \]

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

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

   10. Applied egg-rr 60.2%

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

   if 132000 < im < 2.0599999999999999e142

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. fma-undefine N/A

         \[\leadsto re \cdot \left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2}\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2} + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 67.9%

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

   9. Taylor expanded in re around inf

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

   11. Simplified 54.1%

       \[\leadsto re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(-0.08333333333333333 + \frac{0.5}{re \cdot re}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

   13. Applied egg-rr 78.4%

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

   if 2.0599999999999999e142 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 87.5%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

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

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 87.5%

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

   11. Simplified 87.5%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 132000:\\ \;\;\;\;re + \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.06 \cdot 10^{+142}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(-0.08333333333333333 + \frac{0.5}{re \cdot re}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.1% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot 0.041666666666666664\\ \mathbf{if}\;im \leq 5.8 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + t\_0\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+140}:\\ \;\;\;\;re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) 0.041666666666666664)))
   (if (<= im 5.8e+51)
     (*
      (+ 1.0 (* (* im im) (+ 0.5 t_0)))
      (*
       re
       (+
        1.0
        (*
         (* re re)
         (+ -0.16666666666666666 (* (* re re) 0.008333333333333333))))))
     (if (<= im 4.8e+140)
       (*
        re
        (*
         (+
          2.0
          (*
           im
           (*
            im
            (+
             1.0
             (*
              im
              (*
               im
               (+ 0.08333333333333333 (* (* im im) 0.002777777777777778))))))))
         (+ 0.5 (* (* re re) -0.08333333333333333))))
       (* im (* im (* re t_0)))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double tmp;
	if (im <= 5.8e+51) {
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333)))));
	} else if (im <= 4.8e+140) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)));
	} else {
		tmp = im * (im * (re * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im * im) * 0.041666666666666664d0
    if (im <= 5.8d+51) then
        tmp = (1.0d0 + ((im * im) * (0.5d0 + t_0))) * (re * (1.0d0 + ((re * re) * ((-0.16666666666666666d0) + ((re * re) * 0.008333333333333333d0)))))
    else if (im <= 4.8d+140) then
        tmp = re * ((2.0d0 + (im * (im * (1.0d0 + (im * (im * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0)))))))) * (0.5d0 + ((re * re) * (-0.08333333333333333d0))))
    else
        tmp = im * (im * (re * t_0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double tmp;
	if (im <= 5.8e+51) {
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333)))));
	} else if (im <= 4.8e+140) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)));
	} else {
		tmp = im * (im * (re * t_0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * 0.041666666666666664
	tmp = 0
	if im <= 5.8e+51:
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333)))))
	elif im <= 4.8e+140:
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)))
	else:
		tmp = im * (im * (re * t_0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * 0.041666666666666664)
	tmp = 0.0
	if (im <= 5.8e+51)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + t_0))) * Float64(re * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.008333333333333333))))));
	elseif (im <= 4.8e+140)
		tmp = Float64(re * Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(im * Float64(im * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))))) * Float64(0.5 + Float64(Float64(re * re) * -0.08333333333333333))));
	else
		tmp = Float64(im * Float64(im * Float64(re * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * 0.041666666666666664;
	tmp = 0.0;
	if (im <= 5.8e+51)
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333)))));
	elseif (im <= 4.8e+140)
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)));
	else
		tmp = im * (im * (re * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]}, If[LessEqual[im, 5.8e+51], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.8e+140], N[(re * N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(im * N[(im * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.7999999999999997e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 88.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{120} \cdot {re}^{2} + \frac{-1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{6} + \frac{1}{120} \cdot {re}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\frac{1}{120} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{120}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 59.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{120}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 59.7%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   if 5.7999999999999997e51 < im < 4.7999999999999999e140

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. fma-undefine N/A

         \[\leadsto re \cdot \left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2}\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2} + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 84.6%

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

   if 4.7999999999999999e140 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 87.5%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

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

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 87.5%

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

   11. Simplified 87.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.8 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+140}:\\ \;\;\;\;re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.05 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+140}:\\ \;\;\;\;re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.05e+51)
   (*
    (+ 1.0 (* 0.5 (* im im)))
    (*
     re
     (+
      1.0
      (*
       re
       (* re (+ -0.16666666666666666 (* (* re re) 0.008333333333333333)))))))
   (if (<= im 7e+140)
     (*
      re
      (*
       (+
        2.0
        (*
         im
         (*
          im
          (+
           1.0
           (*
            im
            (*
             im
             (+ 0.08333333333333333 (* (* im im) 0.002777777777777778))))))))
       (+ 0.5 (* (* re re) -0.08333333333333333))))
     (* im (* im (* re (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.05e+51) {
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + (re * (re * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))));
	} else if (im <= 7e+140) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)));
	} else {
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.05d+51) then
        tmp = (1.0d0 + (0.5d0 * (im * im))) * (re * (1.0d0 + (re * (re * ((-0.16666666666666666d0) + ((re * re) * 0.008333333333333333d0))))))
    else if (im <= 7d+140) then
        tmp = re * ((2.0d0 + (im * (im * (1.0d0 + (im * (im * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0)))))))) * (0.5d0 + ((re * re) * (-0.08333333333333333d0))))
    else
        tmp = im * (im * (re * ((im * im) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.05e+51) {
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + (re * (re * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))));
	} else if (im <= 7e+140) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)));
	} else {
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.05e+51:
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + (re * (re * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))))
	elif im <= 7e+140:
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)))
	else:
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.05e+51)
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(im * im))) * Float64(re * Float64(1.0 + Float64(re * Float64(re * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.008333333333333333)))))));
	elseif (im <= 7e+140)
		tmp = Float64(re * Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(im * Float64(im * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))))) * Float64(0.5 + Float64(Float64(re * re) * -0.08333333333333333))));
	else
		tmp = Float64(im * Float64(im * Float64(re * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.05e+51)
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + (re * (re * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))));
	elseif (im <= 7e+140)
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + ((re * re) * -0.08333333333333333)));
	else
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.05e+51], N[(N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(re * N[(re * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7e+140], N[(re * N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(im * N[(im * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(re * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.0500000000000001e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {re}^{2}} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{120} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{120} \cdot {re}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {re}^{2}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{120}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 53.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{120}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 53.9%

       \[\leadsto \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\right)\right)} \]

   if 1.0500000000000001e51 < im < 6.99999999999999978e140

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. fma-undefine N/A

         \[\leadsto re \cdot \left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2}\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2} + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 84.6%

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

   if 6.99999999999999978e140 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 87.5%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

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

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 87.5%

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

   11. Simplified 87.5%

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

Alternative 12: 53.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot 0.041666666666666664\\ \mathbf{if}\;im \leq 3.6 \cdot 10^{+60}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.52 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + t\_0\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) 0.041666666666666664)))
   (if (<= im 3.6e+60)
     (*
      (+ 1.0 (* 0.5 (* im im)))
      (*
       re
       (+
        1.0
        (*
         re
         (* re (+ -0.16666666666666666 (* (* re re) 0.008333333333333333)))))))
     (if (<= im 1.52e+141)
       (*
        (+ 1.0 (* (* im im) (+ 0.5 t_0)))
        (* re (+ 1.0 (* (* re re) -0.16666666666666666))))
       (* im (* im (* re t_0)))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double tmp;
	if (im <= 3.6e+60) {
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + (re * (re * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))));
	} else if (im <= 1.52e+141) {
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	} else {
		tmp = im * (im * (re * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im * im) * 0.041666666666666664d0
    if (im <= 3.6d+60) then
        tmp = (1.0d0 + (0.5d0 * (im * im))) * (re * (1.0d0 + (re * (re * ((-0.16666666666666666d0) + ((re * re) * 0.008333333333333333d0))))))
    else if (im <= 1.52d+141) then
        tmp = (1.0d0 + ((im * im) * (0.5d0 + t_0))) * (re * (1.0d0 + ((re * re) * (-0.16666666666666666d0))))
    else
        tmp = im * (im * (re * t_0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double tmp;
	if (im <= 3.6e+60) {
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + (re * (re * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))));
	} else if (im <= 1.52e+141) {
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	} else {
		tmp = im * (im * (re * t_0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * 0.041666666666666664
	tmp = 0
	if im <= 3.6e+60:
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + (re * (re * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))))
	elif im <= 1.52e+141:
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)))
	else:
		tmp = im * (im * (re * t_0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * 0.041666666666666664)
	tmp = 0.0
	if (im <= 3.6e+60)
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(im * im))) * Float64(re * Float64(1.0 + Float64(re * Float64(re * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.008333333333333333)))))));
	elseif (im <= 1.52e+141)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + t_0))) * Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666))));
	else
		tmp = Float64(im * Float64(im * Float64(re * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * 0.041666666666666664;
	tmp = 0.0;
	if (im <= 3.6e+60)
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + (re * (re * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))));
	elseif (im <= 1.52e+141)
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	else
		tmp = im * (im * (re * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]}, If[LessEqual[im, 3.6e+60], N[(N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(re * N[(re * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.52e+141], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.59999999999999968e60

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {re}^{2}} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{120} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{120} \cdot {re}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {re}^{2}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{120}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 53.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{120}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 53.4%

       \[\leadsto \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\right)\right)} \]

   if 3.59999999999999968e60 < im < 1.52e141

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 84.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 83.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 83.3%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   if 1.52e141 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 87.5%

      \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{re} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)}\right) \cdot re \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right) \cdot re \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re \]

       5. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

       6. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re\right)} \]

       7. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot re\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot re\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)}\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{re}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 87.5%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   11. Simplified 87.5%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+60}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.52 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot 0.041666666666666664\\ \mathbf{if}\;im \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+140}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + t\_0\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) 0.041666666666666664)))
   (if (<= im 5.8e+60)
     (*
      re
      (+
       1.0
       (*
        (* im im)
        (+
         0.5
         (*
          (* im im)
          (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))))))
     (if (<= im 9.6e+140)
       (*
        (+ 1.0 (* (* im im) (+ 0.5 t_0)))
        (* re (+ 1.0 (* (* re re) -0.16666666666666666))))
       (* im (* im (* re t_0)))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double tmp;
	if (im <= 5.8e+60) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	} else if (im <= 9.6e+140) {
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	} else {
		tmp = im * (im * (re * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im * im) * 0.041666666666666664d0
    if (im <= 5.8d+60) then
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0))))))
    else if (im <= 9.6d+140) then
        tmp = (1.0d0 + ((im * im) * (0.5d0 + t_0))) * (re * (1.0d0 + ((re * re) * (-0.16666666666666666d0))))
    else
        tmp = im * (im * (re * t_0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double tmp;
	if (im <= 5.8e+60) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	} else if (im <= 9.6e+140) {
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	} else {
		tmp = im * (im * (re * t_0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * 0.041666666666666664
	tmp = 0
	if im <= 5.8e+60:
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))
	elif im <= 9.6e+140:
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)))
	else:
		tmp = im * (im * (re * t_0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * 0.041666666666666664)
	tmp = 0.0
	if (im <= 5.8e+60)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)))))));
	elseif (im <= 9.6e+140)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + t_0))) * Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666))));
	else
		tmp = Float64(im * Float64(im * Float64(re * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * 0.041666666666666664;
	tmp = 0.0;
	if (im <= 5.8e+60)
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	elseif (im <= 9.6e+140)
		tmp = (1.0 + ((im * im) * (0.5 + t_0))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	else
		tmp = im * (im * (re * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]}, If[LessEqual[im, 5.8e+60], N[(re * 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]), $MachinePrecision], If[LessEqual[im, 9.6e+140], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.79999999999999999e60

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \sin re\right) \]

      7. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \sin \color{blue}{re}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

      13. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}, \mathsf{sin.f64}\left(re\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{re}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      14. *-lowering-*.f64 90.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   7. Simplified 90.9%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 58.9%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 58.9%

       \[\leadsto \color{blue}{re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)} \]

   if 5.79999999999999999e60 < im < 9.5999999999999999e140

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 84.7%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 83.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 83.3%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   if 9.5999999999999999e140 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 87.5%

      \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{re} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)}\right) \cdot re \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right) \cdot re \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re \]

       5. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

       6. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re\right)} \]

       7. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot re\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot re\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)}\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{re}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 87.5%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   11. Simplified 87.5%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+140}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+219}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.5e+219)
   (*
    re
    (+
     1.0
     (*
      (* im im)
      (+
       0.5
       (*
        (* im im)
        (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))))))
   (* re (+ 1.0 (* (* re re) -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+219) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	} else {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.5d+219) then
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0))))))
    else
        tmp = re * (1.0d0 + ((re * re) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+219) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	} else {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.5e+219:
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))
	else:
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.5e+219)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)))))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.5e+219)
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	else
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.5e+219], N[(re * 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]), $MachinePrecision], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.4999999999999999e219

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \sin re\right) \]

      7. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \sin \color{blue}{re}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

      13. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}, \mathsf{sin.f64}\left(re\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{re}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      14. *-lowering-*.f64 93.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   7. Simplified 93.5%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 64.2%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 64.2%

       \[\leadsto \color{blue}{re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)} \]

   if 1.4999999999999999e219 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 27.3%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 27.3%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 30.5%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 30.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 58.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+219}:\\ \;\;\;\;re \cdot \left(1 + \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + im \cdot \left(0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.5e+219)
   (*
    re
    (+
     1.0
     (*
      (* 0.5 (* im im))
      (+ 1.0 (* im (* 0.002777777777777778 (* im (* im im))))))))
   (* re (+ 1.0 (* (* re re) -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+219) {
		tmp = re * (1.0 + ((0.5 * (im * im)) * (1.0 + (im * (0.002777777777777778 * (im * (im * im)))))));
	} else {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.5d+219) then
        tmp = re * (1.0d0 + ((0.5d0 * (im * im)) * (1.0d0 + (im * (0.002777777777777778d0 * (im * (im * im)))))))
    else
        tmp = re * (1.0d0 + ((re * re) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+219) {
		tmp = re * (1.0 + ((0.5 * (im * im)) * (1.0 + (im * (0.002777777777777778 * (im * (im * im)))))));
	} else {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.5e+219:
		tmp = re * (1.0 + ((0.5 * (im * im)) * (1.0 + (im * (0.002777777777777778 * (im * (im * im)))))))
	else:
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.5e+219)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(0.5 * Float64(im * im)) * Float64(1.0 + Float64(im * Float64(0.002777777777777778 * Float64(im * Float64(im * im))))))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.5e+219)
		tmp = re * (1.0 + ((0.5 * (im * im)) * (1.0 + (im * (0.002777777777777778 * (im * (im * im)))))));
	else
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.5e+219], N[(re * N[(1.0 + N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(im * N[(0.002777777777777778 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.4999999999999999e219

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \left({im}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 93.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 93.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(re \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{2} + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)}\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(2 \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \frac{1}{2}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(1 + \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {im}^{2}\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   8. Simplified 64.2%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{360} \cdot {im}^{3}\right)}\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{360}, \color{blue}{\left({im}^{3}\right)}\right)\right)\right)\right)\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{360}, \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right)\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{360}, \left(im \cdot {im}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 64.2%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 64.2%

       \[\leadsto re \cdot \left(1 + \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + im \cdot \color{blue}{\left(0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)}\right)\right) \]

   if 1.4999999999999999e219 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 27.3%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 27.3%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 30.5%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 30.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 46.3% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im + 2\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.9e+60)
   (*
    re
    (+
     1.0
     (*
      (* re re)
      (+ -0.16666666666666666 (* (* re re) 0.008333333333333333)))))
   (if (<= im 2.1e+79)
     (* re (* (+ (* im im) 2.0) (+ 0.5 (* re (* re -0.08333333333333333)))))
     (* re (+ 1.0 (* im (* im (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+60) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	} else if (im <= 2.1e+79) {
		tmp = re * (((im * im) + 2.0) * (0.5 + (re * (re * -0.08333333333333333))));
	} else {
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.9d+60) then
        tmp = re * (1.0d0 + ((re * re) * ((-0.16666666666666666d0) + ((re * re) * 0.008333333333333333d0))))
    else if (im <= 2.1d+79) then
        tmp = re * (((im * im) + 2.0d0) * (0.5d0 + (re * (re * (-0.08333333333333333d0)))))
    else
        tmp = re * (1.0d0 + (im * (im * ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+60) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	} else if (im <= 2.1e+79) {
		tmp = re * (((im * im) + 2.0) * (0.5 + (re * (re * -0.08333333333333333))));
	} else {
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.9e+60:
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))
	elif im <= 2.1e+79:
		tmp = re * (((im * im) + 2.0) * (0.5 + (re * (re * -0.08333333333333333))))
	else:
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.9e+60)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.008333333333333333)))));
	elseif (im <= 2.1e+79)
		tmp = Float64(re * Float64(Float64(Float64(im * im) + 2.0) * Float64(0.5 + Float64(re * Float64(re * -0.08333333333333333)))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.9e+60)
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	elseif (im <= 2.1e+79)
		tmp = re * (((im * im) + 2.0) * (0.5 + (re * (re * -0.08333333333333333))));
	else
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.9e+60], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.1e+79], N[(re * N[(N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.9000000000000003e60

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 59.6%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 59.6%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{120} \cdot {re}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{120} \cdot {re}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{120} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{120} \cdot {re}^{2} + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {re}^{2}}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2}\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 40.9%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 40.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)} \]

   if 4.9000000000000003e60 < im < 2.10000000000000008e79

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \left({im}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)}, re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \frac{-1}{12} \cdot \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{2}}\right), re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(re, \left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot \color{blue}{{re}^{2}}, re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. fma-undefine N/A

         \[\leadsto re \cdot \left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2}\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2} + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + {im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

       2. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \left(2 + {im}^{2}\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto re \cdot \left(2 \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(2 \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)}\right) \]

       5. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right), \color{blue}{\left(2 + {im}^{2}\right)}\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{12} \cdot {re}^{2}\right)\right), \left(\color{blue}{2} + {im}^{2}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{12}\right)\right), \left(2 + {im}^{2}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right), \left(2 + {im}^{2}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \left(re \cdot \frac{-1}{12}\right)\right)\right), \left(2 + {im}^{2}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \frac{-1}{12}\right)\right)\right), \left(2 + {im}^{2}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right), \left(2 + {im}^{2}\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right), \mathsf{+.f64}\left(2, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{re \cdot \left(\left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right) \cdot \left(2 + im \cdot im\right)\right)} \]

   if 2.10000000000000008e79 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 84.3%

      \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 84.3%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   11. Simplified 84.3%

       \[\leadsto re \cdot \left(1 + \color{blue}{im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im + 2\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.4% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im + 2\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+79)
   (* re (* (+ (* im im) 2.0) (+ 0.5 (* re (* re -0.08333333333333333)))))
   (* re (+ 1.0 (* im (* im (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+79) {
		tmp = re * (((im * im) + 2.0) * (0.5 + (re * (re * -0.08333333333333333))));
	} else {
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.1d+79) then
        tmp = re * (((im * im) + 2.0d0) * (0.5d0 + (re * (re * (-0.08333333333333333d0)))))
    else
        tmp = re * (1.0d0 + (im * (im * ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+79) {
		tmp = re * (((im * im) + 2.0) * (0.5 + (re * (re * -0.08333333333333333))));
	} else {
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+79:
		tmp = re * (((im * im) + 2.0) * (0.5 + (re * (re * -0.08333333333333333))))
	else:
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+79)
		tmp = Float64(re * Float64(Float64(Float64(im * im) + 2.0) * Float64(0.5 + Float64(re * Float64(re * -0.08333333333333333)))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+79)
		tmp = re * (((im * im) + 2.0) * (0.5 + (re * (re * -0.08333333333333333))));
	else
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+79], N[(re * N[(N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.10000000000000008e79

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \left({im}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 91.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 91.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)}, re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \frac{-1}{12} \cdot \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{2}}\right), re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(re, \left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot \color{blue}{{re}^{2}}, re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. fma-undefine N/A

         \[\leadsto re \cdot \left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2}\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2} + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 57.9%

      \[\leadsto \color{blue}{re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + {im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

       2. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \left(2 + {im}^{2}\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto re \cdot \left(2 \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(2 \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)}\right) \]

       5. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right), \color{blue}{\left(2 + {im}^{2}\right)}\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{12} \cdot {re}^{2}\right)\right), \left(\color{blue}{2} + {im}^{2}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{12}\right)\right), \left(2 + {im}^{2}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right), \left(2 + {im}^{2}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \left(re \cdot \frac{-1}{12}\right)\right)\right), \left(2 + {im}^{2}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \frac{-1}{12}\right)\right)\right), \left(2 + {im}^{2}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right), \left(2 + {im}^{2}\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right), \mathsf{+.f64}\left(2, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 50.4%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 50.4%

       \[\leadsto \color{blue}{re \cdot \left(\left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right) \cdot \left(2 + im \cdot im\right)\right)} \]

   if 2.10000000000000008e79 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 84.3%

      \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 84.3%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   11. Simplified 84.3%

       \[\leadsto re \cdot \left(1 + \color{blue}{im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im + 2\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.5% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+79)
   (* re (+ 1.0 (* (* re re) -0.16666666666666666)))
   (* re (+ 1.0 (* im (* im (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+79) {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	} else {
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.1d+79) then
        tmp = re * (1.0d0 + ((re * re) * (-0.16666666666666666d0)))
    else
        tmp = re * (1.0d0 + (im * (im * ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+79) {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	} else {
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+79:
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666))
	else:
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+79)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666)));
	else
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+79)
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	else
		tmp = re * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+79], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.10000000000000008e79

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 58.2%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 58.2%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 39.2%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 39.2%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]

   if 2.10000000000000008e79 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 84.3%

      \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 84.3%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   11. Simplified 84.3%

       \[\leadsto re \cdot \left(1 + \color{blue}{im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 44.5% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+79)
   (* re (+ 1.0 (* (* re re) -0.16666666666666666)))
   (* re (* im (* im (* im (* im 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+79) {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	} else {
		tmp = re * (im * (im * (im * (im * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.1d+79) then
        tmp = re * (1.0d0 + ((re * re) * (-0.16666666666666666d0)))
    else
        tmp = re * (im * (im * (im * (im * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+79) {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	} else {
		tmp = re * (im * (im * (im * (im * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+79:
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666))
	else:
		tmp = re * (im * (im * (im * (im * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+79)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666)));
	else
		tmp = Float64(re * Float64(im * Float64(im * Float64(im * Float64(im * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+79)
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	else
		tmp = re * (im * (im * (im * (im * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+79], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.10000000000000008e79

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 58.2%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 58.2%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 39.2%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 39.2%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]

   if 2.10000000000000008e79 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 84.3%

      \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(im \cdot \left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right) \cdot \color{blue}{im}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot \left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right), \color{blue}{im}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right), im\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right), im\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)\right), im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(im \cdot \frac{1}{24}\right)\right)\right)\right), im\right)\right)\right) \]

      8. *-lowering-*.f64 84.3%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right), im\right)\right)\right) \]

   10. Applied egg-rr 84.3%

       \[\leadsto re \cdot \left(1 + \color{blue}{\left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot im}\right) \]

   11. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{re} \]

       2. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

          \[\leadsto re \cdot \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

       4. pow-sqr N/A

          \[\leadsto re \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto re \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right) \]

       6. *-commutative N/A

          \[\leadsto re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{24} \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot im\right)}\right)\right)\right)\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

       18. *-lowering-*.f64 84.3%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   13. Simplified 84.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 43.2% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+79)
   (* re (+ 1.0 (* (* re re) -0.16666666666666666)))
   (* im (* im (* re (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+79) {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	} else {
		tmp = im * (im * (re * ((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 <= 2.1d+79) then
        tmp = re * (1.0d0 + ((re * re) * (-0.16666666666666666d0)))
    else
        tmp = im * (im * (re * ((im * im) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+79) {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	} else {
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+79:
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666))
	else:
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+79)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666)));
	else
		tmp = Float64(im * Float64(im * Float64(re * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+79)
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	else
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+79], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(re * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.10000000000000008e79

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 58.2%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 58.2%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 39.2%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 39.2%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]

   if 2.10000000000000008e79 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 84.3%

      \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{re} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)}\right) \cdot re \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right) \cdot re \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re \]

       5. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

       6. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re\right)} \]

       7. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot re\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot re\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)}\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{re}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 67.8%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   11. Simplified 67.8%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 40.8% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.55 \cdot 10^{+140}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.55e+140)
   (* re (+ 1.0 (* (* re re) -0.16666666666666666)))
   (* re (* 0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.55e+140) {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.55d+140) then
        tmp = re * (1.0d0 + ((re * re) * (-0.16666666666666666d0)))
    else
        tmp = re * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.55e+140) {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.55e+140:
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666))
	else:
		tmp = re * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.55e+140)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666)));
	else
		tmp = Float64(re * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.55e+140)
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	else
		tmp = re * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.55e+140], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.55e140

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 53.5%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 53.5%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 37.8%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 37.8%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]

   if 1.55e140 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 87.5%

      \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re + \frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity N/A

          \[\leadsto 1 \cdot re + \color{blue}{\frac{1}{2}} \cdot \left({im}^{2} \cdot re\right) \]

       2. associate-*r* N/A

          \[\leadsto 1 \cdot re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{re} \]

       3. distribute-rgt-out N/A

          \[\leadsto re \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 84.6%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 84.6%

       \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       3. *-lowering-*.f64 84.6%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   14. Simplified 84.6%

       \[\leadsto re \cdot \color{blue}{\left(0.5 \cdot \left(im \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 37.0% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0185:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0185) re (* re (* 0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0185) {
		tmp = re;
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0185d0) then
        tmp = re
    else
        tmp = re * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0185) {
		tmp = re;
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0185:
		tmp = re
	else:
		tmp = re * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0185)
		tmp = re;
	else
		tmp = Float64(re * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0185)
		tmp = re;
	else
		tmp = re * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0185], re, N[(re * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0184999999999999991

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 63.7%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 63.7%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \]
   7. Step-by-step derivation

   8. Simplified 34.7%

      \[\leadsto \color{blue}{re} \]

   if 0.0184999999999999991 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 76.7%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 64.8%

      \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re + \frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity N/A

          \[\leadsto 1 \cdot re + \color{blue}{\frac{1}{2}} \cdot \left({im}^{2} \cdot re\right) \]

       2. associate-*r* N/A

          \[\leadsto 1 \cdot re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{re} \]

       3. distribute-rgt-out N/A

          \[\leadsto re \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 45.7%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 45.7%

       \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       3. *-lowering-*.f64 45.7%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   14. Simplified 45.7%

       \[\leadsto re \cdot \color{blue}{\left(0.5 \cdot \left(im \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 48.0% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re (+ 1.0 (* 0.5 (* im im)))))

C

double code(double re, double im) {
	return re * (1.0 + (0.5 * (im * im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (1.0d0 + (0.5d0 * (im * im)))
end function

Java

public static double code(double re, double im) {
	return re * (1.0 + (0.5 * (im * im)));
}

Python

def code(re, im):
	return re * (1.0 + (0.5 * (im * im)))

Julia

function code(re, im)
	return Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (1.0 + (0.5 * (im * im)));
end

Wolfram

code[re_, im_] := N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

   2. +-commutative N/A

      \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

   6. distribute-lft-out N/A

      \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

   14. distribute-lft-out N/A

       \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

5. Simplified 89.1%

   \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 61.2%

   \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

9. Taylor expanded in im around 0

   \[\leadsto \color{blue}{re + \frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} \]
10. Step-by-step derivation

    1. *-lft-identity N/A

       \[\leadsto 1 \cdot re + \color{blue}{\frac{1}{2}} \cdot \left({im}^{2} \cdot re\right) \]

    2. associate-*r* N/A

       \[\leadsto 1 \cdot re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{re} \]

    3. distribute-rgt-out N/A

       \[\leadsto re \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

    5. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

    8. *-lowering-*.f64 51.5%

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

11. Simplified 51.5%

    \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]
12. Add Preprocessing

Alternative 24: 26.5% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 re)

C

double code(double re, double im) {
	return re;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re
end function

Java

public static double code(double re, double im) {
	return re;
}

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

function tmp = code(re, im)
	tmp = re;
end

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 47.1%

      \[\leadsto \mathsf{sin.f64}\left(re\right) \]

5. Simplified 47.1%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re} \]
7. Step-by-step derivation

8. Simplified 26.0%

   \[\leadsto \color{blue}{re} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))