math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.9s

Alternatives: 21

Speedup: 1.5×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in N/A

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

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

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

   3. *-commutative N/A

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

   4. sub0-neg N/A

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

   5. exp-neg N/A

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

   6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. exp-lowering-exp.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. sub0-neg N/A

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

   7. cosh-undef N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. exp-0 N/A

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

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

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

   12. exp-0 N/A

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

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

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

   14. sin-lowering-sin.f64 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
5. Step-by-step derivation

   1. *-commutative N/A

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

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

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

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

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

   4. *-lft-identity N/A

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

   5. cosh-lowering-cosh.f64 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 3: 79.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{-18}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2e-18)
   (* re (cosh im))
   (*
    (sin re)
    (+
     1.0
     (*
      (* im im)
      (+
       0.5
       (*
        im
        (*
         im
         (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2e-18) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2d-18) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2e-18) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2e-18:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2e-18)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2e-18)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2e-18], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.0000000000000001e-18

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 76.3%

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

   if 2.0000000000000001e-18 < re

   1. Initial program 99.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 96.5%

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

   7. Simplified 96.5%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{-18}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{-18}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(0.001388888888888889 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e-18)
   (* re (cosh im))
   (*
    (sin re)
    (+
     1.0
     (* (* im im) (+ 0.5 (* im (* 0.001388888888888889 (* im (* im im))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1e-18) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (0.001388888888888889 * (im * (im * im)))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1d-18) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (0.001388888888888889d0 * (im * (im * im)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1e-18) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (0.001388888888888889 * (im * (im * im)))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1e-18:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (0.001388888888888889 * (im * (im * im)))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1e-18)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(0.001388888888888889 * Float64(im * Float64(im * im))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1e-18)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (0.001388888888888889 * (im * (im * im)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1e-18], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(0.001388888888888889 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.0000000000000001e-18

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 76.3%

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

   if 1.0000000000000001e-18 < re

   1. Initial program 99.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in im around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 96.0%

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

   10. Simplified 96.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 10^{-18}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(0.001388888888888889 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 24:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 24.0)
   (* (* 0.5 (sin re)) (+ (* im im) 2.0))
   (if (<= im 2.6e+77)
     (* re (cosh im))
     (* (sin re) (* im (* im (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 24.0) {
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	} else if (im <= 2.6e+77) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 24.0d0) then
        tmp = (0.5d0 * sin(re)) * ((im * im) + 2.0d0)
    else if (im <= 2.6d+77) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (im * (im * ((im * im) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 24.0) {
		tmp = (0.5 * Math.sin(re)) * ((im * im) + 2.0);
	} else if (im <= 2.6e+77) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 24.0:
		tmp = (0.5 * math.sin(re)) * ((im * im) + 2.0)
	elif im <= 2.6e+77:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (im * (im * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 24.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(im * im) + 2.0));
	elseif (im <= 2.6e+77)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 24.0)
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	elseif (im <= 2.6e+77)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (im * (im * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 24.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 24

   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 86.8%

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

   5. Simplified 86.8%

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

   if 24 < 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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 70.0%

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

   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. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 24:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{-18}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.5e-18)
   (* re (cosh im))
   (*
    (sin re)
    (+ 1.0 (* (* im im) (+ 0.5 (* im (* im 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.5e-18) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.5d-18) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.5e-18) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.5e-18:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.5e-18)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.5e-18)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.5e-18], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.50000000000000018e-18

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 76.3%

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

   if 2.50000000000000018e-18 < re

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 93.1%

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

Alternative 7: 78.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{-20}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2e-20)
   (* re (cosh im))
   (* (sin re) (+ 1.0 (* im (* im (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2e-20) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2d-20) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (1.0d0 + (im * (im * ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2e-20) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2e-20:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2e-20)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2e-20)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2e-20], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.99999999999999989e-20

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 76.3%

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

   if 1.99999999999999989e-20 < re

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-out N/A

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

   5. Simplified 93.1%

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

   6. Taylor expanded in im around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 92.3%

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

   8. Simplified 92.3%

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

Alternative 8: 84.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 24:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 24.0)
   (* (* 0.5 (sin re)) (+ (* im im) 2.0))
   (if (<= im 1.35e+154) (* re (cosh im)) (* (sin re) (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 24.0) {
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	} else if (im <= 1.35e+154) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 24.0d0) then
        tmp = (0.5d0 * sin(re)) * ((im * im) + 2.0d0)
    else if (im <= 1.35d+154) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 24.0) {
		tmp = (0.5 * Math.sin(re)) * ((im * im) + 2.0);
	} else if (im <= 1.35e+154) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 24.0:
		tmp = (0.5 * math.sin(re)) * ((im * im) + 2.0)
	elif im <= 1.35e+154:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 24.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(im * im) + 2.0));
	elseif (im <= 1.35e+154)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 24.0)
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	elseif (im <= 1.35e+154)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 24.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 24

   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 86.8%

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

   5. Simplified 86.8%

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

   if 24 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 75.9%

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

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 24:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00015:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00015)
   (sin re)
   (if (<= im 1.5e+154) (* re (cosh im)) (* (sin re) (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00015) {
		tmp = sin(re);
	} else if (im <= 1.5e+154) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00015d0) then
        tmp = sin(re)
    else if (im <= 1.5d+154) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00015) {
		tmp = Math.sin(re);
	} else if (im <= 1.5e+154) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00015:
		tmp = math.sin(re)
	elif im <= 1.5e+154:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00015)
		tmp = sin(re);
	elseif (im <= 1.5e+154)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00015)
		tmp = sin(re);
	elseif (im <= 1.5e+154)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00015], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.5e+154], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.49999999999999987e-4

   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 72.8%

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

   5. Simplified 72.8%

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

   if 1.49999999999999987e-4 < im < 1.50000000000000013e154

   1. Initial program 99.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 71.2%

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

   if 1.50000000000000013e154 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00015:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00028:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+260}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00028)
   (sin re)
   (if (<= im 6e+260)
     (* re (cosh im))
     (*
      (+ 1.0 (* 0.5 (* im im)))
      (* re (+ 1.0 (* (* re re) -0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00028) {
		tmp = sin(re);
	} else if (im <= 6e+260) {
		tmp = re * cosh(im);
	} else {
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00028d0) then
        tmp = sin(re)
    else if (im <= 6d+260) then
        tmp = re * cosh(im)
    else
        tmp = (1.0d0 + (0.5d0 * (im * im))) * (re * (1.0d0 + ((re * re) * (-0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00028) {
		tmp = Math.sin(re);
	} else if (im <= 6e+260) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00028:
		tmp = math.sin(re)
	elif im <= 6e+260:
		tmp = re * math.cosh(im)
	else:
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00028)
		tmp = sin(re);
	elseif (im <= 6e+260)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(im * im))) * Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00028)
		tmp = sin(re);
	elseif (im <= 6e+260)
		tmp = re * cosh(im);
	else
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00028], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6e+260], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.7999999999999998e-4

   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 72.8%

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

   5. Simplified 72.8%

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

   if 2.7999999999999998e-4 < im < 5.9999999999999996e260

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 75.7%

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

   if 5.9999999999999996e260 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

Alternative 11: 68.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 660:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+53}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(\left(re \cdot re\right) \cdot \frac{re \cdot \left(3.936759889140842 \cdot 10^{-8} - \frac{6.944444444444444 \cdot 10^{-5}}{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right)}{-0.0001984126984126984 - \frac{0.008333333333333333}{re \cdot re}}\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+260}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 660.0)
   (sin re)
   (if (<= im 4.8e+53)
     (*
      re
      (+
       1.0
       (*
        re
        (*
         re
         (+
          -0.16666666666666666
          (*
           re
           (*
            (* re re)
            (/
             (*
              re
              (-
               3.936759889140842e-8
               (/ 6.944444444444444e-5 (* (* re re) (* re re)))))
             (-
              -0.0001984126984126984
              (/ 0.008333333333333333 (* re re)))))))))))
     (if (<= im 2e+260)
       (*
        re
        (+
         1.0
         (*
          im
          (*
           im
           (+
            0.5
            (*
             (* im im)
             (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))))))
       (*
        (+ 1.0 (* 0.5 (* im im)))
        (* re (+ 1.0 (* (* re re) -0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 660.0) {
		tmp = sin(re);
	} else if (im <= 4.8e+53) {
		tmp = re * (1.0 + (re * (re * (-0.16666666666666666 + (re * ((re * re) * ((re * (3.936759889140842e-8 - (6.944444444444444e-5 / ((re * re) * (re * re))))) / (-0.0001984126984126984 - (0.008333333333333333 / (re * re))))))))));
	} else if (im <= 2e+260) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 660.0d0) then
        tmp = sin(re)
    else if (im <= 4.8d+53) then
        tmp = re * (1.0d0 + (re * (re * ((-0.16666666666666666d0) + (re * ((re * re) * ((re * (3.936759889140842d-8 - (6.944444444444444d-5 / ((re * re) * (re * re))))) / ((-0.0001984126984126984d0) - (0.008333333333333333d0 / (re * re))))))))))
    else if (im <= 2d+260) then
        tmp = re * (1.0d0 + (im * (im * (0.5d0 + ((im * im) * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))
    else
        tmp = (1.0d0 + (0.5d0 * (im * im))) * (re * (1.0d0 + ((re * re) * (-0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 660.0) {
		tmp = Math.sin(re);
	} else if (im <= 4.8e+53) {
		tmp = re * (1.0 + (re * (re * (-0.16666666666666666 + (re * ((re * re) * ((re * (3.936759889140842e-8 - (6.944444444444444e-5 / ((re * re) * (re * re))))) / (-0.0001984126984126984 - (0.008333333333333333 / (re * re))))))))));
	} else if (im <= 2e+260) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 660.0:
		tmp = math.sin(re)
	elif im <= 4.8e+53:
		tmp = re * (1.0 + (re * (re * (-0.16666666666666666 + (re * ((re * re) * ((re * (3.936759889140842e-8 - (6.944444444444444e-5 / ((re * re) * (re * re))))) / (-0.0001984126984126984 - (0.008333333333333333 / (re * re))))))))))
	elif im <= 2e+260:
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))
	else:
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 660.0)
		tmp = sin(re);
	elseif (im <= 4.8e+53)
		tmp = Float64(re * Float64(1.0 + Float64(re * Float64(re * Float64(-0.16666666666666666 + Float64(re * Float64(Float64(re * re) * Float64(Float64(re * Float64(3.936759889140842e-8 - Float64(6.944444444444444e-5 / Float64(Float64(re * re) * Float64(re * re))))) / Float64(-0.0001984126984126984 - Float64(0.008333333333333333 / Float64(re * re)))))))))));
	elseif (im <= 2e+260)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(im * im))) * Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 660.0)
		tmp = sin(re);
	elseif (im <= 4.8e+53)
		tmp = re * (1.0 + (re * (re * (-0.16666666666666666 + (re * ((re * re) * ((re * (3.936759889140842e-8 - (6.944444444444444e-5 / ((re * re) * (re * re))))) / (-0.0001984126984126984 - (0.008333333333333333 / (re * re))))))))));
	elseif (im <= 2e+260)
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	else
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 660.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4.8e+53], N[(re * N[(1.0 + N[(re * N[(re * N[(-0.16666666666666666 + N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * N[(3.936759889140842e-8 - N[(6.944444444444444e-5 / N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.0001984126984126984 - N[(0.008333333333333333 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2e+260], N[(re * N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 660

   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 72.3%

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

   5. Simplified 72.3%

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

   if 660 < im < 4.8e53

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 2.7%

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

   5. Simplified 2.7%

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

   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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 29.9%

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

   8. Simplified 29.9%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{120} \cdot \frac{1}{{re}^{2}} - \frac{1}{5040}\right)\right)}\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

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

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. +-commutative N/A

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

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

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

       12. associate-*r/ N/A

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

       13. metadata-eval N/A

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

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

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 29.7%

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

   11. Simplified 29.7%

       \[\leadsto re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \left(re \cdot \left(-0.0001984126984126984 + \frac{0.008333333333333333}{re \cdot re}\right)\right)\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. flip-+ N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

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

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

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{25401600}, \left(\frac{\frac{1}{120}}{re \cdot re} \cdot \frac{\frac{1}{120}}{re \cdot re}\right)\right), re\right), \left(\frac{-1}{5040} - \frac{\frac{1}{120}}{re \cdot re}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       8. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{25401600}, \left(\frac{\frac{1}{120} \cdot \frac{1}{120}}{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right)\right), re\right), \left(\frac{-1}{5040} - \frac{\frac{1}{120}}{re \cdot re}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{25401600}, \mathsf{/.f64}\left(\left(\frac{1}{120} \cdot \frac{1}{120}\right), \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)\right), re\right), \left(\frac{-1}{5040} - \frac{\frac{1}{120}}{re \cdot re}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{25401600}, \mathsf{/.f64}\left(\frac{1}{14400}, \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)\right), re\right), \left(\frac{-1}{5040} - \frac{\frac{1}{120}}{re \cdot re}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{25401600}, \mathsf{/.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot re\right)\right)\right)\right), re\right), \left(\frac{-1}{5040} - \frac{\frac{1}{120}}{re \cdot re}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{25401600}, \mathsf{/.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot re\right)\right)\right)\right), re\right), \left(\frac{-1}{5040} - \frac{\frac{1}{120}}{re \cdot re}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{25401600}, \mathsf{/.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), re\right), \left(\frac{-1}{5040} - \frac{\frac{1}{120}}{re \cdot re}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{25401600}, \mathsf{/.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), re\right), \mathsf{\_.f64}\left(\frac{-1}{5040}, \color{blue}{\left(\frac{\frac{1}{120}}{re \cdot re}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{25401600}, \mathsf{/.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), re\right), \mathsf{\_.f64}\left(\frac{-1}{5040}, \mathsf{/.f64}\left(\frac{1}{120}, \color{blue}{\left(re \cdot re\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 57.6%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{25401600}, \mathsf{/.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), re\right), \mathsf{\_.f64}\left(\frac{-1}{5040}, \mathsf{/.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. Applied egg-rr 57.6%

       \[\leadsto re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\frac{\left(3.936759889140842 \cdot 10^{-8} - \frac{6.944444444444444 \cdot 10^{-5}}{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot re}{-0.0001984126984126984 - \frac{0.008333333333333333}{re \cdot re}}}\right)\right)\right)\right) \]

   if 4.8e53 < im < 2.00000000000000013e260

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 80.0%

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

   10. Simplified 80.0%

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

   if 2.00000000000000013e260 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 660:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+53}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(\left(re \cdot re\right) \cdot \frac{re \cdot \left(3.936759889140842 \cdot 10^{-8} - \frac{6.944444444444444 \cdot 10^{-5}}{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right)}{-0.0001984126984126984 - \frac{0.008333333333333333}{re \cdot re}}\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+260}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{if}\;im \leq 1.08 \cdot 10^{+74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+260}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (+ 1.0 (* 0.5 (* im im)))
          (* re (+ 1.0 (* (* re re) -0.16666666666666666))))))
   (if (<= im 1.08e+74)
     t_0
     (if (<= im 2e+260)
       (* re (* im (* im (* (* im im) 0.041666666666666664))))
       t_0))))

C

double code(double re, double im) {
	double t_0 = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	double tmp;
	if (im <= 1.08e+74) {
		tmp = t_0;
	} else if (im <= 2e+260) {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (0.5d0 * (im * im))) * (re * (1.0d0 + ((re * re) * (-0.16666666666666666d0))))
    if (im <= 1.08d+74) then
        tmp = t_0
    else if (im <= 2d+260) then
        tmp = re * (im * (im * ((im * im) * 0.041666666666666664d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	double tmp;
	if (im <= 1.08e+74) {
		tmp = t_0;
	} else if (im <= 2e+260) {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)))
	tmp = 0
	if im <= 1.08e+74:
		tmp = t_0
	elif im <= 2e+260:
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(1.0 + Float64(0.5 * Float64(im * im))) * Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666))))
	tmp = 0.0
	if (im <= 1.08e+74)
		tmp = t_0;
	elseif (im <= 2e+260)
		tmp = Float64(re * Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	tmp = 0.0;
	if (im <= 1.08e+74)
		tmp = t_0;
	elseif (im <= 2e+260)
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.08e+74], t$95$0, If[LessEqual[im, 2e+260], N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.08e74 or 2.00000000000000013e260 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{re}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      4. *-lowering-*.f64 83.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   7. Simplified 83.5%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 56.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right)\right) \]

   10. Simplified 56.1%

       \[\leadsto \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)} \]

   if 1.08e74 < im < 2.00000000000000013e260

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 81.1%

      \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{re} \]

       2. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

          \[\leadsto re \cdot \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

       4. pow-sqr N/A

          \[\leadsto re \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto re \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right) \]

       6. *-commutative N/A

          \[\leadsto re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 81.1%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   11. Simplified 81.1%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.7% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.9 \cdot 10^{+86}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.9e+86)
   (*
    re
    (+
     1.0
     (*
      im
      (*
       im
       (+
        0.5
        (*
         (* im im)
         (+ 0.041666666666666664 (* (* im im) 0.001388888888888889))))))))
   (*
    (+ 1.0 (* 0.5 (* im im)))
    (* re (+ 1.0 (* (* re re) -0.16666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.9e+86) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.9d+86) then
        tmp = re * (1.0d0 + (im * (im * (0.5d0 + ((im * im) * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0)))))))
    else
        tmp = (1.0d0 + (0.5d0 * (im * im))) * (re * (1.0d0 + ((re * re) * (-0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.9e+86) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.9e+86:
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))))
	else:
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.9e+86)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(im * im))) * Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.9e+86)
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889)))))));
	else
		tmp = (1.0 + (0.5 * (im * im))) * (re * (1.0 + ((re * re) * -0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.9e+86], N[(re * N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.8999999999999999e86

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \sin re\right) \]

      7. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \sin \color{blue}{re}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

      13. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}, \mathsf{sin.f64}\left(re\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{re}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      14. *-lowering-*.f64 93.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   7. Simplified 93.3%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 68.7%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 68.7%

       \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   if 2.8999999999999999e86 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \sin re\right) \]

      7. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \sin \color{blue}{re}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

      13. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{1}{2} \cdot {im}^{2}\right)}, \mathsf{sin.f64}\left(re\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{re}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      4. *-lowering-*.f64 75.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   7. Simplified 75.8%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 25.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right)\right) \]

   10. Simplified 25.5%

       \[\leadsto \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 56.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.9 \cdot 10^{+86}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.006944444444444444\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.9e+86)
   (* re (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))
   (*
    re
    (*
     (* (* im im) (* im im))
     (+ 0.041666666666666664 (* (* re re) -0.006944444444444444))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.9e+86) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	} else {
		tmp = re * (((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.006944444444444444)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.9d+86) then
        tmp = re * (1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0)))))
    else
        tmp = re * (((im * im) * (im * im)) * (0.041666666666666664d0 + ((re * re) * (-0.006944444444444444d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.9e+86) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	} else {
		tmp = re * (((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.006944444444444444)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.9e+86:
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))))
	else:
		tmp = re * (((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.006944444444444444)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.9e+86)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664))))));
	else
		tmp = Float64(re * Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.006944444444444444))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.9e+86)
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	else
		tmp = re * (((im * im) * (im * im)) * (0.041666666666666664 + ((re * re) * -0.006944444444444444)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.9e+86], 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], N[(re * N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.8999999999999999e86

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 90.2%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   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 66.5%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 66.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   if 2.8999999999999999e86 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 92.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\sin re} \]

      2. flip-+ N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)}{1 - \left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)} \cdot \sin \color{blue}{re} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(1 \cdot 1 - \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \sin re}{\color{blue}{1 - \left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \sin re\right), \color{blue}{\left(1 - \left(im \cdot im\right) \cdot \left(\frac{1}{2} + im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)}\right) \]

   7. Applied egg-rr 62.5%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \sin re}{1 - im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)}} \]

   8. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \sin re\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\sin re} \]

      2. *-commutative N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \sin re \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \sin re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right) \]

      15. *-lowering-*.f64 37.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right) \]

   10. Simplified 37.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   11. Taylor expanded in re around 0

       \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{144} \cdot \left({im}^{4} \cdot {re}^{2}\right) + \frac{1}{24} \cdot {im}^{4}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{144} \cdot \left({im}^{4} \cdot {re}^{2}\right) + \frac{1}{24} \cdot {im}^{4}\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\frac{1}{24} \cdot {im}^{4} + \color{blue}{\frac{-1}{144} \cdot \left({im}^{4} \cdot {re}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\frac{1}{24} \cdot {im}^{4} + \frac{-1}{144} \cdot \left({re}^{2} \cdot \color{blue}{{im}^{4}}\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\frac{1}{24} \cdot {im}^{4} + \left(\frac{-1}{144} \cdot {re}^{2}\right) \cdot \color{blue}{{im}^{4}}\right)\right) \]

       5. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left({im}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {re}^{2}\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {re}^{2}\right)}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {re}^{2}\right)\right)\right) \]

       8. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {re}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {re}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {re}^{2}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {re}^{2}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {re}^{2}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {re}^{2}\right)\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{144} \cdot {re}^{2}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 25.4%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right) \]

   13. Simplified 25.4%

       \[\leadsto \color{blue}{re \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + -0.006944444444444444 \cdot \left(re \cdot re\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.9 \cdot 10^{+86}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.006944444444444444\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.0% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.65 \cdot 10^{+220}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.65e+220)
   (* re (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))
   (* re (* (* re re) -0.16666666666666666))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.65e+220) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	} 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 <= 1.65d+220) then
        tmp = re * (1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0)))))
    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 <= 1.65e+220) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	} else {
		tmp = re * ((re * re) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.65e+220:
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))))
	else:
		tmp = re * ((re * re) * -0.16666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.65e+220)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664))))));
	else
		tmp = Float64(re * Float64(Float64(re * re) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.65e+220)
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	else
		tmp = re * ((re * re) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.65e+220], 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], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.65000000000000011e220

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 90.6%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 60.6%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 60.6%

      \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   if 1.65000000000000011e220 < re

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 58.2%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 58.2%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)} \]
   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. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{6} + \color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \left({re}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 19.0%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 19.0%

      \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot re\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 19.0%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   11. Simplified 19.0%

       \[\leadsto re \cdot \left(1 + re \cdot \color{blue}{\left(re \cdot -0.16666666666666666\right)}\right) \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   13. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{re}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({re}^{2} \cdot re\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re} \]

       4. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       8. *-lowering-*.f64 19.0%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   14. Simplified 19.0%

       \[\leadsto \color{blue}{re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.65 \cdot 10^{+220}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.0% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.6 \cdot 10^{+74}:\\ \;\;\;\;re + -0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.6e+74)
   (+ re (* -0.16666666666666666 (* re (* re re))))
   (* re (* im (* im (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.6e+74) {
		tmp = re + (-0.16666666666666666 * (re * (re * re)));
	} else {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.6d+74) then
        tmp = re + ((-0.16666666666666666d0) * (re * (re * re)))
    else
        tmp = re * (im * (im * ((im * im) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.6e+74) {
		tmp = re + (-0.16666666666666666 * (re * (re * re)));
	} else {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.6e+74:
		tmp = re + (-0.16666666666666666 * (re * (re * re)))
	else:
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.6e+74)
		tmp = Float64(re + Float64(-0.16666666666666666 * Float64(re * Float64(re * re))));
	else
		tmp = Float64(re * Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.6e+74)
		tmp = re + (-0.16666666666666666 * (re * (re * re)));
	else
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.6e+74], N[(re + N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.59999999999999997e74

   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 69.0%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 69.0%

      \[\leadsto \color{blue}{\sin re} \]

   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. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{6} + \color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \left({re}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 45.8%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 45.8%

      \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot re\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 44.1%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   11. Simplified 44.1%

       \[\leadsto re \cdot \left(1 + re \cdot \color{blue}{\left(re \cdot -0.16666666666666666\right)}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right) + \color{blue}{1}\right) \]

       2. distribute-lft-in N/A

          \[\leadsto re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right) + \color{blue}{re \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right) + re \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right)\right), \color{blue}{re}\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right), re\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \frac{-1}{6}\right), re\right) \]

       7. cube-unmult N/A

          \[\leadsto \mathsf{+.f64}\left(\left({re}^{3} \cdot \frac{-1}{6}\right), re\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({re}^{3}\right), \frac{-1}{6}\right), re\right) \]

       9. cube-unmult N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(re \cdot \left(re \cdot re\right)\right), \frac{-1}{6}\right), re\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot re\right)\right), \frac{-1}{6}\right), re\right) \]

       11. *-lowering-*.f64 44.1%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{-1}{6}\right), re\right) \]

   13. Applied egg-rr 44.1%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot -0.16666666666666666 + re} \]

   if 1.59999999999999997e74 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 73.3%

      \[\leadsto \color{blue}{re} \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{re} \]

       2. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

          \[\leadsto re \cdot \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

       4. pow-sqr N/A

          \[\leadsto re \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto re \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right) \]

       6. *-commutative N/A

          \[\leadsto re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 73.3%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   11. Simplified 73.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.6 \cdot 10^{+74}:\\ \;\;\;\;re + -0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.6% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{+105}:\\ \;\;\;\;re + -0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.5e+105)
   (+ re (* -0.16666666666666666 (* re (* re re))))
   (* re (+ 1.0 (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.5e+105) {
		tmp = re + (-0.16666666666666666 * (re * (re * re)));
	} else {
		tmp = re * (1.0 + (0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7.5d+105) then
        tmp = re + ((-0.16666666666666666d0) * (re * (re * re)))
    else
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.5e+105) {
		tmp = re + (-0.16666666666666666 * (re * (re * re)));
	} else {
		tmp = re * (1.0 + (0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.5e+105:
		tmp = re + (-0.16666666666666666 * (re * (re * re)))
	else:
		tmp = re * (1.0 + (0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.5e+105)
		tmp = Float64(re + Float64(-0.16666666666666666 * Float64(re * Float64(re * re))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.5e+105)
		tmp = re + (-0.16666666666666666 * (re * (re * re)));
	else
		tmp = re * (1.0 + (0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.5e+105], N[(re + N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.5000000000000002e105

   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 67.1%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 67.1%

      \[\leadsto \color{blue}{\sin re} \]

   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. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{6} + \color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \left({re}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 45.5%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 45.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot re\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   11. Simplified 43.8%

       \[\leadsto re \cdot \left(1 + re \cdot \color{blue}{\left(re \cdot -0.16666666666666666\right)}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right) + \color{blue}{1}\right) \]

       2. distribute-lft-in N/A

          \[\leadsto re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right) + \color{blue}{re \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right) + re \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right)\right), \color{blue}{re}\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right), re\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \frac{-1}{6}\right), re\right) \]

       7. cube-unmult N/A

          \[\leadsto \mathsf{+.f64}\left(\left({re}^{3} \cdot \frac{-1}{6}\right), re\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({re}^{3}\right), \frac{-1}{6}\right), re\right) \]

       9. cube-unmult N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(re \cdot \left(re \cdot re\right)\right), \frac{-1}{6}\right), re\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot re\right)\right), \frac{-1}{6}\right), re\right) \]

       11. *-lowering-*.f64 43.8%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{-1}{6}\right), re\right) \]

   13. Applied egg-rr 43.8%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot -0.16666666666666666 + re} \]

   if 7.5000000000000002e105 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \sin re\right) \]

      7. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \sin \color{blue}{re}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

      13. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{1}{2} \cdot {im}^{2}\right)}, \mathsf{sin.f64}\left(re\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{re}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      4. *-lowering-*.f64 68.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   7. Simplified 68.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \color{blue}{re}\right) \]
   9. Step-by-step derivation

   10. Simplified 57.2%

       \[\leadsto \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{+105}:\\ \;\;\;\;re + -0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.6% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8 \cdot 10^{+105}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8e+105)
   (* re (+ 1.0 (* re (* re -0.16666666666666666))))
   (* re (+ 1.0 (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8e+105) {
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	} else {
		tmp = re * (1.0 + (0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 8d+105) then
        tmp = re * (1.0d0 + (re * (re * (-0.16666666666666666d0))))
    else
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8e+105) {
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	} else {
		tmp = re * (1.0 + (0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8e+105:
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)))
	else:
		tmp = re * (1.0 + (0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8e+105)
		tmp = Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8e+105)
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	else
		tmp = re * (1.0 + (0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8e+105], N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.9999999999999995e105

   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 67.1%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 67.1%

      \[\leadsto \color{blue}{\sin re} \]

   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. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{6} + \color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \left({re}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 45.5%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 45.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot re\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   11. Simplified 43.8%

       \[\leadsto re \cdot \left(1 + re \cdot \color{blue}{\left(re \cdot -0.16666666666666666\right)}\right) \]

   if 7.9999999999999995e105 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right), \color{blue}{\sin re}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right), \sin \color{blue}{re}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{0 - im}\right)\right), \sin re\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right), \sin re\right) \]

      7. cosh-undef N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \cosh im\right)\right), \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right), \sin \color{blue}{re}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

      13. cosh-lowering-cosh.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{1}{2} \cdot {im}^{2}\right)}, \mathsf{sin.f64}\left(re\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{re}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      4. *-lowering-*.f64 68.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   7. Simplified 68.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \color{blue}{re}\right) \]
   9. Step-by-step derivation

   10. Simplified 57.2%

       \[\leadsto \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8 \cdot 10^{+105}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 31.2% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.9 \cdot 10^{+86}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.9e+86) re (* re (* (* re re) -0.16666666666666666))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.9e+86) {
		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 <= 2.9d+86) 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 <= 2.9e+86) {
		tmp = re;
	} else {
		tmp = re * ((re * re) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.9e+86:
		tmp = re
	else:
		tmp = re * ((re * re) * -0.16666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.9e+86)
		tmp = re;
	else
		tmp = Float64(re * Float64(Float64(re * re) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.9e+86)
		tmp = re;
	else
		tmp = re * ((re * re) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.9e+86], re, N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.8999999999999999e86

   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 57.2%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 57.2%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \]
   7. Step-by-step derivation

   8. Simplified 39.1%

      \[\leadsto \color{blue}{re} \]

   if 2.8999999999999999e86 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 58.1%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 58.1%

      \[\leadsto \color{blue}{\sin re} \]

   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. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{6} + \color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \left({re}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 25.5%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 25.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot re\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 23.2%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   11. Simplified 23.2%

       \[\leadsto re \cdot \left(1 + re \cdot \color{blue}{\left(re \cdot -0.16666666666666666\right)}\right) \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   13. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{re}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({re}^{2} \cdot re\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re} \]

       4. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       8. *-lowering-*.f64 23.2%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   14. Simplified 23.2%

       \[\leadsto \color{blue}{re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.9 \cdot 10^{+86}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.1% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* re (+ 1.0 (* re (* re -0.16666666666666666)))))

C

double code(double re, double im) {
	return re * (1.0 + (re * (re * -0.16666666666666666)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (1.0d0 + (re * (re * (-0.16666666666666666d0))))
end function

Java

public static double code(double re, double im) {
	return re * (1.0 + (re * (re * -0.16666666666666666)));
}

Python

def code(re, im):
	return re * (1.0 + (re * (re * -0.16666666666666666)))

Julia

function code(re, im)
	return Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666))))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
end

Wolfram

code[re_, im_] := N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 57.3%

      \[\leadsto \mathsf{sin.f64}\left(re\right) \]

5. Simplified 57.3%

   \[\leadsto \color{blue}{\sin re} \]

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. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{6} + \color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {re}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \left({re}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   18. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   19. *-lowering-*.f64 42.2%

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 42.2%

   \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \]

9. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot re\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

    2. *-lowering-*.f64 40.5%

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

11. Simplified 40.5%

    \[\leadsto re \cdot \left(1 + re \cdot \color{blue}{\left(re \cdot -0.16666666666666666\right)}\right) \]
12. Add Preprocessing

Alternative 21: 27.6% 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 57.3%

      \[\leadsto \mathsf{sin.f64}\left(re\right) \]

5. Simplified 57.3%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re} \]
7. Step-by-step derivation

8. Simplified 33.2%

   \[\leadsto \color{blue}{re} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024156 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))