math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.2s

Alternatives: 18

Speedup: 1.5×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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, 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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

      \[\leadsto \sin re \cdot \frac{e^{0 - im} + e^{im}}{\color{blue}{2}} \]

   6. +-commutative N/A

      \[\leadsto \sin re \cdot \frac{e^{im} + e^{0 - im}}{2} \]

   7. sub0-neg N/A

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

   8. cosh-def N/A

      \[\leadsto \sin re \cdot \cosh im \]

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

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

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

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

   11. cosh-lowering-cosh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 73.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.75e-12)
   (sin re)
   (if (<= im 7e+51)
     (* (cosh im) (* re (+ 1.0 (* re (* re -0.16666666666666666)))))
     (*
      (sin re)
      (+
       1.0
       (*
        (* im im)
        (+ 0.5 (* im (* im (* (* im im) 0.001388888888888889))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = sin(re);
	} else if (im <= 7e+51) {
		tmp = cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * ((im * im) * 0.001388888888888889))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.75d-12) then
        tmp = sin(re)
    else if (im <= 7d+51) then
        tmp = cosh(im) * (re * (1.0d0 + (re * (re * (-0.16666666666666666d0)))))
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * ((im * im) * 0.001388888888888889d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = Math.sin(re);
	} else if (im <= 7e+51) {
		tmp = Math.cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * ((im * im) * 0.001388888888888889))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.75e-12:
		tmp = math.sin(re)
	elif im <= 7e+51:
		tmp = math.cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))))
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * ((im * im) * 0.001388888888888889))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.75e-12)
		tmp = sin(re);
	elseif (im <= 7e+51)
		tmp = Float64(cosh(im) * Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666)))));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(Float64(im * im) * 0.001388888888888889)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.75e-12)
		tmp = sin(re);
	elseif (im <= 7e+51)
		tmp = cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * ((im * im) * 0.001388888888888889))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.75e-12], N[Sin[re], $MachinePrecision], If[LessEqual[im, 7e+51], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.75e-12

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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 69.3%

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

   5. Simplified 69.3%

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

   if 1.75e-12 < im < 7e51

   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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

         \[\leadsto \sin re \cdot \frac{e^{0 - im} + e^{im}}{\color{blue}{2}} \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \frac{e^{im} + e^{0 - im}}{2} \]

      7. sub0-neg N/A

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

      8. cosh-def N/A

         \[\leadsto \sin re \cdot \cosh im \]

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

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

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

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

      11. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 88.2%

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

   7. Simplified 88.2%

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

   if 7e51 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

         \[\leadsto \sin re \cdot \frac{e^{0 - im} + e^{im}}{\color{blue}{2}} \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \frac{e^{im} + e^{0 - im}}{2} \]

      7. sub0-neg N/A

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

      8. cosh-def N/A

         \[\leadsto \sin re \cdot \cosh im \]

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

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

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

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

      11. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{720} \cdot {im}^{3}\right)}\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.75e-12)
   (sin re)
   (if (<= im 2.6e+77)
     (* (cosh im) (* re (+ 1.0 (* re (* re -0.16666666666666666)))))
     (*
      (sin re)
      (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = sin(re);
	} else if (im <= 2.6e+77) {
		tmp = cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	} 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 (im <= 1.75d-12) then
        tmp = sin(re)
    else if (im <= 2.6d+77) then
        tmp = cosh(im) * (re * (1.0d0 + (re * (re * (-0.16666666666666666d0)))))
    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 (im <= 1.75e-12) {
		tmp = Math.sin(re);
	} else if (im <= 2.6e+77) {
		tmp = Math.cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	} 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 im <= 1.75e-12:
		tmp = math.sin(re)
	elif im <= 2.6e+77:
		tmp = math.cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))))
	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 (im <= 1.75e-12)
		tmp = sin(re);
	elseif (im <= 2.6e+77)
		tmp = Float64(cosh(im) * Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666)))));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.75e-12)
		tmp = sin(re);
	elseif (im <= 2.6e+77)
		tmp = cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	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[im, 1.75e-12], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.75e-12

   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 69.3%

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

   5. Simplified 69.3%

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

   if 1.75e-12 < 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

         \[\leadsto \sin re \cdot \frac{e^{0 - im} + e^{im}}{\color{blue}{2}} \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \frac{e^{im} + e^{0 - im}}{2} \]

      7. sub0-neg N/A

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

      8. cosh-def N/A

         \[\leadsto \sin re \cdot \cosh im \]

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

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

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

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

      11. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 85.7%

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

   7. Simplified 85.7%

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

   if 2.6000000000000002e77 < 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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.75e-12)
   (sin re)
   (if (<= im 1.35e+154)
     (* (cosh im) (* re (+ 1.0 (* re (* re -0.16666666666666666)))))
     (* (* im im) (* (sin re) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = sin(re);
	} else if (im <= 1.35e+154) {
		tmp = cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	} else {
		tmp = (im * im) * (sin(re) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.75d-12) then
        tmp = sin(re)
    else if (im <= 1.35d+154) then
        tmp = cosh(im) * (re * (1.0d0 + (re * (re * (-0.16666666666666666d0)))))
    else
        tmp = (im * im) * (sin(re) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+154) {
		tmp = Math.cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	} else {
		tmp = (im * im) * (Math.sin(re) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.75e-12:
		tmp = math.sin(re)
	elif im <= 1.35e+154:
		tmp = math.cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))))
	else:
		tmp = (im * im) * (math.sin(re) * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.75e-12)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(cosh(im) * Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666)))));
	else
		tmp = Float64(Float64(im * im) * Float64(sin(re) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.75e-12)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = cosh(im) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	else
		tmp = (im * im) * (sin(re) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.75e-12], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.75e-12

   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 69.3%

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

   5. Simplified 69.3%

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

   if 1.75e-12 < 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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

         \[\leadsto \sin re \cdot \frac{e^{0 - im} + e^{im}}{\color{blue}{2}} \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \frac{e^{im} + e^{0 - im}}{2} \]

      7. sub0-neg N/A

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

      8. cosh-def N/A

         \[\leadsto \sin re \cdot \cosh im \]

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

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

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

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

      11. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 82.5%

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

   7. Simplified 82.5%

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

   if 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 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = sin(re);
	} else if (im <= 1.35e+154) {
		tmp = re * cosh(im);
	} else {
		tmp = (im * im) * (sin(re) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.75d-12) then
        tmp = sin(re)
    else if (im <= 1.35d+154) then
        tmp = re * cosh(im)
    else
        tmp = (im * im) * (sin(re) * 0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if im < 1.75e-12

   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 69.3%

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

   5. Simplified 69.3%

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

   if 1.75e-12 < 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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

         \[\leadsto \sin re \cdot \frac{e^{0 - im} + e^{im}}{\color{blue}{2}} \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \frac{e^{im} + e^{0 - im}}{2} \]

      7. sub0-neg N/A

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

      8. cosh-def N/A

         \[\leadsto \sin re \cdot \cosh im \]

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

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

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

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

      11. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 77.5%

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

   if 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 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 74.1%

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

Alternative 6: 68.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.75e-12) (sin re) (* re (cosh im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = sin(re);
	} else {
		tmp = re * cosh(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.75d-12) then
        tmp = sin(re)
    else
        tmp = re * cosh(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = Math.sin(re);
	} else {
		tmp = re * Math.cosh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.75e-12:
		tmp = math.sin(re)
	else:
		tmp = re * math.cosh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.75e-12)
		tmp = sin(re);
	else
		tmp = Float64(re * cosh(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.75e-12)
		tmp = sin(re);
	else
		tmp = re * cosh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.75e-12], N[Sin[re], $MachinePrecision], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.75e-12

   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 69.3%

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

   5. Simplified 69.3%

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

   if 1.75e-12 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

         \[\leadsto \sin re \cdot \frac{e^{0 - im} + e^{im}}{\color{blue}{2}} \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \frac{e^{im} + e^{0 - im}}{2} \]

      7. sub0-neg N/A

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

      8. cosh-def N/A

         \[\leadsto \sin re \cdot \cosh im \]

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

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

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

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

      11. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 76.8%

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

Alternative 7: 65.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+172}:\\ \;\;\;\;\left(re \cdot \left(0.5 + re \cdot \left(re \cdot \left(-0.08333333333333333 + \left(re \cdot re\right) \cdot \left(0.004166666666666667 + \left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)\right)\right)\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.75e-12)
   (sin re)
   (if (<= im 9e+172)
     (*
      (*
       re
       (+
        0.5
        (*
         re
         (*
          re
          (+
           -0.08333333333333333
           (*
            (* re re)
            (+ 0.004166666666666667 (* (* re re) -9.92063492063492e-5))))))))
      (+
       2.0
       (*
        (* im im)
        (+
         1.0
         (*
          (* im im)
          (+ 0.08333333333333333 (* (* im im) 0.002777777777777778)))))))
     (* (* re 0.5) (+ (* im im) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = sin(re);
	} else if (im <= 9e+172) {
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * (0.004166666666666667 + ((re * re) * -9.92063492063492e-5)))))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	} else {
		tmp = (re * 0.5) * ((im * im) + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.75d-12) then
        tmp = sin(re)
    else if (im <= 9d+172) then
        tmp = (re * (0.5d0 + (re * (re * ((-0.08333333333333333d0) + ((re * re) * (0.004166666666666667d0 + ((re * re) * (-9.92063492063492d-5))))))))) * (2.0d0 + ((im * im) * (1.0d0 + ((im * im) * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0))))))
    else
        tmp = (re * 0.5d0) * ((im * im) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-12) {
		tmp = Math.sin(re);
	} else if (im <= 9e+172) {
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * (0.004166666666666667 + ((re * re) * -9.92063492063492e-5)))))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	} else {
		tmp = (re * 0.5) * ((im * im) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.75e-12:
		tmp = math.sin(re)
	elif im <= 9e+172:
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * (0.004166666666666667 + ((re * re) * -9.92063492063492e-5)))))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))))
	else:
		tmp = (re * 0.5) * ((im * im) + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.75e-12)
		tmp = sin(re);
	elseif (im <= 9e+172)
		tmp = Float64(Float64(re * Float64(0.5 + Float64(re * Float64(re * Float64(-0.08333333333333333 + Float64(Float64(re * re) * Float64(0.004166666666666667 + Float64(Float64(re * re) * -9.92063492063492e-5)))))))) * Float64(2.0 + Float64(Float64(im * im) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))));
	else
		tmp = Float64(Float64(re * 0.5) * Float64(Float64(im * im) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.75e-12)
		tmp = sin(re);
	elseif (im <= 9e+172)
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * (0.004166666666666667 + ((re * re) * -9.92063492063492e-5)))))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	else
		tmp = (re * 0.5) * ((im * im) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.75e-12], N[Sin[re], $MachinePrecision], If[LessEqual[im, 9e+172], N[(N[(re * N[(0.5 + N[(re * N[(re * N[(-0.08333333333333333 + N[(N[(re * re), $MachinePrecision] * N[(0.004166666666666667 + N[(N[(re * re), $MachinePrecision] * -9.92063492063492e-5), $MachinePrecision]), $MachinePrecision]), $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[(re * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.75e-12

   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 69.3%

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

   5. Simplified 69.3%

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

   if 1.75e-12 < im < 9.0000000000000004e172

   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 70.2%

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

   5. Simplified 70.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(re \cdot re\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \frac{-1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{12} + {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{240}, \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{240}, \left({re}^{2} \cdot \frac{-1}{10080}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{10080}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{10080}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 61.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{10080}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 61.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + re \cdot \left(re \cdot \left(-0.08333333333333333 + \left(re \cdot re\right) \cdot \left(0.004166666666666667 + \left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)\right)\right)\right)} \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right) \]

   if 9.0000000000000004e172 < 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 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 79.2%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+172}:\\ \;\;\;\;\left(re \cdot \left(0.5 + re \cdot \left(re \cdot \left(-0.08333333333333333 + \left(re \cdot re\right) \cdot \left(0.004166666666666667 + \left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)\right)\right)\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;\left(re \cdot \left(0.5 + re \cdot \left(re \cdot \left(-0.08333333333333333 + \left(re \cdot re\right) \cdot \left(0.004166666666666667 + \left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)\right)\right)\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+296}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.2e+154)
   (*
    (*
     re
     (+
      0.5
      (*
       re
       (*
        re
        (+
         -0.08333333333333333
         (*
          (* re re)
          (+ 0.004166666666666667 (* (* re re) -9.92063492063492e-5))))))))
    (+
     2.0
     (*
      (* im im)
      (+
       1.0
       (*
        (* im im)
        (+ 0.08333333333333333 (* (* im im) 0.002777777777777778)))))))
   (if (<= re 6.5e+296)
     (*
      re
      (+
       1.0
       (*
        (* re re)
        (+ -0.16666666666666666 (* (* re re) 0.008333333333333333)))))
     (* re (+ 1.0 (* -0.16666666666666666 (* re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.2e+154) {
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * (0.004166666666666667 + ((re * re) * -9.92063492063492e-5)))))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	} else if (re <= 6.5e+296) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (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 <= 1.2d+154) then
        tmp = (re * (0.5d0 + (re * (re * ((-0.08333333333333333d0) + ((re * re) * (0.004166666666666667d0 + ((re * re) * (-9.92063492063492d-5))))))))) * (2.0d0 + ((im * im) * (1.0d0 + ((im * im) * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0))))))
    else if (re <= 6.5d+296) then
        tmp = re * (1.0d0 + ((re * re) * ((-0.16666666666666666d0) + ((re * re) * 0.008333333333333333d0))))
    else
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.2e+154) {
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * (0.004166666666666667 + ((re * re) * -9.92063492063492e-5)))))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	} else if (re <= 6.5e+296) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.2e+154:
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * (0.004166666666666667 + ((re * re) * -9.92063492063492e-5)))))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))))
	elif re <= 6.5e+296:
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))
	else:
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.2e+154)
		tmp = Float64(Float64(re * Float64(0.5 + Float64(re * Float64(re * Float64(-0.08333333333333333 + Float64(Float64(re * re) * Float64(0.004166666666666667 + Float64(Float64(re * re) * -9.92063492063492e-5)))))))) * Float64(2.0 + Float64(Float64(im * im) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))));
	elseif (re <= 6.5e+296)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.008333333333333333)))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.2e+154)
		tmp = (re * (0.5 + (re * (re * (-0.08333333333333333 + ((re * re) * (0.004166666666666667 + ((re * re) * -9.92063492063492e-5)))))))) * (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))));
	elseif (re <= 6.5e+296)
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	else
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.2e+154], N[(N[(re * N[(0.5 + N[(re * N[(re * N[(-0.08333333333333333 + N[(N[(re * re), $MachinePrecision] * N[(0.004166666666666667 + N[(N[(re * re), $MachinePrecision] * -9.92063492063492e-5), $MachinePrecision]), $MachinePrecision]), $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], If[LessEqual[re, 6.5e+296], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.20000000000000007e154

   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 89.9%

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

   5. Simplified 89.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(re \cdot re\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \frac{-1}{12}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{12} + {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{240}, \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{240}, \left({re}^{2} \cdot \frac{-1}{10080}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{10080}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{10080}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 61.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{10080}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 61.0%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + re \cdot \left(re \cdot \left(-0.08333333333333333 + \left(re \cdot re\right) \cdot \left(0.004166666666666667 + \left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)\right)\right)\right)} \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right) \]

   if 1.20000000000000007e154 < re < 6.5e296

   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 72.9%

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

   5. Simplified 72.9%

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

   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 29.5%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 29.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)} \]

   if 6.5e296 < 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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 66.9%

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

   8. Simplified 66.9%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   10. 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. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 66.9%

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

   11. Simplified 66.9%

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

Alternative 9: 59.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2 \cdot 10^{+172}:\\ \;\;\;\;\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2e+172)
   (*
    (+
     2.0
     (*
      (* im im)
      (+
       1.0
       (*
        (* im im)
        (+ 0.08333333333333333 (* (* im im) 0.002777777777777778))))))
    (* re (+ 0.5 (* -0.08333333333333333 (* re re)))))
   (* (* re 0.5) (+ (* im im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2e+172) {
		tmp = (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))) * (re * (0.5 + (-0.08333333333333333 * (re * re))));
	} else {
		tmp = (re * 0.5) * ((im * im) + 2.0);
	}
	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+172) then
        tmp = (2.0d0 + ((im * im) * (1.0d0 + ((im * im) * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0)))))) * (re * (0.5d0 + ((-0.08333333333333333d0) * (re * re))))
    else
        tmp = (re * 0.5d0) * ((im * im) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2e+172) {
		tmp = (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))) * (re * (0.5 + (-0.08333333333333333 * (re * re))));
	} else {
		tmp = (re * 0.5) * ((im * im) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2e+172:
		tmp = (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))) * (re * (0.5 + (-0.08333333333333333 * (re * re))))
	else:
		tmp = (re * 0.5) * ((im * im) + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2e+172)
		tmp = Float64(Float64(2.0 + Float64(Float64(im * im) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))) * Float64(re * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))));
	else
		tmp = Float64(Float64(re * 0.5) * Float64(Float64(im * im) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2e+172)
		tmp = (2.0 + ((im * im) * (1.0 + ((im * im) * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))) * (re * (0.5 + (-0.08333333333333333 * (re * re))));
	else
		tmp = (re * 0.5) * ((im * im) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2e+172], N[(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] * N[(re * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.0000000000000002e172

   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 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 52.0%

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

   8. Simplified 52.0%

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

   if 2.0000000000000002e172 < 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 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 79.2%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2 \cdot 10^{+172}:\\ \;\;\;\;\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.1% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+296}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e+41)
   (*
    re
    (+
     1.0
     (*
      im
      (*
       im
       (+
        0.5
        (*
         im
         (*
          im
          (+ (* (* im im) 0.001388888888888889) 0.041666666666666664))))))))
   (if (<= re 1.2e+154)
     (*
      re
      (+
       1.0
       (*
        (* re re)
        (+
         -0.16666666666666666
         (*
          re
          (*
           re
           (+ 0.008333333333333333 (* (* re re) -0.0001984126984126984))))))))
     (if (<= re 6.5e+296)
       (*
        re
        (+
         1.0
         (*
          (* re re)
          (+ -0.16666666666666666 (* (* re re) 0.008333333333333333)))))
       (* re (+ 1.0 (* -0.16666666666666666 (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+41) {
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))));
	} else if (re <= 1.2e+154) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984)))))));
	} else if (re <= 6.5e+296) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (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 <= 6.2d+41) then
        tmp = re * (1.0d0 + (im * (im * (0.5d0 + (im * (im * (((im * im) * 0.001388888888888889d0) + 0.041666666666666664d0)))))))
    else if (re <= 1.2d+154) then
        tmp = re * (1.0d0 + ((re * re) * ((-0.16666666666666666d0) + (re * (re * (0.008333333333333333d0 + ((re * re) * (-0.0001984126984126984d0))))))))
    else if (re <= 6.5d+296) then
        tmp = re * (1.0d0 + ((re * re) * ((-0.16666666666666666d0) + ((re * re) * 0.008333333333333333d0))))
    else
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+41) {
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))));
	} else if (re <= 1.2e+154) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984)))))));
	} else if (re <= 6.5e+296) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e+41:
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))))
	elif re <= 1.2e+154:
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984)))))))
	elif re <= 6.5e+296:
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))
	else:
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e+41)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * Float64(Float64(Float64(im * im) * 0.001388888888888889) + 0.041666666666666664))))))));
	elseif (re <= 1.2e+154)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.16666666666666666 + Float64(re * Float64(re * Float64(0.008333333333333333 + Float64(Float64(re * re) * -0.0001984126984126984))))))));
	elseif (re <= 6.5e+296)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.008333333333333333)))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.2e+41)
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))));
	elseif (re <= 1.2e+154)
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984)))))));
	elseif (re <= 6.5e+296)
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	else
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e+41], N[(re * N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.2e+154], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.16666666666666666 + N[(re * N[(re * N[(0.008333333333333333 + N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e+296], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < 6.2e41

   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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

         \[\leadsto \sin re \cdot \frac{e^{0 - im} + e^{im}}{\color{blue}{2}} \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \frac{e^{im} + e^{0 - im}}{2} \]

      7. sub0-neg N/A

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

      8. cosh-def N/A

         \[\leadsto \sin re \cdot \cosh im \]

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

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

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

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

      11. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 89.9%

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

   7. Simplified 89.9%

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

   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 63.0%

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

   10. Simplified 63.0%

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

   if 6.2e41 < re < 1.20000000000000007e154

   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 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{6} + \color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \left({re}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 26.0%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 26.0%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)} \]

   if 1.20000000000000007e154 < re < 6.5e296

   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 72.9%

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

   5. Simplified 72.9%

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

   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 29.5%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 29.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)} \]

   if 6.5e296 < 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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 66.9%

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

   8. Simplified 66.9%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   10. 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. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 66.9%

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

   11. Simplified 66.9%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+296}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+296}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e+41)
   (*
    re
    (+
     1.0
     (*
      im
      (*
       im
       (+
        0.5
        (*
         im
         (*
          im
          (+ (* (* im im) 0.001388888888888889) 0.041666666666666664))))))))
   (if (<= re 1.2e+154)
     (* re (* (* im im) (+ 0.5 (* -0.08333333333333333 (* re re)))))
     (if (<= re 6.5e+296)
       (*
        re
        (+
         1.0
         (*
          (* re re)
          (+ -0.16666666666666666 (* (* re re) 0.008333333333333333)))))
       (* re (+ 1.0 (* -0.16666666666666666 (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+41) {
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))));
	} else if (re <= 1.2e+154) {
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else if (re <= 6.5e+296) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (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 <= 6.2d+41) then
        tmp = re * (1.0d0 + (im * (im * (0.5d0 + (im * (im * (((im * im) * 0.001388888888888889d0) + 0.041666666666666664d0)))))))
    else if (re <= 1.2d+154) then
        tmp = re * ((im * im) * (0.5d0 + ((-0.08333333333333333d0) * (re * re))))
    else if (re <= 6.5d+296) then
        tmp = re * (1.0d0 + ((re * re) * ((-0.16666666666666666d0) + ((re * re) * 0.008333333333333333d0))))
    else
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+41) {
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))));
	} else if (re <= 1.2e+154) {
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else if (re <= 6.5e+296) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e+41:
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))))
	elif re <= 1.2e+154:
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))))
	elif re <= 6.5e+296:
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))
	else:
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e+41)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * Float64(Float64(Float64(im * im) * 0.001388888888888889) + 0.041666666666666664))))))));
	elseif (re <= 1.2e+154)
		tmp = Float64(re * Float64(Float64(im * im) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))));
	elseif (re <= 6.5e+296)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.008333333333333333)))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.2e+41)
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (((im * im) * 0.001388888888888889) + 0.041666666666666664)))))));
	elseif (re <= 1.2e+154)
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	elseif (re <= 6.5e+296)
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	else
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e+41], N[(re * N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.2e+154], N[(re * N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e+296], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < 6.2e41

   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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

         \[\leadsto \sin re \cdot \frac{e^{0 - im} + e^{im}}{\color{blue}{2}} \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \frac{e^{im} + e^{0 - im}}{2} \]

      7. sub0-neg N/A

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

      8. cosh-def N/A

         \[\leadsto \sin re \cdot \cosh im \]

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

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

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

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

      11. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 89.9%

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

   7. Simplified 89.9%

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

   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 63.0%

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

   10. Simplified 63.0%

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

   if 6.2e41 < re < 1.20000000000000007e154

   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 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

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

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 26.2%

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

   8. Simplified 26.2%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 27.1%

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

   11. Simplified 27.1%

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

   if 1.20000000000000007e154 < re < 6.5e296

   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 72.9%

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

   5. Simplified 72.9%

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

   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 29.5%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 29.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)} \]

   if 6.5e296 < 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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 66.9%

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

   8. Simplified 66.9%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   10. 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. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 66.9%

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

   11. Simplified 66.9%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 + 0.041666666666666664\right)\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+296}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+296}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e+41)
   (* re (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))
   (if (<= re 1.2e+154)
     (* re (* (* im im) (+ 0.5 (* -0.08333333333333333 (* re re)))))
     (if (<= re 6.5e+296)
       (*
        re
        (+
         1.0
         (*
          (* re re)
          (+ -0.16666666666666666 (* (* re re) 0.008333333333333333)))))
       (* re (+ 1.0 (* -0.16666666666666666 (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+41) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	} else if (re <= 1.2e+154) {
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else if (re <= 6.5e+296) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (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 <= 6.2d+41) then
        tmp = re * (1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0)))))
    else if (re <= 1.2d+154) then
        tmp = re * ((im * im) * (0.5d0 + ((-0.08333333333333333d0) * (re * re))))
    else if (re <= 6.5d+296) then
        tmp = re * (1.0d0 + ((re * re) * ((-0.16666666666666666d0) + ((re * re) * 0.008333333333333333d0))))
    else
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+41) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	} else if (re <= 1.2e+154) {
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else if (re <= 6.5e+296) {
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e+41:
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))))
	elif re <= 1.2e+154:
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))))
	elif re <= 6.5e+296:
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))))
	else:
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e+41)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664))))));
	elseif (re <= 1.2e+154)
		tmp = Float64(re * Float64(Float64(im * im) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))));
	elseif (re <= 6.5e+296)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.008333333333333333)))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.2e+41)
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	elseif (re <= 1.2e+154)
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	elseif (re <= 6.5e+296)
		tmp = re * (1.0 + ((re * re) * (-0.16666666666666666 + ((re * re) * 0.008333333333333333))));
	else
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e+41], 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], If[LessEqual[re, 1.2e+154], N[(re * N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e+296], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < 6.2e41

   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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

   5. Simplified 86.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   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 60.7%

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

   8. Simplified 60.7%

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

   if 6.2e41 < re < 1.20000000000000007e154

   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 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

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

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 26.2%

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

   8. Simplified 26.2%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 27.1%

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

   11. Simplified 27.1%

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

   if 1.20000000000000007e154 < re < 6.5e296

   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 72.9%

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

   5. Simplified 72.9%

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

   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 29.5%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 29.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)} \]

   if 6.5e296 < 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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 66.9%

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

   8. Simplified 66.9%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   10. 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. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 66.9%

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

   11. Simplified 66.9%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+296}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{if}\;re \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+296}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          re
          (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))))
   (if (<= re 6.2e+41)
     t_0
     (if (<= re 1.2e+154)
       (* re (* (* im im) (+ 0.5 (* -0.08333333333333333 (* re re)))))
       (if (<= re 6.5e+296)
         t_0
         (* re (+ 1.0 (* -0.16666666666666666 (* re re)))))))))

C

double code(double re, double im) {
	double t_0 = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	double tmp;
	if (re <= 6.2e+41) {
		tmp = t_0;
	} else if (re <= 1.2e+154) {
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else if (re <= 6.5e+296) {
		tmp = t_0;
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0)))))
    if (re <= 6.2d+41) then
        tmp = t_0
    else if (re <= 1.2d+154) then
        tmp = re * ((im * im) * (0.5d0 + ((-0.08333333333333333d0) * (re * re))))
    else if (re <= 6.5d+296) then
        tmp = t_0
    else
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	double tmp;
	if (re <= 6.2e+41) {
		tmp = t_0;
	} else if (re <= 1.2e+154) {
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else if (re <= 6.5e+296) {
		tmp = t_0;
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))))
	tmp = 0
	if re <= 6.2e+41:
		tmp = t_0
	elif re <= 1.2e+154:
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))))
	elif re <= 6.5e+296:
		tmp = t_0
	else:
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664))))))
	tmp = 0.0
	if (re <= 6.2e+41)
		tmp = t_0;
	elseif (re <= 1.2e+154)
		tmp = Float64(re * Float64(Float64(im * im) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))));
	elseif (re <= 6.5e+296)
		tmp = t_0;
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	tmp = 0.0;
	if (re <= 6.2e+41)
		tmp = t_0;
	elseif (re <= 1.2e+154)
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	elseif (re <= 6.5e+296)
		tmp = t_0;
	else
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = 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]}, If[LessEqual[re, 6.2e+41], t$95$0, If[LessEqual[re, 1.2e+154], N[(re * N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e+296], t$95$0, N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 6.2e41 or 1.20000000000000007e154 < re < 6.5e296

   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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

   5. Simplified 87.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   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 56.9%

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

   8. Simplified 56.9%

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

   if 6.2e41 < re < 1.20000000000000007e154

   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 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

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

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 26.2%

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

   8. Simplified 26.2%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 27.1%

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

   11. Simplified 27.1%

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

   if 6.5e296 < 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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 66.9%

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

   8. Simplified 66.9%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   10. 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. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 66.9%

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

   11. Simplified 66.9%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+296}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{if}\;re \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+296}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re 0.5) (+ (* im im) 2.0))))
   (if (<= re 6.2e+41)
     t_0
     (if (<= re 1.2e+154)
       (* re (* (* im im) (+ 0.5 (* -0.08333333333333333 (* re re)))))
       (if (<= re 6.5e+296)
         t_0
         (* re (+ 1.0 (* -0.16666666666666666 (* re re)))))))))

C

double code(double re, double im) {
	double t_0 = (re * 0.5) * ((im * im) + 2.0);
	double tmp;
	if (re <= 6.2e+41) {
		tmp = t_0;
	} else if (re <= 1.2e+154) {
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else if (re <= 6.5e+296) {
		tmp = t_0;
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re * 0.5d0) * ((im * im) + 2.0d0)
    if (re <= 6.2d+41) then
        tmp = t_0
    else if (re <= 1.2d+154) then
        tmp = re * ((im * im) * (0.5d0 + ((-0.08333333333333333d0) * (re * re))))
    else if (re <= 6.5d+296) then
        tmp = t_0
    else
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * 0.5) * ((im * im) + 2.0);
	double tmp;
	if (re <= 6.2e+41) {
		tmp = t_0;
	} else if (re <= 1.2e+154) {
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else if (re <= 6.5e+296) {
		tmp = t_0;
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * 0.5) * ((im * im) + 2.0)
	tmp = 0
	if re <= 6.2e+41:
		tmp = t_0
	elif re <= 1.2e+154:
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))))
	elif re <= 6.5e+296:
		tmp = t_0
	else:
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * 0.5) * Float64(Float64(im * im) + 2.0))
	tmp = 0.0
	if (re <= 6.2e+41)
		tmp = t_0;
	elseif (re <= 1.2e+154)
		tmp = Float64(re * Float64(Float64(im * im) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))));
	elseif (re <= 6.5e+296)
		tmp = t_0;
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * 0.5) * ((im * im) + 2.0);
	tmp = 0.0;
	if (re <= 6.2e+41)
		tmp = t_0;
	elseif (re <= 1.2e+154)
		tmp = re * ((im * im) * (0.5 + (-0.08333333333333333 * (re * re))));
	elseif (re <= 6.5e+296)
		tmp = t_0;
	else
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 6.2e+41], t$95$0, If[LessEqual[re, 1.2e+154], N[(re * N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e+296], t$95$0, N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 6.2e41 or 1.20000000000000007e154 < re < 6.5e296

   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 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 50.3%

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

   if 6.2e41 < re < 1.20000000000000007e154

   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 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

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

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 26.2%

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

   8. Simplified 26.2%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 27.1%

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

   11. Simplified 27.1%

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

   if 6.5e296 < 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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 66.9%

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

   8. Simplified 66.9%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   10. 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. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 66.9%

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

   11. Simplified 66.9%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+296}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.0% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.8 \cdot 10^{+81}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.8e+81)
   (* re (+ 1.0 (* -0.16666666666666666 (* re re))))
   (* (* re 0.5) (+ (* im im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.8e+81) {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	} else {
		tmp = (re * 0.5) * ((im * im) + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.8d+81) then
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
    else
        tmp = (re * 0.5d0) * ((im * im) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.8e+81) {
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	} else {
		tmp = (re * 0.5) * ((im * im) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.8e+81:
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)))
	else:
		tmp = (re * 0.5) * ((im * im) + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.8e+81)
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))));
	else
		tmp = Float64(Float64(re * 0.5) * Float64(Float64(im * im) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.8e+81)
		tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
	else
		tmp = (re * 0.5) * ((im * im) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.8e+81], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.80000000000000003e81

   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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

   5. Simplified 84.5%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 46.9%

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

   8. Simplified 46.9%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   10. 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. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 34.1%

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

   11. Simplified 34.1%

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

   if 1.80000000000000003e81 < 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 64.9%

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

   5. Simplified 64.9%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 57.2%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.8 \cdot 10^{+81}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.6% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 39000000000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 39000000000000.0) re (* re (* re (* re -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 39000000000000.0) {
		tmp = re;
	} else {
		tmp = re * (re * (re * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 39000000000000.0d0) then
        tmp = re
    else
        tmp = re * (re * (re * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 39000000000000.0) {
		tmp = re;
	} else {
		tmp = re * (re * (re * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 39000000000000.0:
		tmp = re
	else:
		tmp = re * (re * (re * -0.16666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 39000000000000.0)
		tmp = re;
	else
		tmp = Float64(re * Float64(re * Float64(re * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 39000000000000.0)
		tmp = re;
	else
		tmp = re * (re * (re * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 39000000000000.0], re, N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.9e13

   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 48.9%

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

   5. Simplified 48.9%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 30.9%

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

   if 3.9e13 < 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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

   5. Simplified 91.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 14.2%

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

   8. Simplified 14.2%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   10. 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. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 9.1%

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

   11. Simplified 9.1%

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

   12. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\left(re \cdot \frac{-1}{6}\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \frac{-1}{6}\right)}\right)\right) \]

       5. *-lowering-*.f64 9.1%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   14. Simplified 9.1%

       \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)} \]
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

   \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
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. associate-*r* N/A

      \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\frac{1}{2}} \cdot \sin re\right) \]

   3. distribute-rgt-out N/A

      \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \]

   5. distribute-lft-out N/A

      \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}}\right) \]

   6. *-commutative N/A

      \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \frac{1}{2}\right) \]

   7. associate-*l* N/A

      \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \frac{1}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2}\right) \]

   9. associate-*l* N/A

      \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{2}\right)}\right) \]

   10. unpow2 N/A

       \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \sin re \cdot \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{2}\right)}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot \color{blue}{im}\right)\right)\right) \]

   13. distribute-lft-out N/A

       \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + im \cdot \left(\frac{1}{2} \cdot im\right)\right)} \]

5. Simplified 87.3%

   \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 53.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

8. Simplified 53.0%

   \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

9. Taylor expanded in im around 0

   \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
10. 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. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

    5. *-lowering-*.f64 31.4%

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

11. Simplified 31.4%

    \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]
12. 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 52.3%

      \[\leadsto \mathsf{sin.f64}\left(re\right) \]

5. Simplified 52.3%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re} \]
7. Step-by-step derivation

8. Simplified 25.0%

   \[\leadsto \color{blue}{re} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))