math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.7s

Alternatives: 18

Speedup: 1.5×

• 50.1% 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

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

FPCore

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

C

double code(double re, double im) {
	return ((0.5 * sin(re)) / exp(im)) + (sin(re) * (0.5 * 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)) + (sin(re) * (0.5d0 * exp(im)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. exp-lowering-exp.f64 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sin re}{e^{im}} + \sin re \cdot \left(0.5 \cdot e^{im}\right)} \]
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. Step-by-step derivation

   1. *-commutative N/A

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

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

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

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

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

   4. *-lft-identity N/A

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

   5. cosh-lowering-cosh.f64 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
7. Add Preprocessing

Alternative 3: 79.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{-8}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.2e-8)
   (* re (cosh im))
   (*
    (sin re)
    (+
     1.0
     (*
      (* im im)
      (+
       0.5
       (*
        im
        (*
         im
         (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.2e-8) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.2d-8) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.2e-8) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.2e-8:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.2e-8)
		tmp = Float64(re * cosh(im));
	else
		tmp = 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))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.2e-8)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.2e-8], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if re < 1.19999999999999999e-8

   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. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 78.8%

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

   if 1.19999999999999999e-8 < 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. 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 91.7%

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

   7. Simplified 91.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{-8}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{-9}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2e-9)
   (* re (cosh im))
   (*
    (sin re)
    (+ 1.0 (* (* im im) (+ 0.5 (* im (* im 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2e-9) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2d-9) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2e-9) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2e-9:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2e-9)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2e-9)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2e-9], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.00000000000000012e-9

   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. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 78.8%

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

   if 2.00000000000000012e-9 < 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

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   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 90.2%

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

Alternative 5: 85.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{if}\;im \leq 0.016:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (* im im) 2.0))))
   (if (<= im 0.016) t_0 (if (<= im 1.35e+154) (* re (cosh im)) t_0))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * ((im * im) + 2.0);
	double tmp;
	if (im <= 0.016) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		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 = (0.5d0 * sin(re)) * ((im * im) + 2.0d0)
    if (im <= 0.016d0) then
        tmp = t_0
    else if (im <= 1.35d+154) 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 = (0.5 * Math.sin(re)) * ((im * im) + 2.0);
	double tmp;
	if (im <= 0.016) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = re * Math.cosh(im);
	} 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 <= 0.016:
		tmp = t_0
	elif im <= 1.35e+154:
		tmp = re * math.cosh(im)
	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 <= 0.016)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = Float64(re * cosh(im));
	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 <= 0.016)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = re * cosh(im);
	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, 0.016], t$95$0, If[LessEqual[im, 1.35e+154], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.016 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 81.9%

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

   5. Simplified 81.9%

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

   if 0.016 < 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. 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. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 77.4%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.016:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+156}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.2e-7)
   (sin re)
   (if (<= im 2e+156)
     (* re (cosh im))
     (*
      (* im (* re (+ 1.0 (* re (* re -0.16666666666666666)))))
      (* im (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.2e-7) {
		tmp = sin(re);
	} else if (im <= 2e+156) {
		tmp = re * cosh(im);
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (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 <= 9.2d-7) then
        tmp = sin(re)
    else if (im <= 2d+156) then
        tmp = re * cosh(im)
    else
        tmp = (im * (re * (1.0d0 + (re * (re * (-0.16666666666666666d0)))))) * (im * ((im * im) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.2e-7) {
		tmp = Math.sin(re);
	} else if (im <= 2e+156) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (im * ((im * im) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.2e-7:
		tmp = math.sin(re)
	elif im <= 2e+156:
		tmp = re * math.cosh(im)
	else:
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (im * ((im * im) * 0.041666666666666664))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.2e-7)
		tmp = sin(re);
	elseif (im <= 2e+156)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666))))) * Float64(im * Float64(Float64(im * im) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.2e-7)
		tmp = sin(re);
	elseif (im <= 2e+156)
		tmp = re * cosh(im);
	else
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (im * ((im * im) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.2e-7], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2e+156], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 9.1999999999999998e-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 66.0%

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

   5. Simplified 66.0%

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

   if 9.1999999999999998e-7 < im < 2e156

   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. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 79.4%

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

   if 2e156 < 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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 83.3%

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

   8. Simplified 83.3%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 83.3%

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

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)} \]

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

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

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

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

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

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

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

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 83.3%

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

   13. Applied egg-rr 83.3%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+156}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot 0.041666666666666664\\ t_1 := im \cdot \left(0.5 + t\_0\right)\\ t_2 := im \cdot t\_0\\ \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(1 - \left(im \cdot im\right) \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)}{1 - im \cdot t\_1}\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(im \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) 0.041666666666666664))
        (t_1 (* im (+ 0.5 t_0)))
        (t_2 (* im t_0)))
   (if (<= im 9.2e-7)
     (sin re)
     (if (<= im 2.6e+77)
       (/
        (*
         (- 1.0 (* (* im im) (* t_1 t_1)))
         (* re (+ 1.0 (* -0.16666666666666666 (* re re)))))
        (- 1.0 (* im t_1)))
       (if (<= im 4e+154)
         (* re (* im t_2))
         (* (* im (* re (+ 1.0 (* re (* re -0.16666666666666666))))) t_2))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double t_1 = im * (0.5 + t_0);
	double t_2 = im * t_0;
	double tmp;
	if (im <= 9.2e-7) {
		tmp = sin(re);
	} else if (im <= 2.6e+77) {
		tmp = ((1.0 - ((im * im) * (t_1 * t_1))) * (re * (1.0 + (-0.16666666666666666 * (re * re))))) / (1.0 - (im * t_1));
	} else if (im <= 4e+154) {
		tmp = re * (im * t_2);
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (im * im) * 0.041666666666666664d0
    t_1 = im * (0.5d0 + t_0)
    t_2 = im * t_0
    if (im <= 9.2d-7) then
        tmp = sin(re)
    else if (im <= 2.6d+77) then
        tmp = ((1.0d0 - ((im * im) * (t_1 * t_1))) * (re * (1.0d0 + ((-0.16666666666666666d0) * (re * re))))) / (1.0d0 - (im * t_1))
    else if (im <= 4d+154) then
        tmp = re * (im * t_2)
    else
        tmp = (im * (re * (1.0d0 + (re * (re * (-0.16666666666666666d0)))))) * t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double t_1 = im * (0.5 + t_0);
	double t_2 = im * t_0;
	double tmp;
	if (im <= 9.2e-7) {
		tmp = Math.sin(re);
	} else if (im <= 2.6e+77) {
		tmp = ((1.0 - ((im * im) * (t_1 * t_1))) * (re * (1.0 + (-0.16666666666666666 * (re * re))))) / (1.0 - (im * t_1));
	} else if (im <= 4e+154) {
		tmp = re * (im * t_2);
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * 0.041666666666666664
	t_1 = im * (0.5 + t_0)
	t_2 = im * t_0
	tmp = 0
	if im <= 9.2e-7:
		tmp = math.sin(re)
	elif im <= 2.6e+77:
		tmp = ((1.0 - ((im * im) * (t_1 * t_1))) * (re * (1.0 + (-0.16666666666666666 * (re * re))))) / (1.0 - (im * t_1))
	elif im <= 4e+154:
		tmp = re * (im * t_2)
	else:
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * 0.041666666666666664)
	t_1 = Float64(im * Float64(0.5 + t_0))
	t_2 = Float64(im * t_0)
	tmp = 0.0
	if (im <= 9.2e-7)
		tmp = sin(re);
	elseif (im <= 2.6e+77)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(im * im) * Float64(t_1 * t_1))) * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))))) / Float64(1.0 - Float64(im * t_1)));
	elseif (im <= 4e+154)
		tmp = Float64(re * Float64(im * t_2));
	else
		tmp = Float64(Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666))))) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * 0.041666666666666664;
	t_1 = im * (0.5 + t_0);
	t_2 = im * t_0;
	tmp = 0.0;
	if (im <= 9.2e-7)
		tmp = sin(re);
	elseif (im <= 2.6e+77)
		tmp = ((1.0 - ((im * im) * (t_1 * t_1))) * (re * (1.0 + (-0.16666666666666666 * (re * re))))) / (1.0 - (im * t_1));
	elseif (im <= 4e+154)
		tmp = re * (im * t_2);
	else
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im * t$95$0), $MachinePrecision]}, If[LessEqual[im, 9.2e-7], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(N[(N[(1.0 - N[(N[(im * im), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(im * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4e+154], N[(re * N[(im * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 9.1999999999999998e-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 66.0%

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

   5. Simplified 66.0%

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

   if 9.1999999999999998e-7 < im < 2.6000000000000002e77

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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 22.2%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 47.3%

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

   8. Simplified 47.3%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)}{1 - \left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)} \cdot \left(\color{blue}{re} \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(1 \cdot 1 - \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right)}{\color{blue}{1 - \left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)}} \]

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

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

   10. Applied egg-rr 79.5%

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

   if 2.6000000000000002e77 < im < 4.00000000000000015e154

   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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 57.9%

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

   11. Simplified 57.9%

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

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   13. 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. *-lowering-*.f64 N/A

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

       4. metadata-eval N/A

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

       5. pow-sqr N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       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. *-lowering-*.f64 N/A

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 84.2%

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

   14. Simplified 84.2%

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

   if 4.00000000000000015e154 < 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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 83.3%

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

   8. Simplified 83.3%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 83.3%

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

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)} \]

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

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

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

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

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

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

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

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 83.3%

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

   13. Applied egg-rr 83.3%

       \[\leadsto \color{blue}{\left(\left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right) \cdot im\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)}{1 - im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)}\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot 0.041666666666666664\\ t_1 := im \cdot \left(0.5 + t\_0\right)\\ t_2 := im \cdot t\_0\\ \mathbf{if}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(1 - \left(im \cdot im\right) \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)}{1 - im \cdot t\_1}\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(im \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) 0.041666666666666664))
        (t_1 (* im (+ 0.5 t_0)))
        (t_2 (* im t_0)))
   (if (<= im 2.6e+77)
     (/
      (*
       (- 1.0 (* (* im im) (* t_1 t_1)))
       (* re (+ 1.0 (* -0.16666666666666666 (* re re)))))
      (- 1.0 (* im t_1)))
     (if (<= im 4e+154)
       (* re (* im t_2))
       (* (* im (* re (+ 1.0 (* re (* re -0.16666666666666666))))) t_2)))))

C

double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double t_1 = im * (0.5 + t_0);
	double t_2 = im * t_0;
	double tmp;
	if (im <= 2.6e+77) {
		tmp = ((1.0 - ((im * im) * (t_1 * t_1))) * (re * (1.0 + (-0.16666666666666666 * (re * re))))) / (1.0 - (im * t_1));
	} else if (im <= 4e+154) {
		tmp = re * (im * t_2);
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (im * im) * 0.041666666666666664d0
    t_1 = im * (0.5d0 + t_0)
    t_2 = im * t_0
    if (im <= 2.6d+77) then
        tmp = ((1.0d0 - ((im * im) * (t_1 * t_1))) * (re * (1.0d0 + ((-0.16666666666666666d0) * (re * re))))) / (1.0d0 - (im * t_1))
    else if (im <= 4d+154) then
        tmp = re * (im * t_2)
    else
        tmp = (im * (re * (1.0d0 + (re * (re * (-0.16666666666666666d0)))))) * t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * 0.041666666666666664;
	double t_1 = im * (0.5 + t_0);
	double t_2 = im * t_0;
	double tmp;
	if (im <= 2.6e+77) {
		tmp = ((1.0 - ((im * im) * (t_1 * t_1))) * (re * (1.0 + (-0.16666666666666666 * (re * re))))) / (1.0 - (im * t_1));
	} else if (im <= 4e+154) {
		tmp = re * (im * t_2);
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * 0.041666666666666664
	t_1 = im * (0.5 + t_0)
	t_2 = im * t_0
	tmp = 0
	if im <= 2.6e+77:
		tmp = ((1.0 - ((im * im) * (t_1 * t_1))) * (re * (1.0 + (-0.16666666666666666 * (re * re))))) / (1.0 - (im * t_1))
	elif im <= 4e+154:
		tmp = re * (im * t_2)
	else:
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * 0.041666666666666664)
	t_1 = Float64(im * Float64(0.5 + t_0))
	t_2 = Float64(im * t_0)
	tmp = 0.0
	if (im <= 2.6e+77)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(im * im) * Float64(t_1 * t_1))) * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))))) / Float64(1.0 - Float64(im * t_1)));
	elseif (im <= 4e+154)
		tmp = Float64(re * Float64(im * t_2));
	else
		tmp = Float64(Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666))))) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * 0.041666666666666664;
	t_1 = im * (0.5 + t_0);
	t_2 = im * t_0;
	tmp = 0.0;
	if (im <= 2.6e+77)
		tmp = ((1.0 - ((im * im) * (t_1 * t_1))) * (re * (1.0 + (-0.16666666666666666 * (re * re))))) / (1.0 - (im * t_1));
	elseif (im <= 4e+154)
		tmp = re * (im * t_2);
	else
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im * t$95$0), $MachinePrecision]}, If[LessEqual[im, 2.6e+77], N[(N[(N[(1.0 - N[(N[(im * im), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(im * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4e+154], N[(re * N[(im * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.6000000000000002e77

   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 85.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 50.8%

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

   8. Simplified 50.8%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)}{1 - \left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)} \cdot \left(\color{blue}{re} \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(1 \cdot 1 - \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right)}{\color{blue}{1 - \left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)}} \]

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

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

   10. Applied egg-rr 41.4%

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

   if 2.6000000000000002e77 < im < 4.00000000000000015e154

   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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 57.9%

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

   11. Simplified 57.9%

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

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   13. 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. *-lowering-*.f64 N/A

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

       4. metadata-eval N/A

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

       5. pow-sqr N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       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. *-lowering-*.f64 N/A

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 84.2%

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

   14. Simplified 84.2%

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

   if 4.00000000000000015e154 < 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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 83.3%

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

   8. Simplified 83.3%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 83.3%

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

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)} \]

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

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

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

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

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

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

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

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 83.3%

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

   13. Applied egg-rr 83.3%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)}{1 - im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)}\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 10^{+158}:\\ \;\;\;\;\left(re \cdot \left(0.5 + re \cdot \left(re \cdot \left(-0.08333333333333333 + \left(re \cdot re\right) \cdot 0.004166666666666667\right)\right)\right)\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1e+158)
   (*
    (*
     re
     (+
      0.5
      (*
       re
       (* re (+ -0.08333333333333333 (* (* re re) 0.004166666666666667))))))
    (+
     2.0
     (*
      (* im im)
      (+
       1.0
       (*
        (* im im)
        (+ 0.08333333333333333 (* (* im im) 0.002777777777777778)))))))
   (*
    (* im (* re (+ 1.0 (* re (* re -0.16666666666666666)))))
    (* im (* (* im im) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1e+158) {
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * 0.004166666666666667)))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (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 <= 1d+158) then
        tmp = (re * (0.5d0 + (re * (re * ((-0.08333333333333333d0) + ((re * re) * 0.004166666666666667d0)))))) * (2.0d0 + ((im * im) * (1.0d0 + ((im * im) * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0))))))
    else
        tmp = (im * (re * (1.0d0 + (re * (re * (-0.16666666666666666d0)))))) * (im * ((im * im) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1e+158) {
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * 0.004166666666666667)))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (im * ((im * im) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1e+158:
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * 0.004166666666666667)))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))))
	else:
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (im * ((im * im) * 0.041666666666666664))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1e+158)
		tmp = Float64(Float64(re * Float64(0.5 + Float64(re * Float64(re * Float64(-0.08333333333333333 + Float64(Float64(re * re) * 0.004166666666666667)))))) * Float64(2.0 + Float64(Float64(im * im) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))));
	else
		tmp = Float64(Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666))))) * Float64(im * Float64(Float64(im * im) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1e+158)
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * 0.004166666666666667)))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	else
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (im * ((im * im) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1e+158], N[(N[(re * N[(0.5 + N[(re * N[(re * N[(-0.08333333333333333 + N[(N[(re * re), $MachinePrecision] * 0.004166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(im * im), $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.99999999999999953e157

   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 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \left(re \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{240} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{240} \cdot {re}^{2} + \frac{-1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{12} + \frac{1}{240} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \left(\frac{1}{240} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \left({re}^{2} \cdot \frac{1}{240}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{1}{240}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{240}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 57.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{240}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 57.7%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + re \cdot \left(re \cdot \left(-0.08333333333333333 + \left(re \cdot re\right) \cdot 0.004166666666666667\right)\right)\right)\right)} \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right) \]

   if 9.99999999999999953e157 < 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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 83.3%

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

   8. Simplified 83.3%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 83.3%

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

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)} \]

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

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

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

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

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

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

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

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 83.3%

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

   13. Applied egg-rr 83.3%

       \[\leadsto \color{blue}{\left(\left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right) \cdot im\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10^{+158}:\\ \;\;\;\;\left(re \cdot \left(0.5 + re \cdot \left(re \cdot \left(-0.08333333333333333 + \left(re \cdot re\right) \cdot 0.004166666666666667\right)\right)\right)\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.3% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\\ \mathbf{if}\;im \leq 1.1 \cdot 10^{+78}:\\ \;\;\;\;\left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.42 \cdot 10^{+155}:\\ \;\;\;\;re \cdot \left(im \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* (* im im) 0.041666666666666664))))
   (if (<= im 1.1e+78)
     (*
      (* re (+ 1.0 (* -0.16666666666666666 (* re re))))
      (+ 1.0 (* 0.5 (* im im))))
     (if (<= im 1.42e+155)
       (* re (* im t_0))
       (* (* im (* re (+ 1.0 (* re (* re -0.16666666666666666))))) t_0)))))

C

double code(double re, double im) {
	double t_0 = im * ((im * im) * 0.041666666666666664);
	double tmp;
	if (im <= 1.1e+78) {
		tmp = (re * (1.0 + (-0.16666666666666666 * (re * re)))) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 1.42e+155) {
		tmp = re * (im * t_0);
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * 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 * im) * 0.041666666666666664d0)
    if (im <= 1.1d+78) then
        tmp = (re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))) * (1.0d0 + (0.5d0 * (im * im)))
    else if (im <= 1.42d+155) then
        tmp = re * (im * t_0)
    else
        tmp = (im * (re * (1.0d0 + (re * (re * (-0.16666666666666666d0)))))) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * ((im * im) * 0.041666666666666664);
	double tmp;
	if (im <= 1.1e+78) {
		tmp = (re * (1.0 + (-0.16666666666666666 * (re * re)))) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 1.42e+155) {
		tmp = re * (im * t_0);
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * ((im * im) * 0.041666666666666664)
	tmp = 0
	if im <= 1.1e+78:
		tmp = (re * (1.0 + (-0.16666666666666666 * (re * re)))) * (1.0 + (0.5 * (im * im)))
	elif im <= 1.42e+155:
		tmp = re * (im * t_0)
	else:
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(Float64(im * im) * 0.041666666666666664))
	tmp = 0.0
	if (im <= 1.1e+78)
		tmp = Float64(Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re)))) * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	elseif (im <= 1.42e+155)
		tmp = Float64(re * Float64(im * t_0));
	else
		tmp = Float64(Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666))))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * ((im * im) * 0.041666666666666664);
	tmp = 0.0;
	if (im <= 1.1e+78)
		tmp = (re * (1.0 + (-0.16666666666666666 * (re * re)))) * (1.0 + (0.5 * (im * im)));
	elseif (im <= 1.42e+155)
		tmp = re * (im * t_0);
	else
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.1e+78], N[(N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.42e+155], N[(re * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.10000000000000007e78

   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 85.2%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 51.0%

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

   8. Simplified 51.0%

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

   9. Taylor expanded in im around 0

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) + \color{blue}{re} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \]

       2. distribute-lft1-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       3. +-commutative N/A

          \[\leadsto \left(1 + \frac{1}{2} \cdot {im}^{2}\right) \cdot \left(\color{blue}{re} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \]

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

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

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

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

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

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

       7. unpow2 N/A

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

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

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

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

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 45.2%

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

   11. Simplified 45.2%

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

   if 1.10000000000000007e78 < im < 1.41999999999999994e155

   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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 55.6%

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

   8. Simplified 55.6%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 55.6%

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

   11. Simplified 55.6%

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

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   13. 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. *-lowering-*.f64 N/A

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

       4. metadata-eval N/A

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

       5. pow-sqr N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       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. *-lowering-*.f64 N/A

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 88.9%

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

   14. Simplified 88.9%

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

   if 1.41999999999999994e155 < 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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 83.3%

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

   8. Simplified 83.3%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 83.3%

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

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)} \]

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

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

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

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

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

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

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

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 83.3%

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

   13. Applied egg-rr 83.3%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+78}:\\ \;\;\;\;\left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.42 \cdot 10^{+155}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;im \leq 1.1 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (* re (+ 1.0 (* -0.16666666666666666 (* re re))))
          (+ 1.0 (* 0.5 (* im im))))))
   (if (<= im 1.1e+78)
     t_0
     (if (<= im 4e+154)
       (* re (* im (* im (* (* im im) 0.041666666666666664))))
       t_0))))

C

double code(double re, double im) {
	double t_0 = (re * (1.0 + (-0.16666666666666666 * (re * re)))) * (1.0 + (0.5 * (im * im)));
	double tmp;
	if (im <= 1.1e+78) {
		tmp = t_0;
	} else if (im <= 4e+154) {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	} 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 = (re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))) * (1.0d0 + (0.5d0 * (im * im)))
    if (im <= 1.1d+78) then
        tmp = t_0
    else if (im <= 4d+154) then
        tmp = re * (im * (im * ((im * im) * 0.041666666666666664d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * (1.0 + (-0.16666666666666666 * (re * re)))) * (1.0 + (0.5 * (im * im)));
	double tmp;
	if (im <= 1.1e+78) {
		tmp = t_0;
	} else if (im <= 4e+154) {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * (1.0 + (-0.16666666666666666 * (re * re)))) * (1.0 + (0.5 * (im * im)))
	tmp = 0
	if im <= 1.1e+78:
		tmp = t_0
	elif im <= 4e+154:
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re)))) * Float64(1.0 + Float64(0.5 * Float64(im * im))))
	tmp = 0.0
	if (im <= 1.1e+78)
		tmp = t_0;
	elseif (im <= 4e+154)
		tmp = Float64(re * Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * (1.0 + (-0.16666666666666666 * (re * re)))) * (1.0 + (0.5 * (im * im)));
	tmp = 0.0;
	if (im <= 1.1e+78)
		tmp = t_0;
	elseif (im <= 4e+154)
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.1e+78], t$95$0, If[LessEqual[im, 4e+154], N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.10000000000000007e78 or 4.00000000000000015e154 < 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 87.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 55.1%

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

   8. Simplified 55.1%

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

   9. Taylor expanded in im around 0

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) + \color{blue}{re} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \]

       2. distribute-lft1-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       3. +-commutative N/A

          \[\leadsto \left(1 + \frac{1}{2} \cdot {im}^{2}\right) \cdot \left(\color{blue}{re} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \]

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

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

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

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

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

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

       7. unpow2 N/A

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

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

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

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

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 50.0%

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

   11. Simplified 50.0%

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

   if 1.10000000000000007e78 < im < 4.00000000000000015e154

   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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 55.6%

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

   8. Simplified 55.6%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 55.6%

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

   11. Simplified 55.6%

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

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   13. 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. *-lowering-*.f64 N/A

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

       4. metadata-eval N/A

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

       5. pow-sqr N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       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. *-lowering-*.f64 N/A

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 88.9%

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

   14. Simplified 88.9%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+78}:\\ \;\;\;\;\left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2 \cdot 10^{+159}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2e+159)
   (*
    re
    (+
     1.0
     (*
      im
      (*
       im
       (+
        0.5
        (*
         im
         (*
          im
          (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))))))))
   (*
    (* im (* re (+ 1.0 (* re (* re -0.16666666666666666)))))
    (* im (* (* im im) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2e+159) {
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))));
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (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 <= 2d+159) then
        tmp = re * (1.0d0 + (im * (im * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0))))))))
    else
        tmp = (im * (re * (1.0d0 + (re * (re * (-0.16666666666666666d0)))))) * (im * ((im * im) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2e+159) {
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))));
	} else {
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (im * ((im * im) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2e+159:
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))))
	else:
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (im * ((im * im) * 0.041666666666666664))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2e+159)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)))))))));
	else
		tmp = Float64(Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666))))) * Float64(im * Float64(Float64(im * im) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2e+159)
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))));
	else
		tmp = (im * (re * (1.0 + (re * (re * -0.16666666666666666))))) * (im * ((im * im) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2e+159], N[(re * N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.9999999999999999e159

   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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 56.9%

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

   10. Simplified 56.9%

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

   if 1.9999999999999999e159 < 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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 83.3%

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

   8. Simplified 83.3%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 83.3%

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

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)} \]

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

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

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

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

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

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

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

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 83.3%

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

   13. Applied egg-rr 83.3%

       \[\leadsto \color{blue}{\left(\left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right) \cdot im\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2 \cdot 10^{+159}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.8% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+78}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.1e+78)
   (* re (+ 1.0 (* -0.16666666666666666 (* re re))))
   (* re (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.1e+78) {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	} else {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.1d+78) then
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
    else
        tmp = re * (1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.1e+78) {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	} else {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.1e+78:
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)))
	else:
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.1e+78)
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.1e+78)
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	else
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.1e+78], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.10000000000000007e78

   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 62.0%

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

   5. Simplified 62.0%

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

   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 38.2%

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

   8. Simplified 38.2%

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

   if 1.10000000000000007e78 < 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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 83.3%

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

   8. Simplified 83.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+78}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.8% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+78}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.1e+78)
   (* re (+ 1.0 (* -0.16666666666666666 (* re re))))
   (* re (* im (* im (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.1e+78) {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	} 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 <= 1.1d+78) then
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
    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 <= 1.1e+78) {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	} else {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.1e+78:
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)))
	else:
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.1e+78)
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))));
	else
		tmp = Float64(re * 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 <= 1.1e+78)
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	else
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.1e+78], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.10000000000000007e78

   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 62.0%

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

   5. Simplified 62.0%

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

   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 38.2%

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

   8. Simplified 38.2%

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

   if 1.10000000000000007e78 < 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%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 72.9%

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

   8. Simplified 72.9%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 72.9%

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

   11. Simplified 72.9%

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

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   13. 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. *-lowering-*.f64 N/A

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

       4. metadata-eval N/A

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

       5. pow-sqr N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       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. *-lowering-*.f64 N/A

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 83.3%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   14. Simplified 83.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+78}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.3% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.58 \cdot 10^{+78}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.008333333333333333 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.58e+78)
   (* re (+ 1.0 (* -0.16666666666666666 (* re re))))
   (* re (* 0.008333333333333333 (* (* re re) (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.58e+78) {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	} else {
		tmp = re * (0.008333333333333333 * ((re * re) * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.58d+78) then
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
    else
        tmp = re * (0.008333333333333333d0 * ((re * re) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.58e+78) {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	} else {
		tmp = re * (0.008333333333333333 * ((re * re) * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.58e+78:
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)))
	else:
		tmp = re * (0.008333333333333333 * ((re * re) * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.58e+78)
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))));
	else
		tmp = Float64(re * Float64(0.008333333333333333 * Float64(Float64(re * re) * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.58e+78)
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	else
		tmp = re * (0.008333333333333333 * ((re * re) * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.58e+78], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(0.008333333333333333 * N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.58000000000000004e78

   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 62.0%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 62.0%

      \[\leadsto \color{blue}{\sin re} \]

   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 38.2%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 38.2%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]

   if 1.58000000000000004e78 < 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

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 2.6%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 2.6%

      \[\leadsto \color{blue}{\sin re} \]

   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 14.5%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 14.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({re}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{120}, \left({re}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{120}, \left({re}^{2} \cdot \color{blue}{{re}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left({\color{blue}{re}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({\color{blue}{re}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 14.1%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   11. Simplified 14.1%

       \[\leadsto re \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.58 \cdot 10^{+78}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.008333333333333333 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.3% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.12 \cdot 10^{+64}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.12e+64) re (* -0.16666666666666666 (* re (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.12e+64) {
		tmp = re;
	} else {
		tmp = -0.16666666666666666 * (re * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.12d+64) then
        tmp = re
    else
        tmp = (-0.16666666666666666d0) * (re * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.12e+64) {
		tmp = re;
	} else {
		tmp = -0.16666666666666666 * (re * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.12e+64:
		tmp = re
	else:
		tmp = -0.16666666666666666 * (re * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.12e+64)
		tmp = re;
	else
		tmp = Float64(-0.16666666666666666 * Float64(re * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.12e+64)
		tmp = re;
	else
		tmp = -0.16666666666666666 * (re * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.12e+64], re, N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.11999999999999995e64

   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 50.7%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 50.7%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \]
   7. Step-by-step derivation

   8. Simplified 33.4%

      \[\leadsto \color{blue}{re} \]

   if 1.11999999999999995e64 < 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 51.4%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 51.4%

      \[\leadsto \color{blue}{\sin re} \]

   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 22.1%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 22.1%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot {re}^{\color{blue}{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       6. *-lowering-*.f64 22.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   11. Simplified 22.1%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 34.2% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* re (+ 1.0 (* -0.16666666666666666 (* re re)))))

C

double code(double re, double im) {
	return re * (1.0 + (-0.16666666666666666 * (re * re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
end function

Java

public static double code(double re, double im) {
	return re * (1.0 + (-0.16666666666666666 * (re * re)));
}

Python

def code(re, im):
	return re * (1.0 + (-0.16666666666666666 * (re * re)))

Julia

function code(re, im)
	return Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
end

Wolfram

code[re_, im_] := N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $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

   \[\leadsto \color{blue}{\sin re} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 50.9%

      \[\leadsto \mathsf{sin.f64}\left(re\right) \]

5. Simplified 50.9%

   \[\leadsto \color{blue}{\sin re} \]

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 32.9%

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

8. Simplified 32.9%

   \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]

9. Final simplification 32.9%

   \[\leadsto re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right) \]
10. Add Preprocessing

Alternative 18: 26.7% 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 50.9%

      \[\leadsto \mathsf{sin.f64}\left(re\right) \]

5. Simplified 50.9%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re} \]
7. Step-by-step derivation

8. Simplified 26.9%

   \[\leadsto \color{blue}{re} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))