math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.6s

Alternatives: 19

Speedup: 1.5×

• 50.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sin re \cdot \frac{1 - {\left(0 - e^{im\_m \cdot 2}\right)}^{-1}}{2 \cdot e^{0 - im\_m}} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (*
  (sin re)
  (/
   (- 1.0 (pow (- 0.0 (exp (* im_m 2.0))) -1.0))
   (* 2.0 (exp (- 0.0 im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return sin(re) * ((1.0 - pow((0.0 - exp((im_m * 2.0))), -1.0)) / (2.0 * exp((0.0 - im_m))));
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = sin(re) * ((1.0d0 - ((0.0d0 - exp((im_m * 2.0d0))) ** (-1.0d0))) / (2.0d0 * exp((0.0d0 - im_m))))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return Math.sin(re) * ((1.0 - Math.pow((0.0 - Math.exp((im_m * 2.0))), -1.0)) / (2.0 * Math.exp((0.0 - im_m))));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return math.sin(re) * ((1.0 - math.pow((0.0 - math.exp((im_m * 2.0))), -1.0)) / (2.0 * math.exp((0.0 - im_m))))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(sin(re) * Float64(Float64(1.0 - (Float64(0.0 - exp(Float64(im_m * 2.0))) ^ -1.0)) / Float64(2.0 * exp(Float64(0.0 - im_m)))))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = sin(re) * ((1.0 - ((0.0 - exp((im_m * 2.0))) ^ -1.0)) / (2.0 * exp((0.0 - im_m))));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[Sin[re], $MachinePrecision] * N[(N[(1.0 - N[Power[N[(0.0 - N[Exp[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Exp[N[(0.0 - im$95$m), $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. Step-by-step derivation

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. *-commutative N/A

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

   7. distribute-rgt-in N/A

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

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

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

   9. exp-diff N/A

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

   10. associate-*l/ N/A

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

   11. exp-0 N/A

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

   12. metadata-eval N/A

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. exp-lowering-exp.f64 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 73.8%

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

7. Final simplification 73.8%

   \[\leadsto \sin re \cdot \frac{1 - {\left(0 - e^{im \cdot 2}\right)}^{-1}}{2 \cdot e^{0 - im}} \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sin re \cdot \cosh im\_m \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* (sin re) (cosh im_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[Sin[re], $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. *-commutative N/A

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

   7. distribute-rgt-in N/A

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

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

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

   9. exp-diff N/A

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

   10. associate-*l/ N/A

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

   11. exp-0 N/A

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

   12. metadata-eval N/A

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. exp-lowering-exp.f64 100.0%

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

3. Simplified 100.0%

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

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

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

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

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

   3. +-commutative N/A

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

   4. div-inv N/A

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

   5. distribute-lft-out N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. div-inv N/A

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

   10. rec-exp N/A

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

   11. metadata-eval N/A

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

   12. cosh-def N/A

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

   13. rem-log-exp N/A

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

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

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

   15. rem-log-exp 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 3: 79.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 0.0125:\\ \;\;\;\;re \cdot \cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im\_m \cdot im\_m\right) \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot \left(0.041666666666666664 + \left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 0.0125)
   (* re (cosh im_m))
   (*
    (sin re)
    (+
     1.0
     (*
      (* im_m im_m)
      (+
       0.5
       (*
        im_m
        (*
         im_m
         (+
          0.041666666666666664
          (* (* im_m im_m) 0.001388888888888889))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 0.0125) {
		tmp = re * cosh(im_m);
	} else {
		tmp = sin(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889)))))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 0.0125d0) then
        tmp = re * cosh(im_m)
    else
        tmp = sin(re) * (1.0d0 + ((im_m * im_m) * (0.5d0 + (im_m * (im_m * (0.041666666666666664d0 + ((im_m * im_m) * 0.001388888888888889d0)))))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 0.0125) {
		tmp = re * Math.cosh(im_m);
	} else {
		tmp = Math.sin(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889)))))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 0.0125:
		tmp = re * math.cosh(im_m)
	else:
		tmp = math.sin(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889)))))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 0.0125)
		tmp = Float64(re * cosh(im_m));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im_m * im_m) * Float64(0.5 + Float64(im_m * Float64(im_m * Float64(0.041666666666666664 + Float64(Float64(im_m * im_m) * 0.001388888888888889))))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 0.0125)
		tmp = re * cosh(im_m);
	else
		tmp = sin(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889)))))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 0.0125], N[(re * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.5 + N[(im$95$m * N[(im$95$m * N[(0.041666666666666664 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.012500000000000001

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

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

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

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

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

      3. +-commutative N/A

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

      4. div-inv N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. rec-exp N/A

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

      11. metadata-eval N/A

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

      12. cosh-def N/A

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

      13. rem-log-exp N/A

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

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

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

      15. rem-log-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 67.9%

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

   if 0.012500000000000001 < re

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 96.9%

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

   7. Simplified 96.9%

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

Alternative 4: 92.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \sin re \cdot \left(1 + \left(im\_m \cdot im\_m\right) \cdot 0.5\right)\\ \mathbf{if}\;im\_m \leq 360:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.35 \cdot 10^{+67}:\\ \;\;\;\;re \cdot \cosh im\_m\\ \mathbf{elif}\;im\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + \left(im\_m \cdot im\_m\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (sin re) (+ 1.0 (* (* im_m im_m) 0.5)))))
   (if (<= im_m 360.0)
     t_0
     (if (<= im_m 1.35e+67)
       (* re (cosh im_m))
       (if (<= im_m 1.35e+154)
         (*
          (+
           1.0
           (* im_m (* im_m (+ 0.5 (* (* im_m im_m) 0.041666666666666664)))))
          (* re (+ 1.0 (* -0.16666666666666666 (* re re)))))
         t_0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = sin(re) * (1.0 + ((im_m * im_m) * 0.5));
	double tmp;
	if (im_m <= 360.0) {
		tmp = t_0;
	} else if (im_m <= 1.35e+67) {
		tmp = re * cosh(im_m);
	} else if (im_m <= 1.35e+154) {
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + (-0.16666666666666666 * (re * re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(re) * (1.0d0 + ((im_m * im_m) * 0.5d0))
    if (im_m <= 360.0d0) then
        tmp = t_0
    else if (im_m <= 1.35d+67) then
        tmp = re * cosh(im_m)
    else if (im_m <= 1.35d+154) then
        tmp = (1.0d0 + (im_m * (im_m * (0.5d0 + ((im_m * im_m) * 0.041666666666666664d0))))) * (re * (1.0d0 + ((-0.16666666666666666d0) * (re * re))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double t_0 = Math.sin(re) * (1.0 + ((im_m * im_m) * 0.5));
	double tmp;
	if (im_m <= 360.0) {
		tmp = t_0;
	} else if (im_m <= 1.35e+67) {
		tmp = re * Math.cosh(im_m);
	} else if (im_m <= 1.35e+154) {
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + (-0.16666666666666666 * (re * re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = math.sin(re) * (1.0 + ((im_m * im_m) * 0.5))
	tmp = 0
	if im_m <= 360.0:
		tmp = t_0
	elif im_m <= 1.35e+67:
		tmp = re * math.cosh(im_m)
	elif im_m <= 1.35e+154:
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + (-0.16666666666666666 * (re * re))))
	else:
		tmp = t_0
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(sin(re) * Float64(1.0 + Float64(Float64(im_m * im_m) * 0.5)))
	tmp = 0.0
	if (im_m <= 360.0)
		tmp = t_0;
	elseif (im_m <= 1.35e+67)
		tmp = Float64(re * cosh(im_m));
	elseif (im_m <= 1.35e+154)
		tmp = Float64(Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(Float64(im_m * im_m) * 0.041666666666666664))))) * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = sin(re) * (1.0 + ((im_m * im_m) * 0.5));
	tmp = 0.0;
	if (im_m <= 360.0)
		tmp = t_0;
	elseif (im_m <= 1.35e+67)
		tmp = re * cosh(im_m);
	elseif (im_m <= 1.35e+154)
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + (-0.16666666666666666 * (re * re))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 360.0], t$95$0, If[LessEqual[im$95$m, 1.35e+67], N[(re * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.35e+154], N[(N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 360 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 85.5%

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

   7. Simplified 85.5%

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

   if 360 < im < 1.35e67

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

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

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

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

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

      3. +-commutative N/A

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

      4. div-inv N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. rec-exp N/A

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

      11. metadata-eval N/A

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

      12. cosh-def N/A

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

      13. rem-log-exp N/A

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

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

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

      15. rem-log-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 87.5%

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

   if 1.35e67 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

   7. Simplified 89.0%

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

   8. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 84.3%

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

   10. Simplified 84.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 360:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+67}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 0.0126:\\ \;\;\;\;re \cdot \cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + \left(im\_m \cdot im\_m\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 0.0126)
   (* re (cosh im_m))
   (*
    (sin re)
    (+ 1.0 (* im_m (* im_m (+ 0.5 (* (* im_m im_m) 0.041666666666666664))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 0.0126) {
		tmp = re * cosh(im_m);
	} else {
		tmp = sin(re) * (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664)))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 0.0126d0) then
        tmp = re * cosh(im_m)
    else
        tmp = sin(re) * (1.0d0 + (im_m * (im_m * (0.5d0 + ((im_m * im_m) * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 0.0126) {
		tmp = re * Math.cosh(im_m);
	} else {
		tmp = Math.sin(re) * (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664)))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 0.0126:
		tmp = re * math.cosh(im_m)
	else:
		tmp = math.sin(re) * (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664)))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 0.0126)
		tmp = Float64(re * cosh(im_m));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(Float64(im_m * im_m) * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 0.0126)
		tmp = re * cosh(im_m);
	else
		tmp = sin(re) * (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 0.0126], N[(re * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.0126

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

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

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

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

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

      3. +-commutative N/A

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

      4. div-inv N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. rec-exp N/A

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

      11. metadata-eval N/A

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

      12. cosh-def N/A

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

      13. rem-log-exp N/A

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

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

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

      15. rem-log-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 67.9%

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

   if 0.0126 < re

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

   7. Simplified 95.4%

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

Alternative 6: 85.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 0.000112:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im\_m \leq 1.35 \cdot 10^{+67}:\\ \;\;\;\;re \cdot \cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + \left(im\_m \cdot im\_m\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 0.000112)
   (sin re)
   (if (<= im_m 1.35e+67)
     (* re (cosh im_m))
     (*
      (+ 1.0 (* im_m (* im_m (+ 0.5 (* (* im_m im_m) 0.041666666666666664)))))
      (* re (+ 1.0 (* -0.16666666666666666 (* re re))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 0.000112) {
		tmp = sin(re);
	} else if (im_m <= 1.35e+67) {
		tmp = re * cosh(im_m);
	} else {
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + (-0.16666666666666666 * (re * re))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.000112d0) then
        tmp = sin(re)
    else if (im_m <= 1.35d+67) then
        tmp = re * cosh(im_m)
    else
        tmp = (1.0d0 + (im_m * (im_m * (0.5d0 + ((im_m * im_m) * 0.041666666666666664d0))))) * (re * (1.0d0 + ((-0.16666666666666666d0) * (re * re))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 0.000112) {
		tmp = Math.sin(re);
	} else if (im_m <= 1.35e+67) {
		tmp = re * Math.cosh(im_m);
	} else {
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + (-0.16666666666666666 * (re * re))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 0.000112:
		tmp = math.sin(re)
	elif im_m <= 1.35e+67:
		tmp = re * math.cosh(im_m)
	else:
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + (-0.16666666666666666 * (re * re))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 0.000112)
		tmp = sin(re);
	elseif (im_m <= 1.35e+67)
		tmp = Float64(re * cosh(im_m));
	else
		tmp = Float64(Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(Float64(im_m * im_m) * 0.041666666666666664))))) * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 0.000112)
		tmp = sin(re);
	elseif (im_m <= 1.35e+67)
		tmp = re * cosh(im_m);
	else
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + (-0.16666666666666666 * (re * re))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 0.000112], N[Sin[re], $MachinePrecision], If[LessEqual[im$95$m, 1.35e+67], N[(re * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.11999999999999998e-4

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 65.1%

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

   7. Simplified 65.1%

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

   if 1.11999999999999998e-4 < im < 1.35e67

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

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

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

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

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

      3. +-commutative N/A

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

      4. div-inv N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. rec-exp N/A

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

      11. metadata-eval N/A

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

      12. cosh-def N/A

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

      13. rem-log-exp N/A

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

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

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

      15. rem-log-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 77.8%

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

   if 1.35e67 < im

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

   7. Simplified 95.2%

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

   8. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 79.1%

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

   10. Simplified 79.1%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.000112:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+67}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 0.000105:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + \left(im\_m \cdot im\_m\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 0.000105)
   (sin re)
   (*
    (+ 1.0 (* im_m (* im_m (+ 0.5 (* (* im_m im_m) 0.041666666666666664)))))
    (*
     re
     (+
      1.0
      (*
       (* re re)
       (+
        -0.16666666666666666
        (*
         re
         (*
          re
          (+
           0.008333333333333333
           (* (* re re) -0.0001984126984126984)))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 0.000105) {
		tmp = sin(re);
	} else {
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984))))))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.000105d0) then
        tmp = sin(re)
    else
        tmp = (1.0d0 + (im_m * (im_m * (0.5d0 + ((im_m * im_m) * 0.041666666666666664d0))))) * (re * (1.0d0 + ((re * re) * ((-0.16666666666666666d0) + (re * (re * (0.008333333333333333d0 + ((re * re) * (-0.0001984126984126984d0)))))))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 0.000105) {
		tmp = Math.sin(re);
	} else {
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984))))))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 0.000105:
		tmp = math.sin(re)
	else:
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984))))))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 0.000105)
		tmp = sin(re);
	else
		tmp = Float64(Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(Float64(im_m * im_m) * 0.041666666666666664))))) * Float64(re * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.16666666666666666 + Float64(re * Float64(re * Float64(0.008333333333333333 + Float64(Float64(re * re) * -0.0001984126984126984)))))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 0.000105)
		tmp = sin(re);
	else
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984))))))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 0.000105], N[Sin[re], $MachinePrecision], N[(N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.16666666666666666 + N[(re * N[(re * N[(0.008333333333333333 + N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.05e-4

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 65.1%

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

   7. Simplified 65.1%

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

   if 1.05e-4 < im

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

   7. Simplified 82.7%

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

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{6} + {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(re \cdot \left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \left({re}^{2} \cdot \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 70.2%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.000105:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + \left(im\_m \cdot im\_m\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (*
  (+ 1.0 (* im_m (* im_m (+ 0.5 (* (* im_m im_m) 0.041666666666666664)))))
  (*
   re
   (+
    1.0
    (*
     (* re re)
     (+
      -0.16666666666666666
      (*
       re
       (*
        re
        (+ 0.008333333333333333 (* (* re re) -0.0001984126984126984))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984))))))));
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = (1.0d0 + (im_m * (im_m * (0.5d0 + ((im_m * im_m) * 0.041666666666666664d0))))) * (re * (1.0d0 + ((re * re) * ((-0.16666666666666666d0) + (re * (re * (0.008333333333333333d0 + ((re * re) * (-0.0001984126984126984d0)))))))))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984))))))));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984))))))))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(Float64(im_m * im_m) * 0.041666666666666664))))) * Float64(re * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.16666666666666666 + Float64(re * Float64(re * Float64(0.008333333333333333 + Float64(Float64(re * re) * -0.0001984126984126984)))))))))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (re * (1.0 + ((re * re) * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984))))))));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.16666666666666666 + N[(re * N[(re * N[(0.008333333333333333 + N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. *-commutative N/A

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

   7. distribute-rgt-in N/A

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

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

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

   9. exp-diff N/A

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

   10. associate-*l/ N/A

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

   11. exp-0 N/A

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

   12. metadata-eval N/A

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. exp-lowering-exp.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0

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

   1. distribute-lft-in N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt1-in N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. associate-+l+ N/A

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

7. Simplified 91.8%

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

8. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{6} + {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(re \cdot \left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \left({re}^{2} \cdot \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 61.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

10. Simplified 61.9%

    \[\leadsto \color{blue}{\left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \]

11. Final simplification 61.9%

    \[\leadsto \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(re \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 9: 58.1% accurate, 9.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;re \cdot \left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im\_m \cdot im\_m\right) \cdot 0.5\right) \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 2.5e+34)
   (*
    re
    (+
     1.0
     (*
      im_m
      (*
       im_m
       (+ 0.5 (* im_m (* im_m (* (* im_m im_m) 0.001388888888888889))))))))
   (*
    (+ 1.0 (* (* im_m im_m) 0.5))
    (*
     re
     (+
      1.0
      (*
       re
       (*
        re
        (+
         -0.16666666666666666
         (*
          re
          (*
           re
           (+
            0.008333333333333333
            (* (* re re) -0.0001984126984126984))))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 2.5e+34) {
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984)))))))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 2.5d+34) then
        tmp = re * (1.0d0 + (im_m * (im_m * (0.5d0 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889d0)))))))
    else
        tmp = (1.0d0 + ((im_m * im_m) * 0.5d0)) * (re * (1.0d0 + (re * (re * ((-0.16666666666666666d0) + (re * (re * (0.008333333333333333d0 + ((re * re) * (-0.0001984126984126984d0))))))))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 2.5e+34) {
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984)))))))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 2.5e+34:
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))))
	else:
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984)))))))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 2.5e+34)
		tmp = Float64(re * Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(im_m * Float64(im_m * Float64(Float64(im_m * im_m) * 0.001388888888888889))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(im_m * im_m) * 0.5)) * Float64(re * Float64(1.0 + Float64(re * Float64(re * Float64(-0.16666666666666666 + Float64(re * Float64(re * Float64(0.008333333333333333 + Float64(Float64(re * re) * -0.0001984126984126984))))))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 2.5e+34)
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))));
	else
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * (-0.16666666666666666 + (re * (re * (0.008333333333333333 + ((re * re) * -0.0001984126984126984)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 2.5e+34], N[(re * N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(re * N[(re * N[(-0.16666666666666666 + N[(re * N[(re * N[(0.008333333333333333 + N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.4999999999999999e34

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+l+ N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   7. Simplified 93.3%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{re} \]

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

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

   10. Simplified 62.0%

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

   11. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 62.0%

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

   13. Simplified 62.0%

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

   if 2.4999999999999999e34 < re

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

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

      11. exp-0 N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

   7. Simplified 94.7%

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

   8. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. sin-lowering-sin.f64 77.5%

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

   10. Simplified 77.5%

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

   11. Taylor expanded in re around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

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

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

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

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

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

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

       16. *-commutative N/A

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

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

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

       18. unpow2 N/A

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

       19. *-lowering-*.f64 34.2%

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

   13. Simplified 34.2%

       \[\leadsto \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot 0.5\right) \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot \left(-0.16666666666666666 + re \cdot \left(re \cdot \left(0.008333333333333333 + \left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;re \cdot \left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + \left(im\_m \cdot im\_m\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 2.5e+34)
   (*
    re
    (+
     1.0
     (*
      im_m
      (*
       im_m
       (+ 0.5 (* im_m (* im_m (* (* im_m im_m) 0.001388888888888889))))))))
   (*
    (+ 1.0 (* im_m (* im_m (+ 0.5 (* (* im_m im_m) 0.041666666666666664)))))
    (* -0.16666666666666666 (* re (* re re))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 2.5e+34) {
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (-0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 2.5d+34) then
        tmp = re * (1.0d0 + (im_m * (im_m * (0.5d0 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889d0)))))))
    else
        tmp = (1.0d0 + (im_m * (im_m * (0.5d0 + ((im_m * im_m) * 0.041666666666666664d0))))) * ((-0.16666666666666666d0) * (re * (re * re)))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 2.5e+34) {
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (-0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 2.5e+34:
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))))
	else:
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (-0.16666666666666666 * (re * (re * re)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 2.5e+34)
		tmp = Float64(re * Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(im_m * Float64(im_m * Float64(Float64(im_m * im_m) * 0.001388888888888889))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(Float64(im_m * im_m) * 0.041666666666666664))))) * Float64(-0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 2.5e+34)
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))));
	else
		tmp = (1.0 + (im_m * (im_m * (0.5 + ((im_m * im_m) * 0.041666666666666664))))) * (-0.16666666666666666 * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 2.5e+34], N[(re * N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.4999999999999999e34

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. exp-diff N/A

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

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\sin re, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2} \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2} \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \color{blue}{\frac{1}{2} \cdot \sin re}\right)\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) + \color{blue}{\frac{1}{2}} \cdot \sin re\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot \sin re\right) + \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \sin re\right)} + \frac{1}{2} \cdot \sin re\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)} + \frac{1}{2} \cdot \sin re\right)\right)\right)\right) \]

   7. Simplified 93.3%

      \[\leadsto \color{blue}{\sin re + \left(im \cdot im\right) \cdot \left(\sin re \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{re} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right), \color{blue}{re}\right) \]

   10. Simplified 62.0%

       \[\leadsto \color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot re} \]

   11. Taylor expanded in im around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{720} \cdot {im}^{3}\right)}\right)\right)\right)\right)\right), re\right) \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot im\right)\right)\right)\right)\right)\right)\right), re\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\frac{1}{720} \cdot \left({im}^{2} \cdot im\right)\right)\right)\right)\right)\right)\right), re\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot im\right)\right)\right)\right)\right)\right), re\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right)\right), re\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right)\right), re\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), re\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), re\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), re\right) \]

       9. *-lowering-*.f64 62.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), re\right) \]

   13. Simplified 62.0%

       \[\leadsto \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)}\right)\right)\right) \cdot re \]

   if 2.4999999999999999e34 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin \color{blue}{re} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right)}\right) \]

      10. associate-+l+ N/A

          \[\leadsto \sin re \cdot \left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \color{blue}{1}\right) \]

   7. Simplified 94.7%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left({re}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 34.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 34.2%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{3}\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left({re}^{3}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 34.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 34.2%

       \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;re \cdot \left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im\_m \cdot im\_m\right) \cdot 0.5\right) \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 2.5e+34)
   (*
    re
    (+
     1.0
     (*
      im_m
      (*
       im_m
       (+ 0.5 (* im_m (* im_m (* (* im_m im_m) 0.001388888888888889))))))))
   (*
    (+ 1.0 (* (* im_m im_m) 0.5))
    (* re (+ 1.0 (* re (* re -0.16666666666666666)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 2.5e+34) {
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 2.5d+34) then
        tmp = re * (1.0d0 + (im_m * (im_m * (0.5d0 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889d0)))))))
    else
        tmp = (1.0d0 + ((im_m * im_m) * 0.5d0)) * (re * (1.0d0 + (re * (re * (-0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 2.5e+34) {
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 2.5e+34:
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))))
	else:
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * -0.16666666666666666))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 2.5e+34)
		tmp = Float64(re * Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(im_m * Float64(im_m * Float64(Float64(im_m * im_m) * 0.001388888888888889))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(im_m * im_m) * 0.5)) * Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 2.5e+34)
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * ((im_m * im_m) * 0.001388888888888889)))))));
	else
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 2.5e+34], N[(re * N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.4999999999999999e34

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\sin re, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2} \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2} \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \color{blue}{\frac{1}{2} \cdot \sin re}\right)\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) + \color{blue}{\frac{1}{2}} \cdot \sin re\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot \sin re\right) + \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \sin re\right)} + \frac{1}{2} \cdot \sin re\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)} + \frac{1}{2} \cdot \sin re\right)\right)\right)\right) \]

   7. Simplified 93.3%

      \[\leadsto \color{blue}{\sin re + \left(im \cdot im\right) \cdot \left(\sin re \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{re} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right), \color{blue}{re}\right) \]

   10. Simplified 62.0%

       \[\leadsto \color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot re} \]

   11. Taylor expanded in im around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{720} \cdot {im}^{3}\right)}\right)\right)\right)\right)\right), re\right) \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot im\right)\right)\right)\right)\right)\right)\right), re\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\frac{1}{720} \cdot \left({im}^{2} \cdot im\right)\right)\right)\right)\right)\right)\right), re\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot im\right)\right)\right)\right)\right)\right), re\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right)\right), re\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right)\right), re\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), re\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), re\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), re\right) \]

       9. *-lowering-*.f64 62.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), re\right) \]

   13. Simplified 62.0%

       \[\leadsto \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)}\right)\right)\right) \cdot re \]

   if 2.4999999999999999e34 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin \color{blue}{re} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right)}\right) \]

      10. associate-+l+ N/A

          \[\leadsto \sin re \cdot \left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \color{blue}{1}\right) \]

   7. Simplified 94.7%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left({re}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 34.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 34.2%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right) + re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) + \color{blue}{re} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \]

       2. distribute-lft1-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       3. +-commutative N/A

          \[\leadsto \left(1 + \frac{1}{2} \cdot {im}^{2}\right) \cdot \left(\color{blue}{re} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right), \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \left(\color{blue}{re} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 34.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

   13. Simplified 34.2%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot 0.5\right) \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;re \cdot \left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im\_m \cdot im\_m\right) \cdot 0.5\right) \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 2.5e+34)
   (*
    re
    (+ 1.0 (* im_m (* im_m (+ 0.5 (* im_m (* im_m 0.041666666666666664)))))))
   (*
    (+ 1.0 (* (* im_m im_m) 0.5))
    (* re (+ 1.0 (* re (* re -0.16666666666666666)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 2.5e+34) {
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * 0.041666666666666664))))));
	} else {
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 2.5d+34) then
        tmp = re * (1.0d0 + (im_m * (im_m * (0.5d0 + (im_m * (im_m * 0.041666666666666664d0))))))
    else
        tmp = (1.0d0 + ((im_m * im_m) * 0.5d0)) * (re * (1.0d0 + (re * (re * (-0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 2.5e+34) {
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * 0.041666666666666664))))));
	} else {
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 2.5e+34:
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * 0.041666666666666664))))))
	else:
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * -0.16666666666666666))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 2.5e+34)
		tmp = Float64(re * Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(im_m * Float64(im_m * 0.041666666666666664)))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(im_m * im_m) * 0.5)) * Float64(re * Float64(1.0 + Float64(re * Float64(re * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 2.5e+34)
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * 0.041666666666666664))))));
	else
		tmp = (1.0 + ((im_m * im_m) * 0.5)) * (re * (1.0 + (re * (re * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 2.5e+34], N[(re * N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(im$95$m * N[(im$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.4999999999999999e34

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin \color{blue}{re} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right)}\right) \]

      10. associate-+l+ N/A

          \[\leadsto \sin re \cdot \left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \color{blue}{1}\right) \]

   7. Simplified 91.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   9. 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

         \[\leadsto \mathsf{*.f64}\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right), \color{blue}{re}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), re\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), re\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(im \cdot \left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right)\right), re\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      12. *-lowering-*.f64 61.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

   10. Simplified 61.0%

       \[\leadsto \color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \cdot re} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), re\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot \frac{1}{24}\right) \cdot im\right)\right)\right)\right)\right), re\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot \frac{1}{24}\right), im\right)\right)\right)\right)\right), re\right) \]

       4. *-lowering-*.f64 61.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \frac{1}{24}\right), im\right)\right)\right)\right)\right), re\right) \]

   12. Applied egg-rr 61.0%

       \[\leadsto \left(1 + im \cdot \left(im \cdot \left(0.5 + \color{blue}{\left(im \cdot 0.041666666666666664\right) \cdot im}\right)\right)\right) \cdot re \]

   if 2.4999999999999999e34 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin \color{blue}{re} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right)}\right) \]

      10. associate-+l+ N/A

          \[\leadsto \sin re \cdot \left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \color{blue}{1}\right) \]

   7. Simplified 94.7%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {re}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left({re}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 34.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 34.2%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right) + re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) + \color{blue}{re} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \]

       2. distribute-lft1-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       3. +-commutative N/A

          \[\leadsto \left(1 + \frac{1}{2} \cdot {im}^{2}\right) \cdot \left(\color{blue}{re} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right), \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \left(\color{blue}{re} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\left(re \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 34.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

   13. Simplified 34.2%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot 0.5\right) \cdot \left(re \cdot \left(1 + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.6% accurate, 15.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 6.3 \cdot 10^{+156}:\\ \;\;\;\;re \cdot \left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 6.3e+156)
   (*
    re
    (+ 1.0 (* im_m (* im_m (+ 0.5 (* im_m (* im_m 0.041666666666666664)))))))
   (* re (* -0.16666666666666666 (* re re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 6.3e+156) {
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * 0.041666666666666664))))));
	} else {
		tmp = re * (-0.16666666666666666 * (re * re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 6.3d+156) then
        tmp = re * (1.0d0 + (im_m * (im_m * (0.5d0 + (im_m * (im_m * 0.041666666666666664d0))))))
    else
        tmp = re * ((-0.16666666666666666d0) * (re * re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 6.3e+156) {
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * 0.041666666666666664))))));
	} else {
		tmp = re * (-0.16666666666666666 * (re * re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 6.3e+156:
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * 0.041666666666666664))))))
	else:
		tmp = re * (-0.16666666666666666 * (re * re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 6.3e+156)
		tmp = Float64(re * Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(im_m * Float64(im_m * 0.041666666666666664)))))));
	else
		tmp = Float64(re * Float64(-0.16666666666666666 * Float64(re * re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 6.3e+156)
		tmp = re * (1.0 + (im_m * (im_m * (0.5 + (im_m * (im_m * 0.041666666666666664))))));
	else
		tmp = re * (-0.16666666666666666 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 6.3e+156], N[(re * N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(im$95$m * N[(im$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.29999999999999982e156

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin \color{blue}{re} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right)}\right) \]

      10. associate-+l+ N/A

          \[\leadsto \sin re \cdot \left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \color{blue}{1}\right) \]

   7. Simplified 91.2%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   9. 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

         \[\leadsto \mathsf{*.f64}\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right), \color{blue}{re}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), re\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), re\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(im \cdot \left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right)\right), re\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      12. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

   10. Simplified 58.1%

       \[\leadsto \color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \cdot re} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), re\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot \frac{1}{24}\right) \cdot im\right)\right)\right)\right)\right), re\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot \frac{1}{24}\right), im\right)\right)\right)\right)\right), re\right) \]

       4. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \frac{1}{24}\right), im\right)\right)\right)\right)\right), re\right) \]

   12. Applied egg-rr 58.1%

       \[\leadsto \left(1 + im \cdot \left(im \cdot \left(0.5 + \color{blue}{\left(im \cdot 0.041666666666666664\right) \cdot im}\right)\right)\right) \cdot re \]

   if 6.29999999999999982e156 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 58.5%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   7. Simplified 58.5%

      \[\leadsto \color{blue}{\sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 29.7%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   10. Simplified 29.7%

       \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{re}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({re}^{2} \cdot re\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re} \]

       4. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       8. *-lowering-*.f64 29.7%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   13. Simplified 29.7%

       \[\leadsto \color{blue}{re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.3 \cdot 10^{+156}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.4% accurate, 17.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 6.3 \cdot 10^{+156}:\\ \;\;\;\;re \cdot \left(1 + im\_m \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 6.3e+156)
   (* re (+ 1.0 (* im_m (* im_m (* (* im_m im_m) 0.041666666666666664)))))
   (* re (* -0.16666666666666666 (* re re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 6.3e+156) {
		tmp = re * (1.0 + (im_m * (im_m * ((im_m * im_m) * 0.041666666666666664))));
	} else {
		tmp = re * (-0.16666666666666666 * (re * re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 6.3d+156) then
        tmp = re * (1.0d0 + (im_m * (im_m * ((im_m * im_m) * 0.041666666666666664d0))))
    else
        tmp = re * ((-0.16666666666666666d0) * (re * re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 6.3e+156) {
		tmp = re * (1.0 + (im_m * (im_m * ((im_m * im_m) * 0.041666666666666664))));
	} else {
		tmp = re * (-0.16666666666666666 * (re * re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 6.3e+156:
		tmp = re * (1.0 + (im_m * (im_m * ((im_m * im_m) * 0.041666666666666664))))
	else:
		tmp = re * (-0.16666666666666666 * (re * re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 6.3e+156)
		tmp = Float64(re * Float64(1.0 + Float64(im_m * Float64(im_m * Float64(Float64(im_m * im_m) * 0.041666666666666664)))));
	else
		tmp = Float64(re * Float64(-0.16666666666666666 * Float64(re * re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 6.3e+156)
		tmp = re * (1.0 + (im_m * (im_m * ((im_m * im_m) * 0.041666666666666664))));
	else
		tmp = re * (-0.16666666666666666 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 6.3e+156], N[(re * N[(1.0 + N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.29999999999999982e156

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin \color{blue}{re} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right)}\right) \]

      10. associate-+l+ N/A

          \[\leadsto \sin re \cdot \left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \color{blue}{1}\right) \]

   7. Simplified 91.2%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   9. 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

         \[\leadsto \mathsf{*.f64}\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right), \color{blue}{re}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), re\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), re\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(im \cdot \left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right)\right), re\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      12. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

   10. Simplified 58.1%

       \[\leadsto \color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \cdot re} \]

   11. Taylor expanded in im around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{3}\right)}\right)\right), re\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left({im}^{3} \cdot \frac{1}{24}\right)\right)\right), re\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{24}\right)\right)\right), re\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(\left(im \cdot {im}^{2}\right) \cdot \frac{1}{24}\right)\right)\right), re\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), re\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), re\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right), re\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right), re\right) \]

       10. *-lowering-*.f64 57.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right), re\right) \]

   13. Simplified 57.7%

       \[\leadsto \left(1 + im \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)}\right) \cdot re \]

   if 6.29999999999999982e156 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 58.5%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   7. Simplified 58.5%

      \[\leadsto \color{blue}{\sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 29.7%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   10. Simplified 29.7%

       \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{re}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({re}^{2} \cdot re\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re} \]

       4. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       8. *-lowering-*.f64 29.7%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   13. Simplified 29.7%

       \[\leadsto \color{blue}{re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.3 \cdot 10^{+156}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.9% accurate, 22.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 6.3 \cdot 10^{+156}:\\ \;\;\;\;re + \left(im\_m \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 6.3e+156)
   (+ re (* (* im_m im_m) (* re 0.5)))
   (* re (* -0.16666666666666666 (* re re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 6.3e+156) {
		tmp = re + ((im_m * im_m) * (re * 0.5));
	} else {
		tmp = re * (-0.16666666666666666 * (re * re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 6.3d+156) then
        tmp = re + ((im_m * im_m) * (re * 0.5d0))
    else
        tmp = re * ((-0.16666666666666666d0) * (re * re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 6.3e+156) {
		tmp = re + ((im_m * im_m) * (re * 0.5));
	} else {
		tmp = re * (-0.16666666666666666 * (re * re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 6.3e+156:
		tmp = re + ((im_m * im_m) * (re * 0.5))
	else:
		tmp = re * (-0.16666666666666666 * (re * re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 6.3e+156)
		tmp = Float64(re + Float64(Float64(im_m * im_m) * Float64(re * 0.5)));
	else
		tmp = Float64(re * Float64(-0.16666666666666666 * Float64(re * re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 6.3e+156)
		tmp = re + ((im_m * im_m) * (re * 0.5));
	else
		tmp = re * (-0.16666666666666666 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 6.3e+156], N[(re + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.29999999999999982e156

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin \color{blue}{re} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right)}\right) \]

      10. associate-+l+ N/A

          \[\leadsto \sin re \cdot \left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \color{blue}{1}\right) \]

   7. Simplified 91.2%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   9. 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

         \[\leadsto \mathsf{*.f64}\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right), \color{blue}{re}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), re\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), re\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(im \cdot \left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right)\right), re\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      12. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

   10. Simplified 58.1%

       \[\leadsto \color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \cdot re} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(im \cdot \left(im \cdot \left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right) + 1\right) \cdot re \]

       2. distribute-lft1-in N/A

          \[\leadsto \left(im \cdot \left(im \cdot \left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot re + \color{blue}{re} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\left(im \cdot \left(im \cdot \left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot re\right), \color{blue}{re}\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right) \cdot re\right), re\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\left(im \cdot im\right) \cdot \left(\left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right) \cdot re\right)\right), re\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left(\left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right) \cdot re\right)\right), re\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right) \cdot re\right)\right), re\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left(\frac{1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right), re\right)\right), re\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right), re\right)\right), re\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right), re\right)\right), re\right) \]

       11. *-lowering-*.f64 53.6%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right), re\right)\right), re\right) \]

   12. Applied egg-rr 53.6%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right) \cdot re\right) + re} \]

   13. Taylor expanded in im around 0

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right), re\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(re \cdot \frac{1}{2}\right)\right), re\right) \]

       2. *-lowering-*.f64 46.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right), re\right) \]

   15. Simplified 46.4%

       \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(re \cdot 0.5\right)} + re \]

   if 6.29999999999999982e156 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 58.5%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   7. Simplified 58.5%

      \[\leadsto \color{blue}{\sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 29.7%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   10. Simplified 29.7%

       \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{re}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({re}^{2} \cdot re\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re} \]

       4. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       8. *-lowering-*.f64 29.7%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   13. Simplified 29.7%

       \[\leadsto \color{blue}{re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.3 \cdot 10^{+156}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.9% accurate, 22.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 6.3 \cdot 10^{+156}:\\ \;\;\;\;re \cdot \left(1 + im\_m \cdot \left(im\_m \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 6.3e+156)
   (* re (+ 1.0 (* im_m (* im_m 0.5))))
   (* re (* -0.16666666666666666 (* re re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 6.3e+156) {
		tmp = re * (1.0 + (im_m * (im_m * 0.5)));
	} else {
		tmp = re * (-0.16666666666666666 * (re * re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 6.3d+156) then
        tmp = re * (1.0d0 + (im_m * (im_m * 0.5d0)))
    else
        tmp = re * ((-0.16666666666666666d0) * (re * re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 6.3e+156) {
		tmp = re * (1.0 + (im_m * (im_m * 0.5)));
	} else {
		tmp = re * (-0.16666666666666666 * (re * re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 6.3e+156:
		tmp = re * (1.0 + (im_m * (im_m * 0.5)))
	else:
		tmp = re * (-0.16666666666666666 * (re * re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 6.3e+156)
		tmp = Float64(re * Float64(1.0 + Float64(im_m * Float64(im_m * 0.5))));
	else
		tmp = Float64(re * Float64(-0.16666666666666666 * Float64(re * re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 6.3e+156)
		tmp = re * (1.0 + (im_m * (im_m * 0.5)));
	else
		tmp = re * (-0.16666666666666666 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 6.3e+156], N[(re * N[(1.0 + N[(im$95$m * N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.29999999999999982e156

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\sin re + {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sin re + \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{2} \cdot \sin re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin re} \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot \sin re + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin \color{blue}{re} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right) + \frac{1}{2} \cdot {im}^{2}\right)} \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + 1\right)}\right) \]

      10. associate-+l+ N/A

          \[\leadsto \sin re \cdot \left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) + \color{blue}{1}\right) \]

   7. Simplified 91.2%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   9. 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

         \[\leadsto \mathsf{*.f64}\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right), \color{blue}{re}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), re\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), re\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(im \cdot \left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), re\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right)\right), re\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

      12. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right), re\right) \]

   10. Simplified 58.1%

       \[\leadsto \color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right) \cdot re} \]

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right)\right), re\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 46.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{2}, im\right)\right)\right), re\right) \]

   13. Simplified 46.4%

       \[\leadsto \left(1 + im \cdot \color{blue}{\left(0.5 \cdot im\right)}\right) \cdot re \]

   if 6.29999999999999982e156 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 58.5%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   7. Simplified 58.5%

      \[\leadsto \color{blue}{\sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 29.7%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   10. Simplified 29.7%

       \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{re}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({re}^{2} \cdot re\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re} \]

       4. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       8. *-lowering-*.f64 29.7%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   13. Simplified 29.7%

       \[\leadsto \color{blue}{re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.3 \cdot 10^{+156}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.2% accurate, 25.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 1.02 \cdot 10^{+18}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 1.02e+18) re (* re (* -0.16666666666666666 (* re re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 1.02e+18) {
		tmp = re;
	} else {
		tmp = re * (-0.16666666666666666 * (re * re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.02d+18) then
        tmp = re
    else
        tmp = re * ((-0.16666666666666666d0) * (re * re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 1.02e+18) {
		tmp = re;
	} else {
		tmp = re * (-0.16666666666666666 * (re * re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 1.02e+18:
		tmp = re
	else:
		tmp = re * (-0.16666666666666666 * (re * re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 1.02e+18)
		tmp = re;
	else
		tmp = Float64(re * Float64(-0.16666666666666666 * Float64(re * re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 1.02e+18)
		tmp = re;
	else
		tmp = re * (-0.16666666666666666 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 1.02e+18], re, N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.02e18

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 51.0%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   7. Simplified 51.0%

      \[\leadsto \color{blue}{\sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \]
   9. Step-by-step derivation

   10. Simplified 31.8%

       \[\leadsto \color{blue}{re} \]

   if 1.02e18 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. associate-*l* N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      11. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      17. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 41.9%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   7. Simplified 41.9%

      \[\leadsto \color{blue}{\sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 21.6%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   10. Simplified 21.6%

       \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{re}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({re}^{2} \cdot re\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re} \]

       4. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       8. *-lowering-*.f64 21.6%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   13. Simplified 21.6%

       \[\leadsto \color{blue}{re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 34.1% accurate, 34.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (* re (+ 1.0 (* -0.16666666666666666 (* re re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return re * (1.0 + (-0.16666666666666666 * (re * re)));
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = re * (1.0d0 + ((-0.16666666666666666d0) * (re * re)))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return re * (1.0 + (-0.16666666666666666 * (re * re)));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return re * (1.0 + (-0.16666666666666666 * (re * re)))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * Float64(re * re))))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = re * (1.0 + (-0.16666666666666666 * (re * re)));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(re * N[(1.0 + N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   3. associate-*l* N/A

      \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

   9. exp-diff N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

   11. exp-0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

   14. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

   17. exp-lowering-exp.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re} \]
6. Step-by-step derivation

   1. sin-lowering-sin.f64 49.0%

      \[\leadsto \mathsf{sin.f64}\left(re\right) \]

7. Simplified 49.0%

   \[\leadsto \color{blue}{\sin re} \]

8. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 35.7%

      \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

10. Simplified 35.7%

    \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]
11. Add Preprocessing

Alternative 19: 26.1% accurate, 309.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 re)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return re;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = re
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return re;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return re

Julia

im_m = abs(im)
function code(re, im_m)
	return re
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = re;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := re

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\sin re \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   3. associate-*l* N/A

      \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \frac{1}{2}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right)\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(e^{0 - im} \cdot \frac{1}{2} + \color{blue}{e^{im} \cdot \frac{1}{2}}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(e^{0 - im} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right)\right) \]

   9. exp-diff N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0}}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{e^{0} \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

   11. exp-0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right)\right) \]

   14. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} \cdot \color{blue}{e^{im}}\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

   17. exp-lowering-exp.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re} \]
6. Step-by-step derivation

   1. sin-lowering-sin.f64 49.0%

      \[\leadsto \mathsf{sin.f64}\left(re\right) \]

7. Simplified 49.0%

   \[\leadsto \color{blue}{\sin re} \]

8. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re} \]
9. Step-by-step derivation

10. Simplified 25.5%

    \[\leadsto \color{blue}{re} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))