math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 15.7s

Alternatives: 18

Speedup: 1.5×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. sub0-neg N/A

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

   7. cosh-undef N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. exp-0 N/A

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

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

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

   12. exp-0 N/A

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

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

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

   14. sin-lowering-sin.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 73.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.5e-13)
   (sin re)
   (if (<= im 1.8e+51)
     (* (cosh im) re)
     (*
      (* (sin re) 0.5)
      (+
       2.0
       (*
        im
        (*
         im
         (+
          1.0
          (*
           im
           (*
            im
            (+
             0.08333333333333333
             (* (* im im) 0.002777777777777778))))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = sin(re);
	} else if (im <= 1.8e+51) {
		tmp = cosh(im) * re;
	} else {
		tmp = (sin(re) * 0.5) * (2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 9.5d-13) then
        tmp = sin(re)
    else if (im <= 1.8d+51) then
        tmp = cosh(im) * re
    else
        tmp = (sin(re) * 0.5d0) * (2.0d0 + (im * (im * (1.0d0 + (im * (im * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = Math.sin(re);
	} else if (im <= 1.8e+51) {
		tmp = Math.cosh(im) * re;
	} else {
		tmp = (Math.sin(re) * 0.5) * (2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.5e-13:
		tmp = math.sin(re)
	elif im <= 1.8e+51:
		tmp = math.cosh(im) * re
	else:
		tmp = (math.sin(re) * 0.5) * (2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.5e-13)
		tmp = sin(re);
	elseif (im <= 1.8e+51)
		tmp = Float64(cosh(im) * re);
	else
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(im * Float64(im * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.5e-13)
		tmp = sin(re);
	elseif (im <= 1.8e+51)
		tmp = cosh(im) * re;
	else
		tmp = (sin(re) * 0.5) * (2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.5e-13], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.8e+51], N[(N[Cosh[im], $MachinePrecision] * re), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * N[(im * N[(1.0 + N[(im * N[(im * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 9.49999999999999991e-13

   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 69.7%

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

   5. Simplified 69.7%

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

   if 9.49999999999999991e-13 < im < 1.80000000000000005e51

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 80.0%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 80.0%

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

   9. Applied egg-rr 80.0%

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

   if 1.80000000000000005e51 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot re\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.5e-13)
   (sin re)
   (if (<= im 2.6e+77)
     (* (cosh im) re)
     (*
      (sin re)
      (+ 1.0 (* (* im im) (+ 0.5 (* im (* im 0.041666666666666664)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = sin(re);
	} else if (im <= 2.6e+77) {
		tmp = cosh(im) * re;
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 9.5d-13) then
        tmp = sin(re)
    else if (im <= 2.6d+77) then
        tmp = cosh(im) * re
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = Math.sin(re);
	} else if (im <= 2.6e+77) {
		tmp = Math.cosh(im) * re;
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.5e-13:
		tmp = math.sin(re)
	elif im <= 2.6e+77:
		tmp = math.cosh(im) * re
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.5e-13)
		tmp = sin(re);
	elseif (im <= 2.6e+77)
		tmp = Float64(cosh(im) * re);
	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 (im <= 9.5e-13)
		tmp = sin(re);
	elseif (im <= 2.6e+77)
		tmp = cosh(im) * re;
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.5e-13], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(N[Cosh[im], $MachinePrecision] * re), $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 3 regimes

2. if im < 9.49999999999999991e-13

   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 69.7%

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

   5. Simplified 69.7%

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

   if 9.49999999999999991e-13 < 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. Taylor expanded in re around 0

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

   7. Simplified 83.3%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 83.3%

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

   9. Applied egg-rr 83.3%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot re\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+146}:\\ \;\;\;\;\cosh im \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.5e-13)
   (sin re)
   (if (<= im 2.1e+146)
     (* (cosh im) re)
     (* (* (sin re) 0.5) (+ 2.0 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = sin(re);
	} else if (im <= 2.1e+146) {
		tmp = cosh(im) * re;
	} else {
		tmp = (sin(re) * 0.5) * (2.0 + (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 <= 9.5d-13) then
        tmp = sin(re)
    else if (im <= 2.1d+146) then
        tmp = cosh(im) * re
    else
        tmp = (sin(re) * 0.5d0) * (2.0d0 + (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = Math.sin(re);
	} else if (im <= 2.1e+146) {
		tmp = Math.cosh(im) * re;
	} else {
		tmp = (Math.sin(re) * 0.5) * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.5e-13:
		tmp = math.sin(re)
	elif im <= 2.1e+146:
		tmp = math.cosh(im) * re
	else:
		tmp = (math.sin(re) * 0.5) * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.5e-13)
		tmp = sin(re);
	elseif (im <= 2.1e+146)
		tmp = Float64(cosh(im) * re);
	else
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.5e-13)
		tmp = sin(re);
	elseif (im <= 2.1e+146)
		tmp = cosh(im) * re;
	else
		tmp = (sin(re) * 0.5) * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.5e-13], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.1e+146], N[(N[Cosh[im], $MachinePrecision] * re), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 9.49999999999999991e-13

   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 69.7%

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

   5. Simplified 69.7%

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

   if 9.49999999999999991e-13 < im < 2.1000000000000001e146

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 85.7%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 85.7%

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

   9. Applied egg-rr 85.7%

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

   if 2.1000000000000001e146 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 97.1%

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

   5. Simplified 97.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+146}:\\ \;\;\;\;\cosh im \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.5e-13) (sin re) (* (cosh im) re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = sin(re);
	} else {
		tmp = cosh(im) * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 9.5d-13) then
        tmp = sin(re)
    else
        tmp = cosh(im) * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.5e-13) {
		tmp = Math.sin(re);
	} else {
		tmp = Math.cosh(im) * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.5e-13:
		tmp = math.sin(re)
	else:
		tmp = math.cosh(im) * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.5e-13)
		tmp = sin(re);
	else
		tmp = Float64(cosh(im) * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.5e-13)
		tmp = sin(re);
	else
		tmp = cosh(im) * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.5e-13], N[Sin[re], $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.49999999999999991e-13

   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 69.7%

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

   5. Simplified 69.7%

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

   if 9.49999999999999991e-13 < 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 re around 0

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

   7. Simplified 74.1%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 74.1%

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

   9. Applied egg-rr 74.1%

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

Alternative 6: 67.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ t_1 := im \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.041666666666666664\right)\\ \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \frac{\left(64 - \left(im \cdot im\right) \cdot \left(t\_0 \cdot \left(\left(im \cdot im\right) \cdot t\_0\right)\right)\right) \cdot \frac{1}{2 - im \cdot im}}{16 + t\_0 \cdot \left(t\_0 + 4\right)}\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+99}:\\ \;\;\;\;re \cdot \left(1 + \frac{im \cdot \left(\left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot t\_1\right)}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im (* im im))))
        (t_1 (* im (+ 0.5 (* (* im im) -0.041666666666666664)))))
   (if (<= im 9.5e-13)
     (sin re)
     (if (<= im 3.4e+38)
       (*
        (* re 0.5)
        (/
         (*
          (- 64.0 (* (* im im) (* t_0 (* (* im im) t_0))))
          (/ 1.0 (- 2.0 (* im im))))
         (+ 16.0 (* t_0 (+ t_0 4.0)))))
       (if (<= im 3e+99)
         (*
          re
          (+
           1.0
           (/
            (* im (* (* im (+ 0.5 (* im (* im 0.041666666666666664)))) t_1))
            t_1)))
         (* re (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * (im * im));
	double t_1 = im * (0.5 + ((im * im) * -0.041666666666666664));
	double tmp;
	if (im <= 9.5e-13) {
		tmp = sin(re);
	} else if (im <= 3.4e+38) {
		tmp = (re * 0.5) * (((64.0 - ((im * im) * (t_0 * ((im * im) * t_0)))) * (1.0 / (2.0 - (im * im)))) / (16.0 + (t_0 * (t_0 + 4.0))));
	} else if (im <= 3e+99) {
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_1)) / t_1));
	} else {
		tmp = re * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (im * (im * im))
    t_1 = im * (0.5d0 + ((im * im) * (-0.041666666666666664d0)))
    if (im <= 9.5d-13) then
        tmp = sin(re)
    else if (im <= 3.4d+38) then
        tmp = (re * 0.5d0) * (((64.0d0 - ((im * im) * (t_0 * ((im * im) * t_0)))) * (1.0d0 / (2.0d0 - (im * im)))) / (16.0d0 + (t_0 * (t_0 + 4.0d0))))
    else if (im <= 3d+99) then
        tmp = re * (1.0d0 + ((im * ((im * (0.5d0 + (im * (im * 0.041666666666666664d0)))) * t_1)) / t_1))
    else
        tmp = re * (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * (im * im));
	double t_1 = im * (0.5 + ((im * im) * -0.041666666666666664));
	double tmp;
	if (im <= 9.5e-13) {
		tmp = Math.sin(re);
	} else if (im <= 3.4e+38) {
		tmp = (re * 0.5) * (((64.0 - ((im * im) * (t_0 * ((im * im) * t_0)))) * (1.0 / (2.0 - (im * im)))) / (16.0 + (t_0 * (t_0 + 4.0))));
	} else if (im <= 3e+99) {
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_1)) / t_1));
	} else {
		tmp = re * (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * (im * im))
	t_1 = im * (0.5 + ((im * im) * -0.041666666666666664))
	tmp = 0
	if im <= 9.5e-13:
		tmp = math.sin(re)
	elif im <= 3.4e+38:
		tmp = (re * 0.5) * (((64.0 - ((im * im) * (t_0 * ((im * im) * t_0)))) * (1.0 / (2.0 - (im * im)))) / (16.0 + (t_0 * (t_0 + 4.0))))
	elif im <= 3e+99:
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_1)) / t_1))
	else:
		tmp = re * (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * Float64(im * im)))
	t_1 = Float64(im * Float64(0.5 + Float64(Float64(im * im) * -0.041666666666666664)))
	tmp = 0.0
	if (im <= 9.5e-13)
		tmp = sin(re);
	elseif (im <= 3.4e+38)
		tmp = Float64(Float64(re * 0.5) * Float64(Float64(Float64(64.0 - Float64(Float64(im * im) * Float64(t_0 * Float64(Float64(im * im) * t_0)))) * Float64(1.0 / Float64(2.0 - Float64(im * im)))) / Float64(16.0 + Float64(t_0 * Float64(t_0 + 4.0)))));
	elseif (im <= 3e+99)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664)))) * t_1)) / t_1)));
	else
		tmp = Float64(re * Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * (im * im));
	t_1 = im * (0.5 + ((im * im) * -0.041666666666666664));
	tmp = 0.0;
	if (im <= 9.5e-13)
		tmp = sin(re);
	elseif (im <= 3.4e+38)
		tmp = (re * 0.5) * (((64.0 - ((im * im) * (t_0 * ((im * im) * t_0)))) * (1.0 / (2.0 - (im * im)))) / (16.0 + (t_0 * (t_0 + 4.0))));
	elseif (im <= 3e+99)
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_1)) / t_1));
	else
		tmp = re * (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 9.5e-13], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.4e+38], N[(N[(re * 0.5), $MachinePrecision] * N[(N[(N[(64.0 - N[(N[(im * im), $MachinePrecision] * N[(t$95$0 * N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(2.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(16.0 + N[(t$95$0 * N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3e+99], N[(re * N[(1.0 + N[(N[(im * N[(N[(im * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 9.49999999999999991e-13

   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 69.7%

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

   5. Simplified 69.7%

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

   if 9.49999999999999991e-13 < im < 3.39999999999999996e38

   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 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 2.9%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 31.3%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\frac{\left(64 - \left(im \cdot im\right) \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\right)\right) \cdot \frac{1}{2 - im \cdot im}}{16 + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right) + 4\right)}} \]

   if 3.39999999999999996e38 < im < 3.00000000000000014e99

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 30.2%

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

   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. *-commutative N/A

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

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

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

   8. Simplified 20.6%

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

      1. *-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. flip-+ N/A

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

      4. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 81.8%

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

   if 3.00000000000000014e99 < 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   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. *-commutative N/A

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

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

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

   8. Simplified 70.3%

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

   9. Taylor expanded in im around inf

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

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot {im}^{4} + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot {im}^{\left(3 + 1\right)} + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot \left({im}^{3} \cdot im\right) + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2} \cdot 1}{{im}^{2}} \cdot {im}^{4}\right), re\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2}}{{im}^{2}} \cdot {im}^{4}\right), re\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2} \cdot {im}^{4}}{{im}^{2}}\right), re\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2} \cdot {im}^{4}}{im \cdot im}\right), re\right) \]

       9. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2}}{im} \cdot \frac{{im}^{4}}{im}\right), re\right) \]

       10. metadata-eval N/A

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

       11. pow-plus N/A

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

       12. associate-/l* N/A

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

       13. *-inverses N/A

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

       14. *-rgt-identity N/A

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

       15. associate-*l/ N/A

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

       16. *-commutative N/A

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

       17. *-commutative N/A

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

       18. associate-/l* N/A

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

   11. Simplified 70.3%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \frac{\left(64 - \left(im \cdot im\right) \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\right)\right) \cdot \frac{1}{2 - im \cdot im}}{16 + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right) + 4\right)}\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+99}:\\ \;\;\;\;re \cdot \left(1 + \frac{im \cdot \left(\left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.041666666666666664\right)\right)\right)}{im \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.041666666666666664\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ t_1 := im \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.041666666666666664\right)\\ \mathbf{if}\;im \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \frac{\left(64 - \left(im \cdot im\right) \cdot \left(t\_0 \cdot \left(\left(im \cdot im\right) \cdot t\_0\right)\right)\right) \cdot \frac{1}{2 - im \cdot im}}{16 + t\_0 \cdot \left(t\_0 + 4\right)}\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+99}:\\ \;\;\;\;re \cdot \left(1 + \frac{im \cdot \left(\left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot t\_1\right)}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im (* im im))))
        (t_1 (* im (+ 0.5 (* (* im im) -0.041666666666666664)))))
   (if (<= im 3.4e+38)
     (*
      (* re 0.5)
      (/
       (*
        (- 64.0 (* (* im im) (* t_0 (* (* im im) t_0))))
        (/ 1.0 (- 2.0 (* im im))))
       (+ 16.0 (* t_0 (+ t_0 4.0)))))
     (if (<= im 3e+99)
       (*
        re
        (+
         1.0
         (/
          (* im (* (* im (+ 0.5 (* im (* im 0.041666666666666664)))) t_1))
          t_1)))
       (* re (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664)))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * (im * im));
	double t_1 = im * (0.5 + ((im * im) * -0.041666666666666664));
	double tmp;
	if (im <= 3.4e+38) {
		tmp = (re * 0.5) * (((64.0 - ((im * im) * (t_0 * ((im * im) * t_0)))) * (1.0 / (2.0 - (im * im)))) / (16.0 + (t_0 * (t_0 + 4.0))));
	} else if (im <= 3e+99) {
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_1)) / t_1));
	} else {
		tmp = re * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (im * (im * im))
    t_1 = im * (0.5d0 + ((im * im) * (-0.041666666666666664d0)))
    if (im <= 3.4d+38) then
        tmp = (re * 0.5d0) * (((64.0d0 - ((im * im) * (t_0 * ((im * im) * t_0)))) * (1.0d0 / (2.0d0 - (im * im)))) / (16.0d0 + (t_0 * (t_0 + 4.0d0))))
    else if (im <= 3d+99) then
        tmp = re * (1.0d0 + ((im * ((im * (0.5d0 + (im * (im * 0.041666666666666664d0)))) * t_1)) / t_1))
    else
        tmp = re * (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * (im * im));
	double t_1 = im * (0.5 + ((im * im) * -0.041666666666666664));
	double tmp;
	if (im <= 3.4e+38) {
		tmp = (re * 0.5) * (((64.0 - ((im * im) * (t_0 * ((im * im) * t_0)))) * (1.0 / (2.0 - (im * im)))) / (16.0 + (t_0 * (t_0 + 4.0))));
	} else if (im <= 3e+99) {
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_1)) / t_1));
	} else {
		tmp = re * (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * (im * im))
	t_1 = im * (0.5 + ((im * im) * -0.041666666666666664))
	tmp = 0
	if im <= 3.4e+38:
		tmp = (re * 0.5) * (((64.0 - ((im * im) * (t_0 * ((im * im) * t_0)))) * (1.0 / (2.0 - (im * im)))) / (16.0 + (t_0 * (t_0 + 4.0))))
	elif im <= 3e+99:
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_1)) / t_1))
	else:
		tmp = re * (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * Float64(im * im)))
	t_1 = Float64(im * Float64(0.5 + Float64(Float64(im * im) * -0.041666666666666664)))
	tmp = 0.0
	if (im <= 3.4e+38)
		tmp = Float64(Float64(re * 0.5) * Float64(Float64(Float64(64.0 - Float64(Float64(im * im) * Float64(t_0 * Float64(Float64(im * im) * t_0)))) * Float64(1.0 / Float64(2.0 - Float64(im * im)))) / Float64(16.0 + Float64(t_0 * Float64(t_0 + 4.0)))));
	elseif (im <= 3e+99)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664)))) * t_1)) / t_1)));
	else
		tmp = Float64(re * Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * (im * im));
	t_1 = im * (0.5 + ((im * im) * -0.041666666666666664));
	tmp = 0.0;
	if (im <= 3.4e+38)
		tmp = (re * 0.5) * (((64.0 - ((im * im) * (t_0 * ((im * im) * t_0)))) * (1.0 / (2.0 - (im * im)))) / (16.0 + (t_0 * (t_0 + 4.0))));
	elseif (im <= 3e+99)
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_1)) / t_1));
	else
		tmp = re * (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3.4e+38], N[(N[(re * 0.5), $MachinePrecision] * N[(N[(N[(64.0 - N[(N[(im * im), $MachinePrecision] * N[(t$95$0 * N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(2.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(16.0 + N[(t$95$0 * N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3e+99], N[(re * N[(1.0 + N[(N[(im * N[(N[(im * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.39999999999999996e38

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 50.8%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 39.3%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\frac{\left(64 - \left(im \cdot im\right) \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\right)\right) \cdot \frac{1}{2 - im \cdot im}}{16 + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right) + 4\right)}} \]

   if 3.39999999999999996e38 < im < 3.00000000000000014e99

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 30.2%

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

   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. *-commutative N/A

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

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

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

   8. Simplified 20.6%

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

      1. *-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. flip-+ N/A

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

      4. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 81.8%

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

   if 3.00000000000000014e99 < 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   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. *-commutative N/A

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

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

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

   8. Simplified 70.3%

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

   9. Taylor expanded in im around inf

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

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot {im}^{4} + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot {im}^{\left(3 + 1\right)} + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot \left({im}^{3} \cdot im\right) + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2} \cdot 1}{{im}^{2}} \cdot {im}^{4}\right), re\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2}}{{im}^{2}} \cdot {im}^{4}\right), re\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2} \cdot {im}^{4}}{{im}^{2}}\right), re\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2} \cdot {im}^{4}}{im \cdot im}\right), re\right) \]

       9. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2}}{im} \cdot \frac{{im}^{4}}{im}\right), re\right) \]

       10. metadata-eval N/A

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

       11. pow-plus N/A

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

       12. associate-/l* N/A

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

       13. *-inverses N/A

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

       14. *-rgt-identity N/A

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

       15. associate-*l/ N/A

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

       16. *-commutative N/A

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

       17. *-commutative N/A

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

       18. associate-/l* N/A

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

   11. Simplified 70.3%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \frac{\left(64 - \left(im \cdot im\right) \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\right)\right) \cdot \frac{1}{2 - im \cdot im}}{16 + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right) + 4\right)}\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+99}:\\ \;\;\;\;re \cdot \left(1 + \frac{im \cdot \left(\left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.041666666666666664\right)\right)\right)}{im \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.041666666666666664\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.041666666666666664\right)\\ \mathbf{if}\;im \leq 3 \cdot 10^{+99}:\\ \;\;\;\;re \cdot \left(1 + \frac{im \cdot \left(\left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot t\_0\right)}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 0.5 (* (* im im) -0.041666666666666664)))))
   (if (<= im 3e+99)
     (*
      re
      (+
       1.0
       (/
        (* im (* (* im (+ 0.5 (* im (* im 0.041666666666666664)))) t_0))
        t_0)))
     (* re (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 + ((im * im) * -0.041666666666666664));
	double tmp;
	if (im <= 3e+99) {
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_0)) / t_0));
	} else {
		tmp = re * (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) :: t_0
    real(8) :: tmp
    t_0 = im * (0.5d0 + ((im * im) * (-0.041666666666666664d0)))
    if (im <= 3d+99) then
        tmp = re * (1.0d0 + ((im * ((im * (0.5d0 + (im * (im * 0.041666666666666664d0)))) * t_0)) / t_0))
    else
        tmp = re * (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 + ((im * im) * -0.041666666666666664));
	double tmp;
	if (im <= 3e+99) {
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_0)) / t_0));
	} else {
		tmp = re * (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 + ((im * im) * -0.041666666666666664))
	tmp = 0
	if im <= 3e+99:
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_0)) / t_0))
	else:
		tmp = re * (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 + Float64(Float64(im * im) * -0.041666666666666664)))
	tmp = 0.0
	if (im <= 3e+99)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664)))) * t_0)) / t_0)));
	else
		tmp = Float64(re * Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 + ((im * im) * -0.041666666666666664));
	tmp = 0.0;
	if (im <= 3e+99)
		tmp = re * (1.0 + ((im * ((im * (0.5 + (im * (im * 0.041666666666666664)))) * t_0)) / t_0));
	else
		tmp = re * (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3e+99], N[(re * N[(1.0 + N[(N[(im * N[(N[(im * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 3.00000000000000014e99

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 85.9%

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

   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. *-commutative N/A

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

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

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

   8. Simplified 53.8%

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

      1. *-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. flip-+ N/A

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

      4. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 42.1%

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

   if 3.00000000000000014e99 < 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   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. *-commutative N/A

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

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

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

   8. Simplified 70.3%

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

   9. Taylor expanded in im around inf

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

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot {im}^{4} + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot {im}^{\left(3 + 1\right)} + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot \left({im}^{3} \cdot im\right) + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}\right), re\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2} \cdot 1}{{im}^{2}} \cdot {im}^{4}\right), re\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2}}{{im}^{2}} \cdot {im}^{4}\right), re\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2} \cdot {im}^{4}}{{im}^{2}}\right), re\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2} \cdot {im}^{4}}{im \cdot im}\right), re\right) \]

       9. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {im}^{3}\right) \cdot im + \frac{\frac{1}{2}}{im} \cdot \frac{{im}^{4}}{im}\right), re\right) \]

       10. metadata-eval N/A

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

       11. pow-plus N/A

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

       12. associate-/l* N/A

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

       13. *-inverses N/A

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

       14. *-rgt-identity N/A

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

       15. associate-*l/ N/A

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

       16. *-commutative N/A

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

       17. *-commutative N/A

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

       18. associate-/l* N/A

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

   11. Simplified 70.3%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3 \cdot 10^{+99}:\\ \;\;\;\;re \cdot \left(1 + \frac{im \cdot \left(\left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.041666666666666664\right)\right)\right)}{im \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.041666666666666664\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.1% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.6 \cdot 10^{+58}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 9.8 \cdot 10^{+216}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(im \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.6e+58)
   (*
    re
    (+
     1.0
     (*
      (* im im)
      (+
       0.5
       (*
        im
        (* im (+ 0.041666666666666664 (* im (* im 0.001388888888888889)))))))))
   (if (<= re 9.8e+216)
     (* (+ 2.0 (* im im)) (* re (+ 0.5 (* -0.08333333333333333 (* re re)))))
     (* (* im (+ 0.5 (* im (* im 0.041666666666666664)))) (* im re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.6e+58) {
		tmp = re * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889))))))));
	} else if (re <= 9.8e+216) {
		tmp = (2.0 + (im * im)) * (re * (0.5 + (-0.08333333333333333 * (re * re))));
	} else {
		tmp = (im * (0.5 + (im * (im * 0.041666666666666664)))) * (im * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.6d+58) then
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * (0.041666666666666664d0 + (im * (im * 0.001388888888888889d0))))))))
    else if (re <= 9.8d+216) then
        tmp = (2.0d0 + (im * im)) * (re * (0.5d0 + ((-0.08333333333333333d0) * (re * re))))
    else
        tmp = (im * (0.5d0 + (im * (im * 0.041666666666666664d0)))) * (im * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.6e+58) {
		tmp = re * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889))))))));
	} else if (re <= 9.8e+216) {
		tmp = (2.0 + (im * im)) * (re * (0.5 + (-0.08333333333333333 * (re * re))));
	} else {
		tmp = (im * (0.5 + (im * (im * 0.041666666666666664)))) * (im * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.6e+58:
		tmp = re * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889))))))))
	elif re <= 9.8e+216:
		tmp = (2.0 + (im * im)) * (re * (0.5 + (-0.08333333333333333 * (re * re))))
	else:
		tmp = (im * (0.5 + (im * (im * 0.041666666666666664)))) * (im * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.6e+58)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(im * Float64(im * 0.001388888888888889)))))))));
	elseif (re <= 9.8e+216)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(re * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))));
	else
		tmp = Float64(Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664)))) * Float64(im * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.6e+58)
		tmp = re * (1.0 + ((im * im) * (0.5 + (im * (im * (0.041666666666666664 + (im * (im * 0.001388888888888889))))))));
	elseif (re <= 9.8e+216)
		tmp = (2.0 + (im * im)) * (re * (0.5 + (-0.08333333333333333 * (re * re))));
	else
		tmp = (im * (0.5 + (im * (im * 0.041666666666666664)))) * (im * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.6e+58], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.8e+216], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 5.5999999999999996e58

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

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

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

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 75.9%

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

   8. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 66.4%

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

   10. Simplified 66.4%

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

   if 5.5999999999999996e58 < re < 9.80000000000000027e216

   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 78.3%

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

   5. Simplified 78.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 39.5%

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

   8. Simplified 39.5%

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

   if 9.80000000000000027e216 < 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 89.9%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 32.9%

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

   9. Taylor expanded in im around inf

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

   11. Simplified 33.7%

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

       1. associate-*r* N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

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

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

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

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

       8. *-lowering-*.f64 33.7%

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

   13. Applied egg-rr 33.7%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.6 \cdot 10^{+58}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 9.8 \cdot 10^{+216}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(im \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.8% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\\ \mathbf{if}\;re \leq 5.6 \cdot 10^{+58}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot t\_0\right)\\ \mathbf{elif}\;re \leq 9.8 \cdot 10^{+216}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot t\_0\right) \cdot \left(im \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* im (* im 0.041666666666666664)))))
   (if (<= re 5.6e+58)
     (* re (+ 1.0 (* (* im im) t_0)))
     (if (<= re 9.8e+216)
       (* (+ 2.0 (* im im)) (* re (+ 0.5 (* -0.08333333333333333 (* re re)))))
       (* (* im t_0) (* im re))))))

C

double code(double re, double im) {
	double t_0 = 0.5 + (im * (im * 0.041666666666666664));
	double tmp;
	if (re <= 5.6e+58) {
		tmp = re * (1.0 + ((im * im) * t_0));
	} else if (re <= 9.8e+216) {
		tmp = (2.0 + (im * im)) * (re * (0.5 + (-0.08333333333333333 * (re * re))));
	} else {
		tmp = (im * t_0) * (im * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (im * (im * 0.041666666666666664d0))
    if (re <= 5.6d+58) then
        tmp = re * (1.0d0 + ((im * im) * t_0))
    else if (re <= 9.8d+216) then
        tmp = (2.0d0 + (im * im)) * (re * (0.5d0 + ((-0.08333333333333333d0) * (re * re))))
    else
        tmp = (im * t_0) * (im * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + (im * (im * 0.041666666666666664));
	double tmp;
	if (re <= 5.6e+58) {
		tmp = re * (1.0 + ((im * im) * t_0));
	} else if (re <= 9.8e+216) {
		tmp = (2.0 + (im * im)) * (re * (0.5 + (-0.08333333333333333 * (re * re))));
	} else {
		tmp = (im * t_0) * (im * re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + (im * (im * 0.041666666666666664))
	tmp = 0
	if re <= 5.6e+58:
		tmp = re * (1.0 + ((im * im) * t_0))
	elif re <= 9.8e+216:
		tmp = (2.0 + (im * im)) * (re * (0.5 + (-0.08333333333333333 * (re * re))))
	else:
		tmp = (im * t_0) * (im * re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664)))
	tmp = 0.0
	if (re <= 5.6e+58)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * t_0)));
	elseif (re <= 9.8e+216)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(re * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))));
	else
		tmp = Float64(Float64(im * t_0) * Float64(im * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + (im * (im * 0.041666666666666664));
	tmp = 0.0;
	if (re <= 5.6e+58)
		tmp = re * (1.0 + ((im * im) * t_0));
	elseif (re <= 9.8e+216)
		tmp = (2.0 + (im * im)) * (re * (0.5 + (-0.08333333333333333 * (re * re))));
	else
		tmp = (im * t_0) * (im * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 5.6e+58], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.8e+216], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * t$95$0), $MachinePrecision] * N[(im * re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 5.5999999999999996e58

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 87.1%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 64.0%

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

   if 5.5999999999999996e58 < re < 9.80000000000000027e216

   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 78.3%

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

   5. Simplified 78.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 39.5%

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

   8. Simplified 39.5%

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

   if 9.80000000000000027e216 < 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 89.9%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 32.9%

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

   9. Taylor expanded in im around inf

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

   11. Simplified 33.7%

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

       1. associate-*r* N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

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

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

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

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

       8. *-lowering-*.f64 33.7%

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

   13. Applied egg-rr 33.7%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.6 \cdot 10^{+58}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;re \leq 9.8 \cdot 10^{+216}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(im \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.0% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.1 \cdot 10^{+78}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.1e+78)
   (* re (+ 1.0 (* re (* re -0.16666666666666666))))
   (* re (* im (* im (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.1e+78) {
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	} else {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.1d+78) then
        tmp = re * (1.0d0 + (re * (re * (-0.16666666666666666d0))))
    else
        tmp = re * (im * (im * ((im * im) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.1e+78) {
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	} else {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.1e+78:
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)))
	else:
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.1e+78)
		tmp = Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666))));
	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 <= 3.1e+78)
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	else
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.1e+78], N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $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 < 3.1e78

   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 64.1%

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

   5. Simplified 64.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 39.5%

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

   8. Simplified 39.5%

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

   if 3.1e78 < 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   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. *-commutative N/A

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

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

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

   8. Simplified 70.0%

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

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow3 N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 70.0%

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

   11. Simplified 70.0%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.1 \cdot 10^{+78}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.7% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.35 \cdot 10^{+78}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(im \cdot \left(im \cdot 0.041666666666666664\right)\right) \cdot \left(im \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.35e+78)
   (* re (+ 1.0 (* re (* re -0.16666666666666666))))
   (* im (* (* im (* im 0.041666666666666664)) (* im re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.35e+78) {
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	} else {
		tmp = im * ((im * (im * 0.041666666666666664)) * (im * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.35d+78) then
        tmp = re * (1.0d0 + (re * (re * (-0.16666666666666666d0))))
    else
        tmp = im * ((im * (im * 0.041666666666666664d0)) * (im * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.35e+78) {
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	} else {
		tmp = im * ((im * (im * 0.041666666666666664)) * (im * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.35e+78:
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)))
	else:
		tmp = im * ((im * (im * 0.041666666666666664)) * (im * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.35e+78)
		tmp = Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666))));
	else
		tmp = Float64(im * Float64(Float64(im * Float64(im * 0.041666666666666664)) * Float64(im * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.35e+78)
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	else
		tmp = im * ((im * (im * 0.041666666666666664)) * (im * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.35e+78], N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.34999999999999983e78

   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 64.1%

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

   5. Simplified 64.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 39.5%

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

   8. Simplified 39.5%

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

   if 3.34999999999999983e78 < 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 70.0%

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

   9. Taylor expanded in im around inf

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

   11. Simplified 63.0%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 63.0%

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

   14. Simplified 63.0%

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

Alternative 13: 55.7% accurate, 20.6× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* re (+ 1.0 (* (* im im) (+ 0.5 (* im (* im 0.041666666666666664)))))))

C

double code(double re, double im) {
	return re * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (1.0d0 + ((im * im) * (0.5d0 + (im * (im * 0.041666666666666664d0)))))
end function

Java

public static double code(double re, double im) {
	return re * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
}

Python

def code(re, im):
	return re * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))))

Julia

function code(re, im)
	return Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664))))))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (1.0 + ((im * im) * (0.5 + (im * (im * 0.041666666666666664)))));
end

Wolfram

code[re_, im_] := N[(re * 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. 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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+l+ N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. distribute-lft1-in N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

5. Simplified 87.9%

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

6. Taylor expanded in re around 0

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

8. Simplified 56.1%

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

9. Final simplification 56.1%

   \[\leadsto re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \]
10. Add Preprocessing

Alternative 14: 41.5% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.6 \cdot 10^{+105}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.6e+105)
   (* re (+ 1.0 (* re (* re -0.16666666666666666))))
   (* (* im im) (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.6e+105) {
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	} else {
		tmp = (im * im) * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 9.6d+105) then
        tmp = re * (1.0d0 + (re * (re * (-0.16666666666666666d0))))
    else
        tmp = (im * im) * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.6e+105) {
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	} else {
		tmp = (im * im) * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.6e+105:
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)))
	else:
		tmp = (im * im) * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.6e+105)
		tmp = Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666))));
	else
		tmp = Float64(Float64(im * im) * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.6e+105)
		tmp = re * (1.0 + (re * (re * -0.16666666666666666)));
	else
		tmp = (im * im) * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.6e+105], N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.599999999999999e105

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 63.0%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 39.2%

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

   8. Simplified 39.2%

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

   if 9.599999999999999e105 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 82.1%

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

   5. Simplified 82.1%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 66.8%

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

   9. Taylor expanded in im around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 66.8%

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

   11. Simplified 66.8%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.6 \cdot 10^{+105}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.4% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.4) re (* re (* im (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = re;
	} else {
		tmp = re * (im * (im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.4d0) then
        tmp = re
    else
        tmp = re * (im * (im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = re;
	} else {
		tmp = re * (im * (im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.4:
		tmp = re
	else:
		tmp = re * (im * (im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.4)
		tmp = re;
	else
		tmp = Float64(re * Float64(im * Float64(im * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.4)
		tmp = re;
	else
		tmp = re * (im * (im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.4], re, N[(re * N[(im * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3999999999999999

   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 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 38.3%

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

   if 1.3999999999999999 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 52.1%

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

   5. Simplified 52.1%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 44.1%

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

   9. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. *-inverses N/A

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

       7. associate-/l* N/A

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

       8. unpow2 N/A

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

       9. associate-/l* N/A

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

       10. *-commutative N/A

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

       11. associate-*r/ N/A

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

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

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

       13. associate-*r/ N/A

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

       14. *-commutative N/A

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

       15. unpow2 N/A

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

       16. associate-*r* N/A

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

       17. associate-/l* N/A

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

       18. *-inverses N/A

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

       19. *-rgt-identity N/A

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

       20. *-commutative N/A

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

       21. *-lowering-*.f64 31.3%

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

   11. Simplified 31.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot \left(im \cdot re\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

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

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

       7. *-lowering-*.f64 44.1%

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

   13. Applied egg-rr 44.1%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.4% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.4) re (* (* im im) (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = re;
	} else {
		tmp = (im * im) * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.4d0) then
        tmp = re
    else
        tmp = (im * im) * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = re;
	} else {
		tmp = (im * im) * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.4:
		tmp = re
	else:
		tmp = (im * im) * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.4)
		tmp = re;
	else
		tmp = Float64(Float64(im * im) * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.4)
		tmp = re;
	else
		tmp = (im * im) * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.4], re, N[(N[(im * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3999999999999999

   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 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 38.3%

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

   if 1.3999999999999999 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 52.1%

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

   5. Simplified 52.1%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 44.1%

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

   9. Taylor expanded in im around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 44.1%

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

   11. Simplified 44.1%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 34.4% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.4) re (* 0.5 (* im (* im re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = re;
	} else {
		tmp = 0.5 * (im * (im * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.4d0) then
        tmp = re
    else
        tmp = 0.5d0 * (im * (im * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = re;
	} else {
		tmp = 0.5 * (im * (im * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.4:
		tmp = re
	else:
		tmp = 0.5 * (im * (im * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.4)
		tmp = re;
	else
		tmp = Float64(0.5 * Float64(im * Float64(im * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.4)
		tmp = re;
	else
		tmp = 0.5 * (im * (im * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.4], re, N[(0.5 * N[(im * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3999999999999999

   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 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 38.3%

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

   if 1.3999999999999999 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 52.1%

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

   5. Simplified 52.1%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 44.1%

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

   9. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. *-inverses N/A

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

       7. associate-/l* N/A

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

       8. unpow2 N/A

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

       9. associate-/l* N/A

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

       10. *-commutative N/A

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

       11. associate-*r/ N/A

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

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

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

       13. associate-*r/ N/A

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

       14. *-commutative N/A

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

       15. unpow2 N/A

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

       16. associate-*r* N/A

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

       17. associate-/l* N/A

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

       18. *-inverses N/A

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

       19. *-rgt-identity N/A

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

       20. *-commutative N/A

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

       21. *-lowering-*.f64 31.3%

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

   11. Simplified 31.3%

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

Alternative 18: 26.8% 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 54.5%

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

5. Simplified 54.5%

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

6. Taylor expanded in re around 0

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

8. Simplified 30.2%

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

Reproduce

herbie shell --seed 2024186 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))