math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.9s

Alternatives: 26

Speedup: 1.5×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \mathsf{fma}\left(\frac{0.5}{e^{im\_m}}, \cos re, \cos re \cdot \left(0.5 \cdot e^{im\_m}\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (fma (/ 0.5 (exp im_m)) (cos re) (* (cos re) (* 0.5 (exp im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return fma((0.5 / exp(im_m)), cos(re), (cos(re) * (0.5 * exp(im_m))));
}

Julia

im_m = abs(im)
function code(re, im_m)
	return fma(Float64(0.5 / exp(im_m)), cos(re), Float64(cos(re) * Float64(0.5 * exp(im_m))))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(0.5 / N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision] + N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. fma-define N/A

      \[\leadsto \mathsf{fma}\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}, \color{blue}{\cos re}, \left(\frac{1}{2} \cdot \cos re\right) \cdot e^{im}\right) \]

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

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

   6. exp-neg N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

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

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

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

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

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

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

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

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

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

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

   17. *-commutative N/A

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

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

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

   19. exp-lowering-exp.f64 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{e^{im}}, \cos re, \cos re \cdot \left(0.5 \cdot e^{im}\right)\right)} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \cos re \cdot \left(0.5 \cdot e^{im\_m}\right) + \frac{0.5 \cdot \cos re}{e^{im\_m}} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (+ (* (cos re) (* 0.5 (exp im_m))) (/ (* 0.5 (cos re)) (exp im_m))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return (cos(re) * (0.5 * exp(im_m))) + ((0.5 * cos(re)) / exp(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 = (cos(re) * (0.5d0 * exp(im_m))) + ((0.5d0 * cos(re)) / exp(im_m))
end function

Java

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

Python

im_m = math.fabs(im)
def code(re, im_m):
	return (math.cos(re) * (0.5 * math.exp(im_m))) + ((0.5 * math.cos(re)) / math.exp(im_m))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(Float64(cos(re) * Float64(0.5 * exp(im_m))) + Float64(Float64(0.5 * cos(re)) / exp(im_m)))
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] / N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

   5. exp-neg N/A

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

   6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

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

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

   16. *-commutative N/A

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

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

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

   18. exp-lowering-exp.f64 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{e^{im}} + \cos re \cdot \left(0.5 \cdot e^{im}\right)} \]

5. Final simplification 100.0%

   \[\leadsto \cos re \cdot \left(0.5 \cdot e^{im}\right) + \frac{0.5 \cdot \cos re}{e^{im}} \]
6. Add Preprocessing

Alternative 3: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

im_m = fabs(im);
double code(double re, double im_m) {
	return cos(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 = cos(re) * cosh(im_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. cosh-undef N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

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

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

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

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

   11. cos-lowering-cos.f64 100.0%

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

4. Applied egg-rr 100.0%

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

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

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

   2. *-lft-identity N/A

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

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

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

   4. cos-lowering-cos.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \cos re \cdot \cosh im \]
8. Add Preprocessing

Alternative 4: 97.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\\ t_1 := -1 + im\_m \cdot \left(0.5 \cdot im\_m\right)\\ t_2 := 0.5 \cdot \left(im\_m \cdot im\_m\right)\\ t_3 := 1 + t\_2\\ t_4 := 0.041666666666666664 + \left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\\ t_5 := 0.041666666666666664 + im\_m \cdot \left(im\_m \cdot -0.001388888888888889\right)\\ \mathbf{if}\;im\_m \leq 0.98:\\ \;\;\;\;\frac{\cos re \cdot \left(\left(t\_4 \cdot t\_4\right) \cdot \left(t\_0 \cdot t\_0\right) + t\_3 \cdot \left(-1 - t\_2\right)\right)}{t\_4 \cdot t\_0 - t\_3}\\ \mathbf{elif}\;im\_m \leq 3.4 \cdot 10^{+24}:\\ \;\;\;\;\cosh im\_m\\ \mathbf{elif}\;im\_m \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\cos re \cdot \left(\left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot 0.25\right)\right) \cdot t\_5 + t\_0 \cdot \left(t\_1 \cdot \left(0.001736111111111111 + t\_0 \cdot -1.9290123456790124 \cdot 10^{-6}\right)\right)\right)}{t\_5}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im\_m \cdot im\_m + 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* im_m (* im_m (* im_m im_m))))
        (t_1 (+ -1.0 (* im_m (* 0.5 im_m))))
        (t_2 (* 0.5 (* im_m im_m)))
        (t_3 (+ 1.0 t_2))
        (t_4 (+ 0.041666666666666664 (* (* im_m im_m) 0.001388888888888889)))
        (t_5 (+ 0.041666666666666664 (* im_m (* im_m -0.001388888888888889)))))
   (if (<= im_m 0.98)
     (/
      (* (cos re) (+ (* (* t_4 t_4) (* t_0 t_0)) (* t_3 (- -1.0 t_2))))
      (- (* t_4 t_0) t_3))
     (if (<= im_m 3.4e+24)
       (cosh im_m)
       (if (<= im_m 2e+154)
         (/
          (/
           (*
            (cos re)
            (+
             (* (+ -1.0 (* (* im_m im_m) (* (* im_m im_m) 0.25))) t_5)
             (*
              t_0
              (*
               t_1
               (+ 0.001736111111111111 (* t_0 -1.9290123456790124e-6))))))
           t_5)
          t_1)
         (* (* 0.5 (cos re)) (+ (* im_m im_m) 2.0)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = im_m * (im_m * (im_m * im_m));
	double t_1 = -1.0 + (im_m * (0.5 * im_m));
	double t_2 = 0.5 * (im_m * im_m);
	double t_3 = 1.0 + t_2;
	double t_4 = 0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889);
	double t_5 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889));
	double tmp;
	if (im_m <= 0.98) {
		tmp = (cos(re) * (((t_4 * t_4) * (t_0 * t_0)) + (t_3 * (-1.0 - t_2)))) / ((t_4 * t_0) - t_3);
	} else if (im_m <= 3.4e+24) {
		tmp = cosh(im_m);
	} else if (im_m <= 2e+154) {
		tmp = ((cos(re) * (((-1.0 + ((im_m * im_m) * ((im_m * im_m) * 0.25))) * t_5) + (t_0 * (t_1 * (0.001736111111111111 + (t_0 * -1.9290123456790124e-6)))))) / t_5) / t_1;
	} else {
		tmp = (0.5 * cos(re)) * ((im_m * im_m) + 2.0);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = im_m * (im_m * (im_m * im_m))
    t_1 = (-1.0d0) + (im_m * (0.5d0 * im_m))
    t_2 = 0.5d0 * (im_m * im_m)
    t_3 = 1.0d0 + t_2
    t_4 = 0.041666666666666664d0 + ((im_m * im_m) * 0.001388888888888889d0)
    t_5 = 0.041666666666666664d0 + (im_m * (im_m * (-0.001388888888888889d0)))
    if (im_m <= 0.98d0) then
        tmp = (cos(re) * (((t_4 * t_4) * (t_0 * t_0)) + (t_3 * ((-1.0d0) - t_2)))) / ((t_4 * t_0) - t_3)
    else if (im_m <= 3.4d+24) then
        tmp = cosh(im_m)
    else if (im_m <= 2d+154) then
        tmp = ((cos(re) * ((((-1.0d0) + ((im_m * im_m) * ((im_m * im_m) * 0.25d0))) * t_5) + (t_0 * (t_1 * (0.001736111111111111d0 + (t_0 * (-1.9290123456790124d-6))))))) / t_5) / t_1
    else
        tmp = (0.5d0 * cos(re)) * ((im_m * im_m) + 2.0d0)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double t_0 = im_m * (im_m * (im_m * im_m));
	double t_1 = -1.0 + (im_m * (0.5 * im_m));
	double t_2 = 0.5 * (im_m * im_m);
	double t_3 = 1.0 + t_2;
	double t_4 = 0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889);
	double t_5 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889));
	double tmp;
	if (im_m <= 0.98) {
		tmp = (Math.cos(re) * (((t_4 * t_4) * (t_0 * t_0)) + (t_3 * (-1.0 - t_2)))) / ((t_4 * t_0) - t_3);
	} else if (im_m <= 3.4e+24) {
		tmp = Math.cosh(im_m);
	} else if (im_m <= 2e+154) {
		tmp = ((Math.cos(re) * (((-1.0 + ((im_m * im_m) * ((im_m * im_m) * 0.25))) * t_5) + (t_0 * (t_1 * (0.001736111111111111 + (t_0 * -1.9290123456790124e-6)))))) / t_5) / t_1;
	} else {
		tmp = (0.5 * Math.cos(re)) * ((im_m * im_m) + 2.0);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = im_m * (im_m * (im_m * im_m))
	t_1 = -1.0 + (im_m * (0.5 * im_m))
	t_2 = 0.5 * (im_m * im_m)
	t_3 = 1.0 + t_2
	t_4 = 0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889)
	t_5 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889))
	tmp = 0
	if im_m <= 0.98:
		tmp = (math.cos(re) * (((t_4 * t_4) * (t_0 * t_0)) + (t_3 * (-1.0 - t_2)))) / ((t_4 * t_0) - t_3)
	elif im_m <= 3.4e+24:
		tmp = math.cosh(im_m)
	elif im_m <= 2e+154:
		tmp = ((math.cos(re) * (((-1.0 + ((im_m * im_m) * ((im_m * im_m) * 0.25))) * t_5) + (t_0 * (t_1 * (0.001736111111111111 + (t_0 * -1.9290123456790124e-6)))))) / t_5) / t_1
	else:
		tmp = (0.5 * math.cos(re)) * ((im_m * im_m) + 2.0)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(im_m * Float64(im_m * Float64(im_m * im_m)))
	t_1 = Float64(-1.0 + Float64(im_m * Float64(0.5 * im_m)))
	t_2 = Float64(0.5 * Float64(im_m * im_m))
	t_3 = Float64(1.0 + t_2)
	t_4 = Float64(0.041666666666666664 + Float64(Float64(im_m * im_m) * 0.001388888888888889))
	t_5 = Float64(0.041666666666666664 + Float64(im_m * Float64(im_m * -0.001388888888888889)))
	tmp = 0.0
	if (im_m <= 0.98)
		tmp = Float64(Float64(cos(re) * Float64(Float64(Float64(t_4 * t_4) * Float64(t_0 * t_0)) + Float64(t_3 * Float64(-1.0 - t_2)))) / Float64(Float64(t_4 * t_0) - t_3));
	elseif (im_m <= 3.4e+24)
		tmp = cosh(im_m);
	elseif (im_m <= 2e+154)
		tmp = Float64(Float64(Float64(cos(re) * Float64(Float64(Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * 0.25))) * t_5) + Float64(t_0 * Float64(t_1 * Float64(0.001736111111111111 + Float64(t_0 * -1.9290123456790124e-6)))))) / t_5) / t_1);
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(im_m * im_m) + 2.0));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = im_m * (im_m * (im_m * im_m));
	t_1 = -1.0 + (im_m * (0.5 * im_m));
	t_2 = 0.5 * (im_m * im_m);
	t_3 = 1.0 + t_2;
	t_4 = 0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889);
	t_5 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889));
	tmp = 0.0;
	if (im_m <= 0.98)
		tmp = (cos(re) * (((t_4 * t_4) * (t_0 * t_0)) + (t_3 * (-1.0 - t_2)))) / ((t_4 * t_0) - t_3);
	elseif (im_m <= 3.4e+24)
		tmp = cosh(im_m);
	elseif (im_m <= 2e+154)
		tmp = ((cos(re) * (((-1.0 + ((im_m * im_m) * ((im_m * im_m) * 0.25))) * t_5) + (t_0 * (t_1 * (0.001736111111111111 + (t_0 * -1.9290123456790124e-6)))))) / t_5) / t_1;
	else
		tmp = (0.5 * cos(re)) * ((im_m * im_m) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 + N[(im$95$m * N[(0.5 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(0.041666666666666664 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.041666666666666664 + N[(im$95$m * N[(im$95$m * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 0.98], N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$4 * t$95$0), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.4e+24], N[Cosh[im$95$m], $MachinePrecision], If[LessEqual[im$95$m, 2e+154], N[(N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] + N[(t$95$0 * N[(t$95$1 * N[(0.001736111111111111 + N[(t$95$0 * -1.9290123456790124e-6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.97999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 91.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{24} + \left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + \frac{1}{2} \cdot \left(im \cdot im\right)\right)\right) \cdot \color{blue}{\cos re} \]

      2. flip-+ N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} + \left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right) \cdot \left(\left(\frac{1}{24} + \left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right) - \left(1 + \frac{1}{2} \cdot \left(im \cdot im\right)\right) \cdot \left(1 + \frac{1}{2} \cdot \left(im \cdot im\right)\right)}{\left(\frac{1}{24} + \left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) - \left(1 + \frac{1}{2} \cdot \left(im \cdot im\right)\right)} \cdot \cos \color{blue}{re} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(\left(\left(\frac{1}{24} + \left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right) \cdot \left(\left(\frac{1}{24} + \left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right) - \left(1 + \frac{1}{2} \cdot \left(im \cdot im\right)\right) \cdot \left(1 + \frac{1}{2} \cdot \left(im \cdot im\right)\right)\right) \cdot \cos re}{\color{blue}{\left(\frac{1}{24} + \left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) - \left(1 + \frac{1}{2} \cdot \left(im \cdot im\right)\right)}} \]

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

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

   7. Applied egg-rr 64.7%

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

   if 0.97999999999999998 < im < 3.4000000000000001e24

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

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

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

      2. *-lft-identity N/A

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

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

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

      4. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 100.0%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   if 3.4000000000000001e24 < im < 2.00000000000000007e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 83.0%

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

   7. Applied egg-rr 39.5%

      \[\leadsto \cos re \cdot \color{blue}{\frac{\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25 - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right) + \left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(\left(0.001736111111111111 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 1.9290123456790124 \cdot 10^{-6}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)}{\left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right)}} \]

   9. Applied egg-rr 97.1%

      \[\leadsto \color{blue}{\frac{\frac{\cos re \cdot \left(\left(-1 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.25\right)\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)\right) + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(im \cdot \left(im \cdot 0.5\right) + -1\right) \cdot \left(0.001736111111111111 + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot -1.9290123456790124 \cdot 10^{-6}\right)\right)\right)}{0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)}}{im \cdot \left(im \cdot 0.5\right) + -1}} \]

   if 2.00000000000000007e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.98:\\ \;\;\;\;\frac{\cos re \cdot \left(\left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right) \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(-1 - 0.5 \cdot \left(im \cdot im\right)\right)\right)}{\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) - \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)}\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+24}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\cos re \cdot \left(\left(-1 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.25\right)\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)\right) + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(-1 + im \cdot \left(0.5 \cdot im\right)\right) \cdot \left(0.001736111111111111 + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot -1.9290123456790124 \cdot 10^{-6}\right)\right)\right)}{0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)}}{-1 + im \cdot \left(0.5 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\\ t_1 := -1 + im\_m \cdot \left(0.5 \cdot im\_m\right)\\ t_2 := 0.041666666666666664 + im\_m \cdot \left(im\_m \cdot -0.001388888888888889\right)\\ \mathbf{if}\;im\_m \leq 0.99:\\ \;\;\;\;\cos re \cdot \frac{\left(im\_m \cdot im\_m\right) \cdot \left(0.001388888888888889 + im\_m \cdot \left(im\_m \cdot \left(0.008680555555555556 + \left(im\_m \cdot im\_m\right) \cdot 0.0005208333333333333\right)\right)\right) + -0.041666666666666664}{\left(0.041666666666666664 + \left(im\_m \cdot im\_m\right) \cdot -0.001388888888888889\right) \cdot \left(0.5 \cdot \left(im\_m \cdot im\_m\right) + -1\right)}\\ \mathbf{elif}\;im\_m \leq 3.4 \cdot 10^{+24}:\\ \;\;\;\;\cosh im\_m\\ \mathbf{elif}\;im\_m \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\cos re \cdot \left(\left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot 0.25\right)\right) \cdot t\_2 + t\_0 \cdot \left(t\_1 \cdot \left(0.001736111111111111 + t\_0 \cdot -1.9290123456790124 \cdot 10^{-6}\right)\right)\right)}{t\_2}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im\_m \cdot im\_m + 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* im_m (* im_m (* im_m im_m))))
        (t_1 (+ -1.0 (* im_m (* 0.5 im_m))))
        (t_2 (+ 0.041666666666666664 (* im_m (* im_m -0.001388888888888889)))))
   (if (<= im_m 0.99)
     (*
      (cos re)
      (/
       (+
        (*
         (* im_m im_m)
         (+
          0.001388888888888889
          (*
           im_m
           (*
            im_m
            (+
             0.008680555555555556
             (* (* im_m im_m) 0.0005208333333333333))))))
        -0.041666666666666664)
       (*
        (+ 0.041666666666666664 (* (* im_m im_m) -0.001388888888888889))
        (+ (* 0.5 (* im_m im_m)) -1.0))))
     (if (<= im_m 3.4e+24)
       (cosh im_m)
       (if (<= im_m 2e+154)
         (/
          (/
           (*
            (cos re)
            (+
             (* (+ -1.0 (* (* im_m im_m) (* (* im_m im_m) 0.25))) t_2)
             (*
              t_0
              (*
               t_1
               (+ 0.001736111111111111 (* t_0 -1.9290123456790124e-6))))))
           t_2)
          t_1)
         (* (* 0.5 (cos re)) (+ (* im_m im_m) 2.0)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = im_m * (im_m * (im_m * im_m));
	double t_1 = -1.0 + (im_m * (0.5 * im_m));
	double t_2 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889));
	double tmp;
	if (im_m <= 0.99) {
		tmp = cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * ((0.5 * (im_m * im_m)) + -1.0)));
	} else if (im_m <= 3.4e+24) {
		tmp = cosh(im_m);
	} else if (im_m <= 2e+154) {
		tmp = ((cos(re) * (((-1.0 + ((im_m * im_m) * ((im_m * im_m) * 0.25))) * t_2) + (t_0 * (t_1 * (0.001736111111111111 + (t_0 * -1.9290123456790124e-6)))))) / t_2) / t_1;
	} else {
		tmp = (0.5 * cos(re)) * ((im_m * im_m) + 2.0);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = im_m * (im_m * (im_m * im_m))
    t_1 = (-1.0d0) + (im_m * (0.5d0 * im_m))
    t_2 = 0.041666666666666664d0 + (im_m * (im_m * (-0.001388888888888889d0)))
    if (im_m <= 0.99d0) then
        tmp = cos(re) * ((((im_m * im_m) * (0.001388888888888889d0 + (im_m * (im_m * (0.008680555555555556d0 + ((im_m * im_m) * 0.0005208333333333333d0)))))) + (-0.041666666666666664d0)) / ((0.041666666666666664d0 + ((im_m * im_m) * (-0.001388888888888889d0))) * ((0.5d0 * (im_m * im_m)) + (-1.0d0))))
    else if (im_m <= 3.4d+24) then
        tmp = cosh(im_m)
    else if (im_m <= 2d+154) then
        tmp = ((cos(re) * ((((-1.0d0) + ((im_m * im_m) * ((im_m * im_m) * 0.25d0))) * t_2) + (t_0 * (t_1 * (0.001736111111111111d0 + (t_0 * (-1.9290123456790124d-6))))))) / t_2) / t_1
    else
        tmp = (0.5d0 * cos(re)) * ((im_m * im_m) + 2.0d0)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double t_0 = im_m * (im_m * (im_m * im_m));
	double t_1 = -1.0 + (im_m * (0.5 * im_m));
	double t_2 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889));
	double tmp;
	if (im_m <= 0.99) {
		tmp = Math.cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * ((0.5 * (im_m * im_m)) + -1.0)));
	} else if (im_m <= 3.4e+24) {
		tmp = Math.cosh(im_m);
	} else if (im_m <= 2e+154) {
		tmp = ((Math.cos(re) * (((-1.0 + ((im_m * im_m) * ((im_m * im_m) * 0.25))) * t_2) + (t_0 * (t_1 * (0.001736111111111111 + (t_0 * -1.9290123456790124e-6)))))) / t_2) / t_1;
	} else {
		tmp = (0.5 * Math.cos(re)) * ((im_m * im_m) + 2.0);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = im_m * (im_m * (im_m * im_m))
	t_1 = -1.0 + (im_m * (0.5 * im_m))
	t_2 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889))
	tmp = 0
	if im_m <= 0.99:
		tmp = math.cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * ((0.5 * (im_m * im_m)) + -1.0)))
	elif im_m <= 3.4e+24:
		tmp = math.cosh(im_m)
	elif im_m <= 2e+154:
		tmp = ((math.cos(re) * (((-1.0 + ((im_m * im_m) * ((im_m * im_m) * 0.25))) * t_2) + (t_0 * (t_1 * (0.001736111111111111 + (t_0 * -1.9290123456790124e-6)))))) / t_2) / t_1
	else:
		tmp = (0.5 * math.cos(re)) * ((im_m * im_m) + 2.0)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(im_m * Float64(im_m * Float64(im_m * im_m)))
	t_1 = Float64(-1.0 + Float64(im_m * Float64(0.5 * im_m)))
	t_2 = Float64(0.041666666666666664 + Float64(im_m * Float64(im_m * -0.001388888888888889)))
	tmp = 0.0
	if (im_m <= 0.99)
		tmp = Float64(cos(re) * Float64(Float64(Float64(Float64(im_m * im_m) * Float64(0.001388888888888889 + Float64(im_m * Float64(im_m * Float64(0.008680555555555556 + Float64(Float64(im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / Float64(Float64(0.041666666666666664 + Float64(Float64(im_m * im_m) * -0.001388888888888889)) * Float64(Float64(0.5 * Float64(im_m * im_m)) + -1.0))));
	elseif (im_m <= 3.4e+24)
		tmp = cosh(im_m);
	elseif (im_m <= 2e+154)
		tmp = Float64(Float64(Float64(cos(re) * Float64(Float64(Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * 0.25))) * t_2) + Float64(t_0 * Float64(t_1 * Float64(0.001736111111111111 + Float64(t_0 * -1.9290123456790124e-6)))))) / t_2) / t_1);
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(im_m * im_m) + 2.0));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = im_m * (im_m * (im_m * im_m));
	t_1 = -1.0 + (im_m * (0.5 * im_m));
	t_2 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889));
	tmp = 0.0;
	if (im_m <= 0.99)
		tmp = cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * ((0.5 * (im_m * im_m)) + -1.0)));
	elseif (im_m <= 3.4e+24)
		tmp = cosh(im_m);
	elseif (im_m <= 2e+154)
		tmp = ((cos(re) * (((-1.0 + ((im_m * im_m) * ((im_m * im_m) * 0.25))) * t_2) + (t_0 * (t_1 * (0.001736111111111111 + (t_0 * -1.9290123456790124e-6)))))) / t_2) / t_1;
	else
		tmp = (0.5 * cos(re)) * ((im_m * im_m) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 + N[(im$95$m * N[(0.5 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.041666666666666664 + N[(im$95$m * N[(im$95$m * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 0.99], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.001388888888888889 + N[(im$95$m * N[(im$95$m * N[(0.008680555555555556 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.0005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] / N[(N[(0.041666666666666664 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.4e+24], N[Cosh[im$95$m], $MachinePrecision], If[LessEqual[im$95$m, 2e+154], N[(N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$0 * N[(t$95$1 * N[(0.001736111111111111 + N[(t$95$0 * -1.9290123456790124e-6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.98999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 91.3%

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

   7. Applied egg-rr 67.3%

      \[\leadsto \cos re \cdot \color{blue}{\frac{\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25 - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right) + \left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(\left(0.001736111111111111 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 1.9290123456790124 \cdot 10^{-6}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)}{\left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right)}} \]

   8. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right) - \frac{1}{24}\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right)}, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right)}, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)}, 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \left({im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \left(\left(im \cdot im\right) \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \left(im \cdot \left(im \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \left(\frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \left({im}^{2} \cdot \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      16. metadata-eval 60.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \frac{-1}{24}\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \color{blue}{1}\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

   10. Simplified 60.5%

       \[\leadsto \cos re \cdot \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(0.001388888888888889 + im \cdot \left(im \cdot \left(0.008680555555555556 + \left(im \cdot im\right) \cdot 0.0005208333333333333\right)\right)\right) + -0.041666666666666664}}{\left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right)} \]

   if 0.98999999999999999 < im < 3.4000000000000001e24

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

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

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

      2. *-lft-identity N/A

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

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

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

      4. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 100.0%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   if 3.4000000000000001e24 < im < 2.00000000000000007e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 83.0%

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

   7. Applied egg-rr 39.5%

      \[\leadsto \cos re \cdot \color{blue}{\frac{\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25 - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right) + \left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(\left(0.001736111111111111 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 1.9290123456790124 \cdot 10^{-6}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)}{\left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right)}} \]

   9. Applied egg-rr 97.1%

      \[\leadsto \color{blue}{\frac{\frac{\cos re \cdot \left(\left(-1 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.25\right)\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)\right) + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(im \cdot \left(im \cdot 0.5\right) + -1\right) \cdot \left(0.001736111111111111 + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot -1.9290123456790124 \cdot 10^{-6}\right)\right)\right)}{0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)}}{im \cdot \left(im \cdot 0.5\right) + -1}} \]

   if 2.00000000000000007e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.99:\\ \;\;\;\;\cos re \cdot \frac{\left(im \cdot im\right) \cdot \left(0.001388888888888889 + im \cdot \left(im \cdot \left(0.008680555555555556 + \left(im \cdot im\right) \cdot 0.0005208333333333333\right)\right)\right) + -0.041666666666666664}{\left(0.041666666666666664 + \left(im \cdot im\right) \cdot -0.001388888888888889\right) \cdot \left(0.5 \cdot \left(im \cdot im\right) + -1\right)}\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+24}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\cos re \cdot \left(\left(-1 + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.25\right)\right) \cdot \left(0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)\right) + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(-1 + im \cdot \left(0.5 \cdot im\right)\right) \cdot \left(0.001736111111111111 + \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot -1.9290123456790124 \cdot 10^{-6}\right)\right)\right)}{0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)}}{-1 + im \cdot \left(0.5 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\\ t_1 := 0.5 \cdot \left(im\_m \cdot im\_m\right)\\ t_2 := im\_m \cdot \left(im\_m \cdot im\_m\right)\\ \mathbf{if}\;im\_m \leq 1.16:\\ \;\;\;\;\cos re \cdot \frac{\left(im\_m \cdot im\_m\right) \cdot \left(0.001388888888888889 + im\_m \cdot \left(im\_m \cdot \left(0.008680555555555556 + \left(im\_m \cdot im\_m\right) \cdot 0.0005208333333333333\right)\right)\right) + -0.041666666666666664}{\left(0.041666666666666664 + \left(im\_m \cdot im\_m\right) \cdot -0.001388888888888889\right) \cdot \left(t\_1 + -1\right)}\\ \mathbf{elif}\;im\_m \leq 3.4 \cdot 10^{+24}:\\ \;\;\;\;\cosh im\_m\\ \mathbf{elif}\;im\_m \leq 10^{+77}:\\ \;\;\;\;\cos re \cdot \left(\left(1 + t\_1\right) + \frac{\left(im\_m \cdot t\_2\right) \cdot \left(7.233796296296296 \cdot 10^{-5} + t\_2 \cdot \left(t\_2 \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right)}{0.001736111111111111 + t\_0 \cdot \left(t\_0 - 0.041666666666666664\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im\_m \cdot im\_m\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* im_m im_m) 0.001388888888888889))
        (t_1 (* 0.5 (* im_m im_m)))
        (t_2 (* im_m (* im_m im_m))))
   (if (<= im_m 1.16)
     (*
      (cos re)
      (/
       (+
        (*
         (* im_m im_m)
         (+
          0.001388888888888889
          (*
           im_m
           (*
            im_m
            (+
             0.008680555555555556
             (* (* im_m im_m) 0.0005208333333333333))))))
        -0.041666666666666664)
       (*
        (+ 0.041666666666666664 (* (* im_m im_m) -0.001388888888888889))
        (+ t_1 -1.0))))
     (if (<= im_m 3.4e+24)
       (cosh im_m)
       (if (<= im_m 1e+77)
         (*
          (cos re)
          (+
           (+ 1.0 t_1)
           (/
            (*
             (* im_m t_2)
             (+ 7.233796296296296e-5 (* t_2 (* t_2 2.6791838134430728e-9))))
            (+ 0.001736111111111111 (* t_0 (- t_0 0.041666666666666664))))))
         (*
          (cos re)
          (+
           1.0
           (*
            (* im_m im_m)
            (+ 0.5 (* 0.041666666666666664 (* im_m im_m)))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (im_m * im_m) * 0.001388888888888889;
	double t_1 = 0.5 * (im_m * im_m);
	double t_2 = im_m * (im_m * im_m);
	double tmp;
	if (im_m <= 1.16) {
		tmp = cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * (t_1 + -1.0)));
	} else if (im_m <= 3.4e+24) {
		tmp = cosh(im_m);
	} else if (im_m <= 1e+77) {
		tmp = cos(re) * ((1.0 + t_1) + (((im_m * t_2) * (7.233796296296296e-5 + (t_2 * (t_2 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_0 * (t_0 - 0.041666666666666664)))));
	} else {
		tmp = cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (im_m * im_m) * 0.001388888888888889d0
    t_1 = 0.5d0 * (im_m * im_m)
    t_2 = im_m * (im_m * im_m)
    if (im_m <= 1.16d0) then
        tmp = cos(re) * ((((im_m * im_m) * (0.001388888888888889d0 + (im_m * (im_m * (0.008680555555555556d0 + ((im_m * im_m) * 0.0005208333333333333d0)))))) + (-0.041666666666666664d0)) / ((0.041666666666666664d0 + ((im_m * im_m) * (-0.001388888888888889d0))) * (t_1 + (-1.0d0))))
    else if (im_m <= 3.4d+24) then
        tmp = cosh(im_m)
    else if (im_m <= 1d+77) then
        tmp = cos(re) * ((1.0d0 + t_1) + (((im_m * t_2) * (7.233796296296296d-5 + (t_2 * (t_2 * 2.6791838134430728d-9)))) / (0.001736111111111111d0 + (t_0 * (t_0 - 0.041666666666666664d0)))))
    else
        tmp = cos(re) * (1.0d0 + ((im_m * im_m) * (0.5d0 + (0.041666666666666664d0 * (im_m * im_m)))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double t_0 = (im_m * im_m) * 0.001388888888888889;
	double t_1 = 0.5 * (im_m * im_m);
	double t_2 = im_m * (im_m * im_m);
	double tmp;
	if (im_m <= 1.16) {
		tmp = Math.cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * (t_1 + -1.0)));
	} else if (im_m <= 3.4e+24) {
		tmp = Math.cosh(im_m);
	} else if (im_m <= 1e+77) {
		tmp = Math.cos(re) * ((1.0 + t_1) + (((im_m * t_2) * (7.233796296296296e-5 + (t_2 * (t_2 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_0 * (t_0 - 0.041666666666666664)))));
	} else {
		tmp = Math.cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = (im_m * im_m) * 0.001388888888888889
	t_1 = 0.5 * (im_m * im_m)
	t_2 = im_m * (im_m * im_m)
	tmp = 0
	if im_m <= 1.16:
		tmp = math.cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * (t_1 + -1.0)))
	elif im_m <= 3.4e+24:
		tmp = math.cosh(im_m)
	elif im_m <= 1e+77:
		tmp = math.cos(re) * ((1.0 + t_1) + (((im_m * t_2) * (7.233796296296296e-5 + (t_2 * (t_2 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_0 * (t_0 - 0.041666666666666664)))))
	else:
		tmp = math.cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m)))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(im_m * im_m) * 0.001388888888888889)
	t_1 = Float64(0.5 * Float64(im_m * im_m))
	t_2 = Float64(im_m * Float64(im_m * im_m))
	tmp = 0.0
	if (im_m <= 1.16)
		tmp = Float64(cos(re) * Float64(Float64(Float64(Float64(im_m * im_m) * Float64(0.001388888888888889 + Float64(im_m * Float64(im_m * Float64(0.008680555555555556 + Float64(Float64(im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / Float64(Float64(0.041666666666666664 + Float64(Float64(im_m * im_m) * -0.001388888888888889)) * Float64(t_1 + -1.0))));
	elseif (im_m <= 3.4e+24)
		tmp = cosh(im_m);
	elseif (im_m <= 1e+77)
		tmp = Float64(cos(re) * Float64(Float64(1.0 + t_1) + Float64(Float64(Float64(im_m * t_2) * Float64(7.233796296296296e-5 + Float64(t_2 * Float64(t_2 * 2.6791838134430728e-9)))) / Float64(0.001736111111111111 + Float64(t_0 * Float64(t_0 - 0.041666666666666664))))));
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im_m * im_m) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im_m * im_m))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = (im_m * im_m) * 0.001388888888888889;
	t_1 = 0.5 * (im_m * im_m);
	t_2 = im_m * (im_m * im_m);
	tmp = 0.0;
	if (im_m <= 1.16)
		tmp = cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * (t_1 + -1.0)));
	elseif (im_m <= 3.4e+24)
		tmp = cosh(im_m);
	elseif (im_m <= 1e+77)
		tmp = cos(re) * ((1.0 + t_1) + (((im_m * t_2) * (7.233796296296296e-5 + (t_2 * (t_2 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_0 * (t_0 - 0.041666666666666664)))));
	else
		tmp = cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 1.16], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.001388888888888889 + N[(im$95$m * N[(im$95$m * N[(0.008680555555555556 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.0005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] / N[(N[(0.041666666666666664 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.4e+24], N[Cosh[im$95$m], $MachinePrecision], If[LessEqual[im$95$m, 1e+77], N[(N[Cos[re], $MachinePrecision] * N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(N[(N[(im$95$m * t$95$2), $MachinePrecision] * N[(7.233796296296296e-5 + N[(t$95$2 * N[(t$95$2 * 2.6791838134430728e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.001736111111111111 + N[(t$95$0 * N[(t$95$0 - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.5 + N[(0.041666666666666664 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.15999999999999992

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 91.3%

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

   7. Applied egg-rr 67.3%

      \[\leadsto \cos re \cdot \color{blue}{\frac{\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25 - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right) + \left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(\left(0.001736111111111111 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 1.9290123456790124 \cdot 10^{-6}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)}{\left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right)}} \]

   8. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right) - \frac{1}{24}\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right)}, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right)}, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)}, 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \left({im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \left(\left(im \cdot im\right) \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \left(im \cdot \left(im \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \left(\frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \left({im}^{2} \cdot \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      16. metadata-eval 60.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \frac{-1}{24}\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \color{blue}{1}\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

   10. Simplified 60.5%

       \[\leadsto \cos re \cdot \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(0.001388888888888889 + im \cdot \left(im \cdot \left(0.008680555555555556 + \left(im \cdot im\right) \cdot 0.0005208333333333333\right)\right)\right) + -0.041666666666666664}}{\left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right)} \]

   if 1.15999999999999992 < im < 3.4000000000000001e24

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

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

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

      2. *-lft-identity N/A

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

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

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

      4. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 100.0%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   if 3.4000000000000001e24 < im < 9.99999999999999983e76

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 59.9%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{{\frac{1}{24}}^{3} + {\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)}^{3}}{\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right)} \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\left(\frac{\left({\frac{1}{24}}^{3} + {\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)}^{3}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)}{\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right)}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left({\frac{1}{24}}^{3} + {\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)}^{3}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right), \left(\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 93.2%

      \[\leadsto \cos re \cdot \left(\color{blue}{\frac{\left(7.233796296296296 \cdot 10^{-5} + \left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)}{0.001736111111111111 + \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)}} + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right) \]

   if 9.99999999999999983e76 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.16:\\ \;\;\;\;\cos re \cdot \frac{\left(im \cdot im\right) \cdot \left(0.001388888888888889 + im \cdot \left(im \cdot \left(0.008680555555555556 + \left(im \cdot im\right) \cdot 0.0005208333333333333\right)\right)\right) + -0.041666666666666664}{\left(0.041666666666666664 + \left(im \cdot im\right) \cdot -0.001388888888888889\right) \cdot \left(0.5 \cdot \left(im \cdot im\right) + -1\right)}\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+24}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 10^{+77}:\\ \;\;\;\;\cos re \cdot \left(\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) + \frac{\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(7.233796296296296 \cdot 10^{-5} + \left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right)}{0.001736111111111111 + \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.25:\\ \;\;\;\;\cos re \cdot \frac{\left(im\_m \cdot im\_m\right) \cdot \left(0.001388888888888889 + im\_m \cdot \left(im\_m \cdot \left(0.008680555555555556 + \left(im\_m \cdot im\_m\right) \cdot 0.0005208333333333333\right)\right)\right) + -0.041666666666666664}{\left(0.041666666666666664 + \left(im\_m \cdot im\_m\right) \cdot -0.001388888888888889\right) \cdot \left(0.5 \cdot \left(im\_m \cdot im\_m\right) + -1\right)}\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im\_m \cdot im\_m\right) \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot \left(0.041666666666666664 + \left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.25)
   (*
    (cos re)
    (/
     (+
      (*
       (* im_m im_m)
       (+
        0.001388888888888889
        (*
         im_m
         (*
          im_m
          (+ 0.008680555555555556 (* (* im_m im_m) 0.0005208333333333333))))))
      -0.041666666666666664)
     (*
      (+ 0.041666666666666664 (* (* im_m im_m) -0.001388888888888889))
      (+ (* 0.5 (* im_m im_m)) -1.0))))
   (if (<= im_m 7.2e+51)
     (cosh im_m)
     (*
      (cos 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 (im_m <= 1.25) {
		tmp = cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * ((0.5 * (im_m * im_m)) + -1.0)));
	} else if (im_m <= 7.2e+51) {
		tmp = cosh(im_m);
	} else {
		tmp = cos(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 (im_m <= 1.25d0) then
        tmp = cos(re) * ((((im_m * im_m) * (0.001388888888888889d0 + (im_m * (im_m * (0.008680555555555556d0 + ((im_m * im_m) * 0.0005208333333333333d0)))))) + (-0.041666666666666664d0)) / ((0.041666666666666664d0 + ((im_m * im_m) * (-0.001388888888888889d0))) * ((0.5d0 * (im_m * im_m)) + (-1.0d0))))
    else if (im_m <= 7.2d+51) then
        tmp = cosh(im_m)
    else
        tmp = cos(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 (im_m <= 1.25) {
		tmp = Math.cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * ((0.5 * (im_m * im_m)) + -1.0)));
	} else if (im_m <= 7.2e+51) {
		tmp = Math.cosh(im_m);
	} else {
		tmp = Math.cos(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 im_m <= 1.25:
		tmp = math.cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * ((0.5 * (im_m * im_m)) + -1.0)))
	elif im_m <= 7.2e+51:
		tmp = math.cosh(im_m)
	else:
		tmp = math.cos(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 (im_m <= 1.25)
		tmp = Float64(cos(re) * Float64(Float64(Float64(Float64(im_m * im_m) * Float64(0.001388888888888889 + Float64(im_m * Float64(im_m * Float64(0.008680555555555556 + Float64(Float64(im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / Float64(Float64(0.041666666666666664 + Float64(Float64(im_m * im_m) * -0.001388888888888889)) * Float64(Float64(0.5 * Float64(im_m * im_m)) + -1.0))));
	elseif (im_m <= 7.2e+51)
		tmp = cosh(im_m);
	else
		tmp = Float64(cos(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 (im_m <= 1.25)
		tmp = cos(re) * ((((im_m * im_m) * (0.001388888888888889 + (im_m * (im_m * (0.008680555555555556 + ((im_m * im_m) * 0.0005208333333333333)))))) + -0.041666666666666664) / ((0.041666666666666664 + ((im_m * im_m) * -0.001388888888888889)) * ((0.5 * (im_m * im_m)) + -1.0)));
	elseif (im_m <= 7.2e+51)
		tmp = cosh(im_m);
	else
		tmp = cos(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[im$95$m, 1.25], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.001388888888888889 + N[(im$95$m * N[(im$95$m * N[(0.008680555555555556 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.0005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] / N[(N[(0.041666666666666664 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 7.2e+51], N[Cosh[im$95$m], $MachinePrecision], N[(N[Cos[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 3 regimes

2. if im < 1.25

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 91.3%

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

   7. Applied egg-rr 67.3%

      \[\leadsto \cos re \cdot \color{blue}{\frac{\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25 - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right) + \left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(\left(0.001736111111111111 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 1.9290123456790124 \cdot 10^{-6}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)}{\left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right)}} \]

   8. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right) - \frac{1}{24}\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right)}, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right)}, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)}, 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{720} + {im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \left({im}^{2} \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \left(\left(im \cdot im\right) \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \left(im \cdot \left(im \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{5}{576} + \frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \left(\frac{1}{1920} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \left({im}^{2} \cdot \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), 1\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

      16. metadata-eval 60.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{5}{576}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{1920}\right)\right)\right)\right)\right)\right), \frac{-1}{24}\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \color{blue}{1}\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]

   10. Simplified 60.5%

       \[\leadsto \cos re \cdot \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(0.001388888888888889 + im \cdot \left(im \cdot \left(0.008680555555555556 + \left(im \cdot im\right) \cdot 0.0005208333333333333\right)\right)\right) + -0.041666666666666664}}{\left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right)} \]

   if 1.25 < im < 7.20000000000000022e51

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

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

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

      2. *-lft-identity N/A

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

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

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

      4. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 78.6%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   if 7.20000000000000022e51 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

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

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

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

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

      18. *-commutative N/A

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

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

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

   8. Simplified 100.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.25:\\ \;\;\;\;\cos re \cdot \frac{\left(im \cdot im\right) \cdot \left(0.001388888888888889 + im \cdot \left(im \cdot \left(0.008680555555555556 + \left(im \cdot im\right) \cdot 0.0005208333333333333\right)\right)\right) + -0.041666666666666664}{\left(0.041666666666666664 + \left(im \cdot im\right) \cdot -0.001388888888888889\right) \cdot \left(0.5 \cdot \left(im \cdot im\right) + -1\right)}\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := im\_m \cdot \left(0.041666666666666664 + \left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\right)\\ \mathbf{if}\;im\_m \leq 0.98:\\ \;\;\;\;\cos re \cdot \left(\left(1 + 0.5 \cdot \left(im\_m \cdot im\_m\right)\right) + \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right) \cdot t\_0\right)\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im\_m \cdot im\_m\right) \cdot \left(0.5 + im\_m \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0
         (*
          im_m
          (+ 0.041666666666666664 (* (* im_m im_m) 0.001388888888888889)))))
   (if (<= im_m 0.98)
     (*
      (cos re)
      (+ (+ 1.0 (* 0.5 (* im_m im_m))) (* (* im_m (* im_m im_m)) t_0)))
     (if (<= im_m 7.2e+51)
       (cosh im_m)
       (* (cos re) (+ 1.0 (* (* im_m im_m) (+ 0.5 (* im_m t_0)))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889));
	double tmp;
	if (im_m <= 0.98) {
		tmp = cos(re) * ((1.0 + (0.5 * (im_m * im_m))) + ((im_m * (im_m * im_m)) * t_0));
	} else if (im_m <= 7.2e+51) {
		tmp = cosh(im_m);
	} else {
		tmp = cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * 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 = im_m * (0.041666666666666664d0 + ((im_m * im_m) * 0.001388888888888889d0))
    if (im_m <= 0.98d0) then
        tmp = cos(re) * ((1.0d0 + (0.5d0 * (im_m * im_m))) + ((im_m * (im_m * im_m)) * t_0))
    else if (im_m <= 7.2d+51) then
        tmp = cosh(im_m)
    else
        tmp = cos(re) * (1.0d0 + ((im_m * im_m) * (0.5d0 + (im_m * 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 = im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889));
	double tmp;
	if (im_m <= 0.98) {
		tmp = Math.cos(re) * ((1.0 + (0.5 * (im_m * im_m))) + ((im_m * (im_m * im_m)) * t_0));
	} else if (im_m <= 7.2e+51) {
		tmp = Math.cosh(im_m);
	} else {
		tmp = Math.cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * t_0))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889))
	tmp = 0
	if im_m <= 0.98:
		tmp = math.cos(re) * ((1.0 + (0.5 * (im_m * im_m))) + ((im_m * (im_m * im_m)) * t_0))
	elif im_m <= 7.2e+51:
		tmp = math.cosh(im_m)
	else:
		tmp = math.cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * t_0))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(im_m * Float64(0.041666666666666664 + Float64(Float64(im_m * im_m) * 0.001388888888888889)))
	tmp = 0.0
	if (im_m <= 0.98)
		tmp = Float64(cos(re) * Float64(Float64(1.0 + Float64(0.5 * Float64(im_m * im_m))) + Float64(Float64(im_m * Float64(im_m * im_m)) * t_0)));
	elseif (im_m <= 7.2e+51)
		tmp = cosh(im_m);
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im_m * im_m) * Float64(0.5 + Float64(im_m * t_0)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889));
	tmp = 0.0;
	if (im_m <= 0.98)
		tmp = cos(re) * ((1.0 + (0.5 * (im_m * im_m))) + ((im_m * (im_m * im_m)) * t_0));
	elseif (im_m <= 7.2e+51)
		tmp = cosh(im_m);
	else
		tmp = cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[(0.041666666666666664 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 0.98], N[(N[Cos[re], $MachinePrecision] * N[(N[(1.0 + N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 7.2e+51], N[Cosh[im$95$m], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.5 + N[(im$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.97999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 91.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 91.3%

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

   7. Applied egg-rr 91.3%

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

   if 0.97999999999999998 < im < 7.20000000000000022e51

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

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

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

      2. *-lft-identity N/A

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

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

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

      4. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 78.6%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   if 7.20000000000000022e51 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

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

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

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

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

      18. *-commutative N/A

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

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

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

   8. Simplified 100.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.98:\\ \;\;\;\;\cos re \cdot \left(\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) + \left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \cos re \cdot \left(1 + \left(im\_m \cdot im\_m\right) \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot \left(0.041666666666666664 + \left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{if}\;im\_m \leq 0.98:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0
         (*
          (cos re)
          (+
           1.0
           (*
            (* im_m im_m)
            (+
             0.5
             (*
              im_m
              (*
               im_m
               (+
                0.041666666666666664
                (* (* im_m im_m) 0.001388888888888889))))))))))
   (if (<= im_m 0.98) t_0 (if (<= im_m 7.2e+51) (cosh im_m) t_0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889)))))));
	double tmp;
	if (im_m <= 0.98) {
		tmp = t_0;
	} else if (im_m <= 7.2e+51) {
		tmp = cosh(im_m);
	} 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 = cos(re) * (1.0d0 + ((im_m * im_m) * (0.5d0 + (im_m * (im_m * (0.041666666666666664d0 + ((im_m * im_m) * 0.001388888888888889d0)))))))
    if (im_m <= 0.98d0) then
        tmp = t_0
    else if (im_m <= 7.2d+51) then
        tmp = cosh(im_m)
    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.cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889)))))));
	double tmp;
	if (im_m <= 0.98) {
		tmp = t_0;
	} else if (im_m <= 7.2e+51) {
		tmp = Math.cosh(im_m);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = math.cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889)))))))
	tmp = 0
	if im_m <= 0.98:
		tmp = t_0
	elif im_m <= 7.2e+51:
		tmp = math.cosh(im_m)
	else:
		tmp = t_0
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(cos(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))))))))
	tmp = 0.0
	if (im_m <= 0.98)
		tmp = t_0;
	elseif (im_m <= 7.2e+51)
		tmp = cosh(im_m);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889)))))));
	tmp = 0.0;
	if (im_m <= 0.98)
		tmp = t_0;
	elseif (im_m <= 7.2e+51)
		tmp = cosh(im_m);
	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[Cos[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]}, If[LessEqual[im$95$m, 0.98], t$95$0, If[LessEqual[im$95$m, 7.2e+51], N[Cosh[im$95$m], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.97999999999999998 or 7.20000000000000022e51 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 93.1%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{1} + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

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

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

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

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

      18. *-commutative N/A

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

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

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

   8. Simplified 93.1%

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

   if 0.97999999999999998 < im < 7.20000000000000022e51

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

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

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

      2. *-lft-identity N/A

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

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

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

      4. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 78.6%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.98:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.041666666666666664 \cdot \left(im\_m \cdot im\_m\right)\\ \mathbf{if}\;im\_m \leq 0.98:\\ \;\;\;\;\cos re \cdot \left(\left(1 + 0.5 \cdot \left(im\_m \cdot im\_m\right)\right) + im\_m \cdot \left(im\_m \cdot t\_0\right)\right)\\ \mathbf{elif}\;im\_m \leq 6.8 \cdot 10^{+74}:\\ \;\;\;\;\cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im\_m \cdot im\_m\right) \cdot \left(0.5 + t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.041666666666666664 (* im_m im_m))))
   (if (<= im_m 0.98)
     (* (cos re) (+ (+ 1.0 (* 0.5 (* im_m im_m))) (* im_m (* im_m t_0))))
     (if (<= im_m 6.8e+74)
       (cosh im_m)
       (* (cos re) (+ 1.0 (* (* im_m im_m) (+ 0.5 t_0))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.041666666666666664 * (im_m * im_m);
	double tmp;
	if (im_m <= 0.98) {
		tmp = cos(re) * ((1.0 + (0.5 * (im_m * im_m))) + (im_m * (im_m * t_0)));
	} else if (im_m <= 6.8e+74) {
		tmp = cosh(im_m);
	} else {
		tmp = cos(re) * (1.0 + ((im_m * im_m) * (0.5 + 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 = 0.041666666666666664d0 * (im_m * im_m)
    if (im_m <= 0.98d0) then
        tmp = cos(re) * ((1.0d0 + (0.5d0 * (im_m * im_m))) + (im_m * (im_m * t_0)))
    else if (im_m <= 6.8d+74) then
        tmp = cosh(im_m)
    else
        tmp = cos(re) * (1.0d0 + ((im_m * im_m) * (0.5d0 + 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 = 0.041666666666666664 * (im_m * im_m);
	double tmp;
	if (im_m <= 0.98) {
		tmp = Math.cos(re) * ((1.0 + (0.5 * (im_m * im_m))) + (im_m * (im_m * t_0)));
	} else if (im_m <= 6.8e+74) {
		tmp = Math.cosh(im_m);
	} else {
		tmp = Math.cos(re) * (1.0 + ((im_m * im_m) * (0.5 + t_0)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = 0.041666666666666664 * (im_m * im_m)
	tmp = 0
	if im_m <= 0.98:
		tmp = math.cos(re) * ((1.0 + (0.5 * (im_m * im_m))) + (im_m * (im_m * t_0)))
	elif im_m <= 6.8e+74:
		tmp = math.cosh(im_m)
	else:
		tmp = math.cos(re) * (1.0 + ((im_m * im_m) * (0.5 + t_0)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.041666666666666664 * Float64(im_m * im_m))
	tmp = 0.0
	if (im_m <= 0.98)
		tmp = Float64(cos(re) * Float64(Float64(1.0 + Float64(0.5 * Float64(im_m * im_m))) + Float64(im_m * Float64(im_m * t_0))));
	elseif (im_m <= 6.8e+74)
		tmp = cosh(im_m);
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(Float64(im_m * im_m) * Float64(0.5 + t_0))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = 0.041666666666666664 * (im_m * im_m);
	tmp = 0.0;
	if (im_m <= 0.98)
		tmp = cos(re) * ((1.0 + (0.5 * (im_m * im_m))) + (im_m * (im_m * t_0)));
	elseif (im_m <= 6.8e+74)
		tmp = cosh(im_m);
	else
		tmp = cos(re) * (1.0 + ((im_m * im_m) * (0.5 + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 0.98], N[(N[Cos[re], $MachinePrecision] * N[(N[(1.0 + N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * N[(im$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 6.8e+74], N[Cosh[im$95$m], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.97999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 88.4%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

      9. associate-*l* N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 88.4%

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

   7. Applied egg-rr 88.4%

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

   if 0.97999999999999998 < im < 6.7999999999999998e74

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

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

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

      2. *-lft-identity N/A

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

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

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

      4. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 81.0%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   if 6.7999999999999998e74 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 97.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.98:\\ \;\;\;\;\cos re \cdot \left(\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) + im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+74}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \cos re \cdot \left(1 + \left(im\_m \cdot im\_m\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{if}\;im\_m \leq 0.98:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 6.8 \cdot 10^{+74}:\\ \;\;\;\;\cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0
         (*
          (cos re)
          (+
           1.0
           (* (* im_m im_m) (+ 0.5 (* 0.041666666666666664 (* im_m im_m))))))))
   (if (<= im_m 0.98) t_0 (if (<= im_m 6.8e+74) (cosh im_m) t_0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	double tmp;
	if (im_m <= 0.98) {
		tmp = t_0;
	} else if (im_m <= 6.8e+74) {
		tmp = cosh(im_m);
	} 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 = cos(re) * (1.0d0 + ((im_m * im_m) * (0.5d0 + (0.041666666666666664d0 * (im_m * im_m)))))
    if (im_m <= 0.98d0) then
        tmp = t_0
    else if (im_m <= 6.8d+74) then
        tmp = cosh(im_m)
    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.cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	double tmp;
	if (im_m <= 0.98) {
		tmp = t_0;
	} else if (im_m <= 6.8e+74) {
		tmp = Math.cosh(im_m);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = math.cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m)))))
	tmp = 0
	if im_m <= 0.98:
		tmp = t_0
	elif im_m <= 6.8e+74:
		tmp = math.cosh(im_m)
	else:
		tmp = t_0
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(cos(re) * Float64(1.0 + Float64(Float64(im_m * im_m) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im_m * im_m))))))
	tmp = 0.0
	if (im_m <= 0.98)
		tmp = t_0;
	elseif (im_m <= 6.8e+74)
		tmp = cosh(im_m);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = cos(re) * (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	tmp = 0.0;
	if (im_m <= 0.98)
		tmp = t_0;
	elseif (im_m <= 6.8e+74)
		tmp = cosh(im_m);
	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[Cos[re], $MachinePrecision] * N[(1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.5 + N[(0.041666666666666664 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 0.98], t$95$0, If[LessEqual[im$95$m, 6.8e+74], N[Cosh[im$95$m], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.97999999999999998 or 6.7999999999999998e74 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 90.1%

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

   if 0.97999999999999998 < im < 6.7999999999999998e74

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

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

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

      2. *-lft-identity N/A

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

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

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

      4. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 81.0%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.98:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+74}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(im\_m \cdot im\_m + 2\right)\\ \mathbf{if}\;im\_m \leq 0.98:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\cosh im\_m\\ \mathbf{elif}\;im\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (* im_m im_m) 2.0))))
   (if (<= im_m 0.98)
     t_0
     (if (<= im_m 2e+107)
       (cosh im_m)
       (if (<= im_m 1.35e+154)
         (*
          (* im_m im_m)
          (*
           (* im_m im_m)
           (+ 0.041666666666666664 (* (* re re) -0.020833333333333332))))
         t_0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (0.5 * cos(re)) * ((im_m * im_m) + 2.0);
	double tmp;
	if (im_m <= 0.98) {
		tmp = t_0;
	} else if (im_m <= 2e+107) {
		tmp = cosh(im_m);
	} else if (im_m <= 1.35e+154) {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	} 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 = (0.5d0 * cos(re)) * ((im_m * im_m) + 2.0d0)
    if (im_m <= 0.98d0) then
        tmp = t_0
    else if (im_m <= 2d+107) then
        tmp = cosh(im_m)
    else if (im_m <= 1.35d+154) then
        tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0))))
    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 = (0.5 * Math.cos(re)) * ((im_m * im_m) + 2.0);
	double tmp;
	if (im_m <= 0.98) {
		tmp = t_0;
	} else if (im_m <= 2e+107) {
		tmp = Math.cosh(im_m);
	} else if (im_m <= 1.35e+154) {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = (0.5 * math.cos(re)) * ((im_m * im_m) + 2.0)
	tmp = 0
	if im_m <= 0.98:
		tmp = t_0
	elif im_m <= 2e+107:
		tmp = math.cosh(im_m)
	elif im_m <= 1.35e+154:
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)))
	else:
		tmp = t_0
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(Float64(im_m * im_m) + 2.0))
	tmp = 0.0
	if (im_m <= 0.98)
		tmp = t_0;
	elseif (im_m <= 2e+107)
		tmp = cosh(im_m);
	elseif (im_m <= 1.35e+154)
		tmp = Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = (0.5 * cos(re)) * ((im_m * im_m) + 2.0);
	tmp = 0.0;
	if (im_m <= 0.98)
		tmp = t_0;
	elseif (im_m <= 2e+107)
		tmp = cosh(im_m);
	elseif (im_m <= 1.35e+154)
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	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[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 0.98], t$95$0, If[LessEqual[im$95$m, 2e+107], N[Cosh[im$95$m], $MachinePrecision], If[LessEqual[im$95$m, 1.35e+154], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.97999999999999998 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 82.1%

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

   5. Simplified 82.1%

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

   if 0.97999999999999998 < im < 1.9999999999999999e107

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

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

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

      2. *-lft-identity N/A

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

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

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

      4. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 82.8%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   if 1.9999999999999999e107 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 83.3%

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

   8. Simplified 83.3%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

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

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

       15. unpow2 N/A

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

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

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

       17. distribute-rgt-in N/A

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

       18. metadata-eval N/A

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

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

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

       20. *-commutative N/A

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

       21. associate-*l* N/A

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

       22. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{48}\right)\right)\right)\right) \]

       23. metadata-eval N/A

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

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

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

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.98:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\cosh im\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 0.0152:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im\_m \leq 4.1 \cdot 10^{+107}:\\ \;\;\;\;\cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 0.0152)
   (cos re)
   (if (<= im_m 4.1e+107)
     (cosh im_m)
     (*
      (* im_m im_m)
      (*
       (* im_m im_m)
       (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 0.0152) {
		tmp = cos(re);
	} else if (im_m <= 4.1e+107) {
		tmp = cosh(im_m);
	} else {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	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 <= 0.0152d0) then
        tmp = cos(re)
    else if (im_m <= 4.1d+107) then
        tmp = cosh(im_m)
    else
        tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0))))
    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 <= 0.0152) {
		tmp = Math.cos(re);
	} else if (im_m <= 4.1e+107) {
		tmp = Math.cosh(im_m);
	} else {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 0.0152:
		tmp = math.cos(re)
	elif im_m <= 4.1e+107:
		tmp = math.cosh(im_m)
	else:
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 0.0152)
		tmp = cos(re);
	elseif (im_m <= 4.1e+107)
		tmp = cosh(im_m);
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 0.0152)
		tmp = cos(re);
	elseif (im_m <= 4.1e+107)
		tmp = cosh(im_m);
	else
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 0.0152], N[Cos[re], $MachinePrecision], If[LessEqual[im$95$m, 4.1e+107], N[Cosh[im$95$m], $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0152

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 60.6%

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

   5. Simplified 60.6%

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

   if 0.0152 < im < 4.0999999999999999e107

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. cos-lowering-cos.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

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

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

      2. *-lft-identity N/A

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

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

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

      4. cos-lowering-cos.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 79.5%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   if 4.0999999999999999e107 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 78.8%

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

   8. Simplified 78.8%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

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

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

       15. unpow2 N/A

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

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

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

       17. distribute-rgt-in N/A

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

       18. metadata-eval N/A

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

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

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

       20. *-commutative N/A

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

       21. associate-*l* N/A

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

       22. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{48}\right)\right)\right)\right) \]

       23. metadata-eval N/A

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

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

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

   11. Simplified 78.8%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0152:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+107}:\\ \;\;\;\;\cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 145000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im\_m \leq 2.05 \cdot 10^{+71}:\\ \;\;\;\;\left(1 + \left(im\_m \cdot im\_m\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im\_m \cdot im\_m\right)\right)\right) \cdot \left(\left(1 - 0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right) \cdot \frac{2}{re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 145000.0)
   (cos re)
   (if (<= im_m 2.05e+71)
     (*
      (+ 1.0 (* (* im_m im_m) (+ 0.5 (* 0.041666666666666664 (* im_m im_m)))))
      (* (- 1.0 (* 0.25 (* (* re re) (* re re)))) (/ 2.0 (* re re))))
     (*
      (* im_m im_m)
      (*
       (* im_m im_m)
       (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 145000.0) {
		tmp = cos(re);
	} else if (im_m <= 2.05e+71) {
		tmp = (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m))))) * ((1.0 - (0.25 * ((re * re) * (re * re)))) * (2.0 / (re * re)));
	} else {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	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 <= 145000.0d0) then
        tmp = cos(re)
    else if (im_m <= 2.05d+71) then
        tmp = (1.0d0 + ((im_m * im_m) * (0.5d0 + (0.041666666666666664d0 * (im_m * im_m))))) * ((1.0d0 - (0.25d0 * ((re * re) * (re * re)))) * (2.0d0 / (re * re)))
    else
        tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0))))
    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 <= 145000.0) {
		tmp = Math.cos(re);
	} else if (im_m <= 2.05e+71) {
		tmp = (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m))))) * ((1.0 - (0.25 * ((re * re) * (re * re)))) * (2.0 / (re * re)));
	} else {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 145000.0:
		tmp = math.cos(re)
	elif im_m <= 2.05e+71:
		tmp = (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m))))) * ((1.0 - (0.25 * ((re * re) * (re * re)))) * (2.0 / (re * re)))
	else:
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 145000.0)
		tmp = cos(re);
	elseif (im_m <= 2.05e+71)
		tmp = Float64(Float64(1.0 + Float64(Float64(im_m * im_m) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im_m * im_m))))) * Float64(Float64(1.0 - Float64(0.25 * Float64(Float64(re * re) * Float64(re * re)))) * Float64(2.0 / Float64(re * re))));
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 145000.0)
		tmp = cos(re);
	elseif (im_m <= 2.05e+71)
		tmp = (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m))))) * ((1.0 - (0.25 * ((re * re) * (re * re)))) * (2.0 / (re * re)));
	else
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 145000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im$95$m, 2.05e+71], N[(N[(1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.5 + N[(0.041666666666666664 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.25 * N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 145000

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. cos-lowering-cos.f64 59.8%

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

   5. Simplified 59.8%

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

   if 145000 < im < 2.0500000000000001e71

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 5.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 18.7%

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

   8. Simplified 18.7%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. swap-sqr N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

      13. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(re \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(re \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      15. metadata-eval N/A

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

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

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

      17. *-lowering-*.f64 8.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), 1\right)\right) \]

   10. Applied egg-rr 8.2%

       \[\leadsto \color{blue}{\left(\left(1 - 0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right) \cdot \frac{1}{1 + 0.5 \cdot \left(re \cdot re\right)}\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   11. Taylor expanded in re around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 53.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   13. Simplified 53.0%

       \[\leadsto \left(\left(1 - 0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right) \cdot \color{blue}{\frac{2}{re \cdot re}}\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   if 2.0500000000000001e71 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

   5. Simplified 95.7%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 76.2%

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

   8. Simplified 76.2%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       7. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left({im}^{2} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       17. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(1 \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{24}}\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)} \cdot \frac{1}{24}\right)\right)\right) \]

       19. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{24}\right)}\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left({re}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{48}\right)\right)\right)\right) \]

       23. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

       24. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{2}\right)}\right)\right)\right)\right) \]

   11. Simplified 76.2%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 145000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{+71}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(1 - 0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right) \cdot \frac{2}{re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\\ t_1 := 0.041666666666666664 + im\_m \cdot \left(im\_m \cdot -0.001388888888888889\right)\\ t_2 := 0.5 \cdot \left(im\_m \cdot im\_m\right) + -1\\ \mathbf{if}\;im\_m \leq 2.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{t\_1 \cdot \left(-1 + 0.25 \cdot t\_0\right) + t\_0 \cdot \left(\left(0.001736111111111111 + -1.9290123456790124 \cdot 10^{-6} \cdot t\_0\right) \cdot t\_2\right)}{t\_1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* im_m im_m) (* im_m im_m)))
        (t_1 (+ 0.041666666666666664 (* im_m (* im_m -0.001388888888888889))))
        (t_2 (+ (* 0.5 (* im_m im_m)) -1.0)))
   (if (<= im_m 2.05e+71)
     (/
      (+
       (* t_1 (+ -1.0 (* 0.25 t_0)))
       (* t_0 (* (+ 0.001736111111111111 (* -1.9290123456790124e-6 t_0)) t_2)))
      (* t_1 t_2))
     (*
      (* im_m im_m)
      (*
       (* im_m im_m)
       (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (im_m * im_m) * (im_m * im_m);
	double t_1 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889));
	double t_2 = (0.5 * (im_m * im_m)) + -1.0;
	double tmp;
	if (im_m <= 2.05e+71) {
		tmp = ((t_1 * (-1.0 + (0.25 * t_0))) + (t_0 * ((0.001736111111111111 + (-1.9290123456790124e-6 * t_0)) * t_2))) / (t_1 * t_2);
	} else {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (im_m * im_m) * (im_m * im_m)
    t_1 = 0.041666666666666664d0 + (im_m * (im_m * (-0.001388888888888889d0)))
    t_2 = (0.5d0 * (im_m * im_m)) + (-1.0d0)
    if (im_m <= 2.05d+71) then
        tmp = ((t_1 * ((-1.0d0) + (0.25d0 * t_0))) + (t_0 * ((0.001736111111111111d0 + ((-1.9290123456790124d-6) * t_0)) * t_2))) / (t_1 * t_2)
    else
        tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double t_0 = (im_m * im_m) * (im_m * im_m);
	double t_1 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889));
	double t_2 = (0.5 * (im_m * im_m)) + -1.0;
	double tmp;
	if (im_m <= 2.05e+71) {
		tmp = ((t_1 * (-1.0 + (0.25 * t_0))) + (t_0 * ((0.001736111111111111 + (-1.9290123456790124e-6 * t_0)) * t_2))) / (t_1 * t_2);
	} else {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = (im_m * im_m) * (im_m * im_m)
	t_1 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889))
	t_2 = (0.5 * (im_m * im_m)) + -1.0
	tmp = 0
	if im_m <= 2.05e+71:
		tmp = ((t_1 * (-1.0 + (0.25 * t_0))) + (t_0 * ((0.001736111111111111 + (-1.9290123456790124e-6 * t_0)) * t_2))) / (t_1 * t_2)
	else:
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(im_m * im_m) * Float64(im_m * im_m))
	t_1 = Float64(0.041666666666666664 + Float64(im_m * Float64(im_m * -0.001388888888888889)))
	t_2 = Float64(Float64(0.5 * Float64(im_m * im_m)) + -1.0)
	tmp = 0.0
	if (im_m <= 2.05e+71)
		tmp = Float64(Float64(Float64(t_1 * Float64(-1.0 + Float64(0.25 * t_0))) + Float64(t_0 * Float64(Float64(0.001736111111111111 + Float64(-1.9290123456790124e-6 * t_0)) * t_2))) / Float64(t_1 * t_2));
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = (im_m * im_m) * (im_m * im_m);
	t_1 = 0.041666666666666664 + (im_m * (im_m * -0.001388888888888889));
	t_2 = (0.5 * (im_m * im_m)) + -1.0;
	tmp = 0.0;
	if (im_m <= 2.05e+71)
		tmp = ((t_1 * (-1.0 + (0.25 * t_0))) + (t_0 * ((0.001736111111111111 + (-1.9290123456790124e-6 * t_0)) * t_2))) / (t_1 * t_2);
	else
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.041666666666666664 + N[(im$95$m * N[(im$95$m * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[im$95$m, 2.05e+71], N[(N[(N[(t$95$1 * N[(-1.0 + N[(0.25 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(0.001736111111111111 + N[(-1.9290123456790124e-6 * t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if im < 2.0500000000000001e71

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 86.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   7. Applied egg-rr 66.4%

      \[\leadsto \cos re \cdot \color{blue}{\frac{\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25 - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right) + \left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(\left(0.001736111111111111 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 1.9290123456790124 \cdot 10^{-6}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)}{\left(0.5 \cdot \left(im \cdot im\right) - 1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(im \cdot im\right)\right)}} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot \left(\frac{1}{4} \cdot {im}^{4} - 1\right) + {im}^{4} \cdot \left(\left(\frac{1}{576} - \frac{1}{518400} \cdot {im}^{4}\right) \cdot \left(\frac{1}{2} \cdot {im}^{2} - 1\right)\right)}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot \left(\frac{1}{2} \cdot {im}^{2} - 1\right)}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot \left(\frac{1}{4} \cdot {im}^{4} - 1\right) + {im}^{4} \cdot \left(\left(\frac{1}{576} - \frac{1}{518400} \cdot {im}^{4}\right) \cdot \left(\frac{1}{2} \cdot {im}^{2} - 1\right)\right)\right), \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot \left(\frac{1}{2} \cdot {im}^{2} - 1\right)\right)}\right) \]

   10. Simplified 40.4%

       \[\leadsto \color{blue}{\frac{\left(0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.25 + -1\right) + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(0.001736111111111111 + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -1.9290123456790124 \cdot 10^{-6}\right) \cdot \left(0.5 \cdot \left(im \cdot im\right) + -1\right)\right)}{\left(0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right) + -1\right)}} \]

   if 2.0500000000000001e71 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 95.7%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 76.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 76.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       7. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left({im}^{2} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       17. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(1 \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{24}}\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)} \cdot \frac{1}{24}\right)\right)\right) \]

       19. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{24}\right)}\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left({re}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{48}\right)\right)\right)\right) \]

       23. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

       24. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{2}\right)}\right)\right)\right)\right) \]

   11. Simplified 76.2%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)\right) \cdot \left(-1 + 0.25 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(0.001736111111111111 + -1.9290123456790124 \cdot 10^{-6} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right) + -1\right)\right)}{\left(0.041666666666666664 + im \cdot \left(im \cdot -0.001388888888888889\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 145000:\\ \;\;\;\;1 + \left(im\_m \cdot im\_m\right) \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot \left(0.041666666666666664 + \left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 2.05 \cdot 10^{+71}:\\ \;\;\;\;\left(1 + \left(im\_m \cdot im\_m\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im\_m \cdot im\_m\right)\right)\right) \cdot \left(\left(1 - 0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right) \cdot \frac{2}{re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 145000.0)
   (+
    1.0
    (*
     (* im_m im_m)
     (+
      0.5
      (*
       im_m
       (*
        im_m
        (+ 0.041666666666666664 (* (* im_m im_m) 0.001388888888888889)))))))
   (if (<= im_m 2.05e+71)
     (*
      (+ 1.0 (* (* im_m im_m) (+ 0.5 (* 0.041666666666666664 (* im_m im_m)))))
      (* (- 1.0 (* 0.25 (* (* re re) (* re re)))) (/ 2.0 (* re re))))
     (*
      (* im_m im_m)
      (*
       (* im_m im_m)
       (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 145000.0) {
		tmp = 1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889))))));
	} else if (im_m <= 2.05e+71) {
		tmp = (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m))))) * ((1.0 - (0.25 * ((re * re) * (re * re)))) * (2.0 / (re * re)));
	} else {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	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 <= 145000.0d0) then
        tmp = 1.0d0 + ((im_m * im_m) * (0.5d0 + (im_m * (im_m * (0.041666666666666664d0 + ((im_m * im_m) * 0.001388888888888889d0))))))
    else if (im_m <= 2.05d+71) then
        tmp = (1.0d0 + ((im_m * im_m) * (0.5d0 + (0.041666666666666664d0 * (im_m * im_m))))) * ((1.0d0 - (0.25d0 * ((re * re) * (re * re)))) * (2.0d0 / (re * re)))
    else
        tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0))))
    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 <= 145000.0) {
		tmp = 1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889))))));
	} else if (im_m <= 2.05e+71) {
		tmp = (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m))))) * ((1.0 - (0.25 * ((re * re) * (re * re)))) * (2.0 / (re * re)));
	} else {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 145000.0:
		tmp = 1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889))))))
	elif im_m <= 2.05e+71:
		tmp = (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m))))) * ((1.0 - (0.25 * ((re * re) * (re * re)))) * (2.0 / (re * re)))
	else:
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 145000.0)
		tmp = 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)))))));
	elseif (im_m <= 2.05e+71)
		tmp = Float64(Float64(1.0 + Float64(Float64(im_m * im_m) * Float64(0.5 + Float64(0.041666666666666664 * Float64(im_m * im_m))))) * Float64(Float64(1.0 - Float64(0.25 * Float64(Float64(re * re) * Float64(re * re)))) * Float64(2.0 / Float64(re * re))));
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 145000.0)
		tmp = 1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889))))));
	elseif (im_m <= 2.05e+71)
		tmp = (1.0 + ((im_m * im_m) * (0.5 + (0.041666666666666664 * (im_m * im_m))))) * ((1.0 - (0.25 * ((re * re) * (re * re)))) * (2.0 / (re * re)));
	else
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 145000.0], 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], If[LessEqual[im$95$m, 2.05e+71], N[(N[(1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.5 + N[(0.041666666666666664 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.25 * N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 145000

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 90.9%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2} + {im}^{\left(2 \cdot 2\right)} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{2} \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2} + {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot \color{blue}{{im}^{2}}\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 58.3%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 58.3%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)} \]

   if 145000 < im < 2.0500000000000001e71

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 5.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 18.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 18.7%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]
   9. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 \cdot 1 - \left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right)}{1 - \frac{-1}{2} \cdot \left(re \cdot re\right)}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(1 \cdot 1 - \left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right) \cdot \frac{1}{1 - \frac{-1}{2} \cdot \left(re \cdot re\right)}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(1 \cdot 1 - \left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right), \left(\frac{1}{1 - \frac{-1}{2} \cdot \left(re \cdot re\right)}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(1 - \left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right), \left(\frac{1}{1 - \frac{-1}{2} \cdot \left(re \cdot re\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{im}, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right)\right), \left(\frac{1}{1 - \frac{-1}{2} \cdot \left(re \cdot re\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(im, im\right)}, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      6. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)\right), \left(\frac{1}{1 - \frac{-1}{2} \cdot \left(re \cdot re\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \color{blue}{im}\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1}{4} \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)\right), \left(\frac{1}{1 - \frac{-1}{2} \cdot \left(re \cdot re\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right)\right), \left(\frac{1}{1 - \frac{-1}{2} \cdot \left(re \cdot re\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \color{blue}{im}\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot re\right)\right)\right)\right), \left(\frac{1}{1 - \frac{-1}{2} \cdot \left(re \cdot re\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot re\right)\right)\right)\right), \left(\frac{1}{1 - \frac{-1}{2} \cdot \left(re \cdot re\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \left(\frac{1}{1 - \frac{-1}{2} \cdot \left(re \cdot re\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 - \frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      13. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(re \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(re \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(re \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\color{blue}{im}, im\right)\right)\right)\right), 1\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\mathsf{*.f64}\left(im, im\right)}\right)\right)\right), 1\right)\right) \]

      17. *-lowering-*.f64 8.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right), 1\right)\right) \]

   10. Applied egg-rr 8.2%

       \[\leadsto \color{blue}{\left(\left(1 - 0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right) \cdot \frac{1}{1 + 0.5 \cdot \left(re \cdot re\right)}\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \color{blue}{\left(\frac{2}{{re}^{2}}\right)}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(2, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(2, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

       3. *-lowering-*.f64 53.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, re\right)\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   13. Simplified 53.0%

       \[\leadsto \left(\left(1 - 0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right) \cdot \color{blue}{\frac{2}{re \cdot re}}\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   if 2.0500000000000001e71 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 95.7%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 76.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 76.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       7. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left({im}^{2} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       17. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(1 \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{24}}\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)} \cdot \frac{1}{24}\right)\right)\right) \]

       19. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{24}\right)}\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left({re}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{48}\right)\right)\right)\right) \]

       23. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

       24. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{2}\right)}\right)\right)\right)\right) \]

   11. Simplified 76.2%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 145000:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{+71}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(1 - 0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\right) \cdot \frac{2}{re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.3% accurate, 10.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 1 + \left(im\_m \cdot im\_m\right) \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot \left(0.041666666666666664 + \left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{if}\;re \leq 4.4 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+198}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (*
           (* im_m im_m)
           (+
            0.5
            (*
             im_m
             (*
              im_m
              (+
               0.041666666666666664
               (* (* im_m im_m) 0.001388888888888889)))))))))
   (if (<= re 4.4e+101)
     t_0
     (if (<= re 2.1e+198)
       (*
        (* im_m im_m)
        (*
         (* im_m im_m)
         (+ 0.041666666666666664 (* (* re re) -0.020833333333333332))))
       t_0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889))))));
	double tmp;
	if (re <= 4.4e+101) {
		tmp = t_0;
	} else if (re <= 2.1e+198) {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	} 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 = 1.0d0 + ((im_m * im_m) * (0.5d0 + (im_m * (im_m * (0.041666666666666664d0 + ((im_m * im_m) * 0.001388888888888889d0))))))
    if (re <= 4.4d+101) then
        tmp = t_0
    else if (re <= 2.1d+198) then
        tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0))))
    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 = 1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889))))));
	double tmp;
	if (re <= 4.4e+101) {
		tmp = t_0;
	} else if (re <= 2.1e+198) {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = 1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889))))))
	tmp = 0
	if re <= 4.4e+101:
		tmp = t_0
	elif re <= 2.1e+198:
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)))
	else:
		tmp = t_0
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = 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)))))))
	tmp = 0.0
	if (re <= 4.4e+101)
		tmp = t_0;
	elseif (re <= 2.1e+198)
		tmp = Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = 1.0 + ((im_m * im_m) * (0.5 + (im_m * (im_m * (0.041666666666666664 + ((im_m * im_m) * 0.001388888888888889))))));
	tmp = 0.0;
	if (re <= 4.4e+101)
		tmp = t_0;
	elseif (re <= 2.1e+198)
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	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[(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]}, If[LessEqual[re, 4.4e+101], t$95$0, If[LessEqual[re, 2.1e+198], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < 4.4000000000000001e101 or 2.10000000000000013e198 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\cos re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \cos re \]

      3. distribute-lft-in N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + \cos \color{blue}{re} \]

      4. associate-+l+ N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \cos re\right)} \]

   5. Simplified 88.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2} + {im}^{\left(2 \cdot 2\right)} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{2} \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2} + {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot \color{blue}{{im}^{2}}\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 61.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 61.2%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)} \]

   if 4.4000000000000001e101 < re < 2.10000000000000013e198

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 73.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 39.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 39.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       7. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left({im}^{2} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       17. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(1 \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{24}}\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)} \cdot \frac{1}{24}\right)\right)\right) \]

       19. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{24}\right)}\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left({re}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{48}\right)\right)\right)\right) \]

       23. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

       24. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{2}\right)}\right)\right)\right)\right) \]

   11. Simplified 39.3%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 58.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 410:\\ \;\;\;\;1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 3.3 \cdot 10^{+24}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 410.0)
   (+ 1.0 (* im_m (* im_m (+ 0.5 (* 0.041666666666666664 (* im_m im_m))))))
   (if (<= im_m 3.3e+24)
     (+ 1.0 (* (* re re) (+ -0.5 (* re (* re 0.041666666666666664)))))
     (*
      (* im_m im_m)
      (*
       (* im_m im_m)
       (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 410.0) {
		tmp = 1.0 + (im_m * (im_m * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	} else if (im_m <= 3.3e+24) {
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))));
	} else {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	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 <= 410.0d0) then
        tmp = 1.0d0 + (im_m * (im_m * (0.5d0 + (0.041666666666666664d0 * (im_m * im_m)))))
    else if (im_m <= 3.3d+24) then
        tmp = 1.0d0 + ((re * re) * ((-0.5d0) + (re * (re * 0.041666666666666664d0))))
    else
        tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0))))
    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 <= 410.0) {
		tmp = 1.0 + (im_m * (im_m * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	} else if (im_m <= 3.3e+24) {
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))));
	} else {
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 410.0:
		tmp = 1.0 + (im_m * (im_m * (0.5 + (0.041666666666666664 * (im_m * im_m)))))
	elif im_m <= 3.3e+24:
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))))
	else:
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 410.0)
		tmp = Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(0.041666666666666664 * Float64(im_m * im_m))))));
	elseif (im_m <= 3.3e+24)
		tmp = Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(re * Float64(re * 0.041666666666666664)))));
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 410.0)
		tmp = 1.0 + (im_m * (im_m * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	elseif (im_m <= 3.3e+24)
		tmp = 1.0 + ((re * re) * (-0.5 + (re * (re * 0.041666666666666664))));
	else
		tmp = (im_m * im_m) * ((im_m * im_m) * (0.041666666666666664 + ((re * re) * -0.020833333333333332)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 410.0], N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(0.041666666666666664 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.3e+24], N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(re * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 410

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 88.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 55.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 55.7%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

   if 410 < im < 3.2999999999999999e24

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{24} \cdot {re}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} \cdot {re}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {re}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {re}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(re \cdot re\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \color{blue}{\left(re \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 44.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 44.7%

      \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot \left(-0.5 + re \cdot \left(re \cdot 0.041666666666666664\right)\right)} \]

   if 3.2999999999999999e24 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 75.7%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 65.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot 2\right)}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       7. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\left({im}^{2} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)\right) \]

       17. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(1 \cdot \frac{1}{24} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{24}}\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} + \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)} \cdot \frac{1}{24}\right)\right)\right) \]

       19. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\left(\frac{-1}{2} \cdot {re}^{2}\right) \cdot \frac{1}{24}\right)}\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left({re}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{48}\right)\right)\right)\right) \]

       23. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \left(\frac{1}{24} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

       24. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{2}\right)}\right)\right)\right)\right) \]

   11. Simplified 65.6%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 56.6% accurate, 14.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{+101}:\\ \;\;\;\;1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+198}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;1 - re \cdot \left(re \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 4.4e+101)
   (+ 1.0 (* im_m (* im_m (+ 0.5 (* 0.041666666666666664 (* im_m im_m))))))
   (if (<= re 2.1e+198)
     (* (* im_m im_m) (+ 0.5 (* (* re re) -0.25)))
     (- 1.0 (* re (* re -0.5))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 4.4e+101) {
		tmp = 1.0 + (im_m * (im_m * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	} else if (re <= 2.1e+198) {
		tmp = (im_m * im_m) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 1.0 - (re * (re * -0.5));
	}
	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 <= 4.4d+101) then
        tmp = 1.0d0 + (im_m * (im_m * (0.5d0 + (0.041666666666666664d0 * (im_m * im_m)))))
    else if (re <= 2.1d+198) then
        tmp = (im_m * im_m) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = 1.0d0 - (re * (re * (-0.5d0)))
    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 <= 4.4e+101) {
		tmp = 1.0 + (im_m * (im_m * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	} else if (re <= 2.1e+198) {
		tmp = (im_m * im_m) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 1.0 - (re * (re * -0.5));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 4.4e+101:
		tmp = 1.0 + (im_m * (im_m * (0.5 + (0.041666666666666664 * (im_m * im_m)))))
	elif re <= 2.1e+198:
		tmp = (im_m * im_m) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = 1.0 - (re * (re * -0.5))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 4.4e+101)
		tmp = Float64(1.0 + Float64(im_m * Float64(im_m * Float64(0.5 + Float64(0.041666666666666664 * Float64(im_m * im_m))))));
	elseif (re <= 2.1e+198)
		tmp = Float64(Float64(im_m * im_m) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = Float64(1.0 - Float64(re * Float64(re * -0.5)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 4.4e+101)
		tmp = 1.0 + (im_m * (im_m * (0.5 + (0.041666666666666664 * (im_m * im_m)))));
	elseif (re <= 2.1e+198)
		tmp = (im_m * im_m) * (0.5 + ((re * re) * -0.25));
	else
		tmp = 1.0 - (re * (re * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 4.4e+101], N[(1.0 + N[(im$95$m * N[(im$95$m * N[(0.5 + N[(0.041666666666666664 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+198], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 4.4000000000000001e101

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 83.4%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 59.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 59.8%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

   if 4.4000000000000001e101 < re < 2.10000000000000013e198

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 73.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 39.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 39.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{2} \cdot {re}^{2} + \frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       3. distribute-rgt1-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       4. lft-mult-inverse N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + \frac{1}{{im}^{2}} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{{im}^{2}} + \frac{1}{2}\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{im}^{2}} \cdot {im}^{2} + \frac{1}{2} \cdot {im}^{2}\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       9. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right)\right) \]

       18. *-lowering-*.f64 39.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right)\right) \]

   11. Simplified 39.2%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       3. associate-*l* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       7. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} \cdot 1 + \color{blue}{\frac{1}{2} \cdot \left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + \color{blue}{\frac{1}{2}} \cdot \left(\frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{-1}{2}\right) \cdot \color{blue}{{re}^{2}}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + \frac{-1}{4} \cdot {\color{blue}{re}}^{2}\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2}\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 39.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   14. Simplified 39.3%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   if 2.10000000000000013e198 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 87.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 11.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 11.6%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right) \]

       5. *-lowering-*.f64 11.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right) \]

   11. Simplified 11.6%

       \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot -0.5} \]

   13. Applied egg-rr 33.4%

       \[\leadsto \color{blue}{1 - re \cdot \left(re \cdot -0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{+101}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+198}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;1 - re \cdot \left(re \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.6% accurate, 14.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 430:\\ \;\;\;\;1 + 0.5 \cdot \left(im\_m \cdot im\_m\right)\\ \mathbf{elif}\;im\_m \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;1 - re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 430.0)
   (+ 1.0 (* 0.5 (* im_m im_m)))
   (if (<= im_m 2.6e+24)
     (- 1.0 (* re (* re -0.5)))
     (* (* im_m im_m) (+ 0.5 (* (* re re) -0.25))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 430.0) {
		tmp = 1.0 + (0.5 * (im_m * im_m));
	} else if (im_m <= 2.6e+24) {
		tmp = 1.0 - (re * (re * -0.5));
	} else {
		tmp = (im_m * im_m) * (0.5 + ((re * re) * -0.25));
	}
	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 <= 430.0d0) then
        tmp = 1.0d0 + (0.5d0 * (im_m * im_m))
    else if (im_m <= 2.6d+24) then
        tmp = 1.0d0 - (re * (re * (-0.5d0)))
    else
        tmp = (im_m * im_m) * (0.5d0 + ((re * re) * (-0.25d0)))
    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 <= 430.0) {
		tmp = 1.0 + (0.5 * (im_m * im_m));
	} else if (im_m <= 2.6e+24) {
		tmp = 1.0 - (re * (re * -0.5));
	} else {
		tmp = (im_m * im_m) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 430.0:
		tmp = 1.0 + (0.5 * (im_m * im_m))
	elif im_m <= 2.6e+24:
		tmp = 1.0 - (re * (re * -0.5))
	else:
		tmp = (im_m * im_m) * (0.5 + ((re * re) * -0.25))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 430.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im_m * im_m)));
	elseif (im_m <= 2.6e+24)
		tmp = Float64(1.0 - Float64(re * Float64(re * -0.5)));
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 430.0)
		tmp = 1.0 + (0.5 * (im_m * im_m));
	elseif (im_m <= 2.6e+24)
		tmp = 1.0 - (re * (re * -0.5));
	else
		tmp = (im_m * im_m) * (0.5 + ((re * re) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 430.0], N[(1.0 + N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.6e+24], N[(1.0 - N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 430

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 88.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 56.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 56.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{2} \cdot {re}^{2} + \frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       3. distribute-rgt1-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       4. lft-mult-inverse N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + \frac{1}{{im}^{2}} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{{im}^{2}} + \frac{1}{2}\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{im}^{2}} \cdot {im}^{2} + \frac{1}{2} \cdot {im}^{2}\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       9. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right)\right) \]

       18. *-lowering-*.f64 50.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right)\right) \]

   11. Simplified 50.4%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       4. *-lowering-*.f64 49.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   14. Simplified 49.9%

       \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   if 430 < im < 2.5999999999999998e24

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 3.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 1.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 1.6%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right) \]

       5. *-lowering-*.f64 1.5%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right) \]

   11. Simplified 1.5%

       \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot -0.5} \]

   13. Applied egg-rr 31.2%

       \[\leadsto \color{blue}{1 - re \cdot \left(re \cdot -0.5\right)} \]

   if 2.5999999999999998e24 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 75.7%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 65.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{2} \cdot {re}^{2} + \frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       3. distribute-rgt1-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       4. lft-mult-inverse N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + \frac{1}{{im}^{2}} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{{im}^{2}} + \frac{1}{2}\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{im}^{2}} \cdot {im}^{2} + \frac{1}{2} \cdot {im}^{2}\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       9. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right)\right) \]

       18. *-lowering-*.f64 48.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right)\right) \]

   11. Simplified 48.1%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       3. associate-*l* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       7. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} \cdot 1 + \color{blue}{\frac{1}{2} \cdot \left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + \color{blue}{\frac{1}{2}} \cdot \left(\frac{-1}{2} \cdot {re}^{2}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{-1}{2}\right) \cdot \color{blue}{{re}^{2}}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + \frac{-1}{4} \cdot {\color{blue}{re}}^{2}\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2}\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 48.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   14. Simplified 48.1%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 430:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;1 - re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 48.5% accurate, 14.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 4.2 \cdot 10^{+33}:\\ \;\;\;\;1 + 0.5 \cdot \left(im\_m \cdot im\_m\right)\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+198}:\\ \;\;\;\;re \cdot \left(re \cdot \left(-0.5 + \left(im\_m \cdot im\_m\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - re \cdot \left(re \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 4.2e+33)
   (+ 1.0 (* 0.5 (* im_m im_m)))
   (if (<= re 2.1e+198)
     (* re (* re (+ -0.5 (* (* im_m im_m) -0.25))))
     (- 1.0 (* re (* re -0.5))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 4.2e+33) {
		tmp = 1.0 + (0.5 * (im_m * im_m));
	} else if (re <= 2.1e+198) {
		tmp = re * (re * (-0.5 + ((im_m * im_m) * -0.25)));
	} else {
		tmp = 1.0 - (re * (re * -0.5));
	}
	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 <= 4.2d+33) then
        tmp = 1.0d0 + (0.5d0 * (im_m * im_m))
    else if (re <= 2.1d+198) then
        tmp = re * (re * ((-0.5d0) + ((im_m * im_m) * (-0.25d0))))
    else
        tmp = 1.0d0 - (re * (re * (-0.5d0)))
    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 <= 4.2e+33) {
		tmp = 1.0 + (0.5 * (im_m * im_m));
	} else if (re <= 2.1e+198) {
		tmp = re * (re * (-0.5 + ((im_m * im_m) * -0.25)));
	} else {
		tmp = 1.0 - (re * (re * -0.5));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 4.2e+33:
		tmp = 1.0 + (0.5 * (im_m * im_m))
	elif re <= 2.1e+198:
		tmp = re * (re * (-0.5 + ((im_m * im_m) * -0.25)))
	else:
		tmp = 1.0 - (re * (re * -0.5))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 4.2e+33)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im_m * im_m)));
	elseif (re <= 2.1e+198)
		tmp = Float64(re * Float64(re * Float64(-0.5 + Float64(Float64(im_m * im_m) * -0.25))));
	else
		tmp = Float64(1.0 - Float64(re * Float64(re * -0.5)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 4.2e+33)
		tmp = 1.0 + (0.5 * (im_m * im_m));
	elseif (re <= 2.1e+198)
		tmp = re * (re * (-0.5 + ((im_m * im_m) * -0.25)));
	else
		tmp = 1.0 - (re * (re * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 4.2e+33], N[(1.0 + N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+198], N[(re * N[(re * N[(-0.5 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 4.2000000000000001e33

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 83.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 67.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 67.4%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{2} \cdot {re}^{2} + \frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       3. distribute-rgt1-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       4. lft-mult-inverse N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + \frac{1}{{im}^{2}} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{{im}^{2}} + \frac{1}{2}\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{im}^{2}} \cdot {im}^{2} + \frac{1}{2} \cdot {im}^{2}\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       9. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right)\right) \]

       18. *-lowering-*.f64 57.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right)\right) \]

   11. Simplified 57.4%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       4. *-lowering-*.f64 52.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   14. Simplified 52.0%

       \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   if 4.2000000000000001e33 < re < 2.10000000000000013e198

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 28.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 28.1%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{2} \cdot {re}^{2} + \frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       3. distribute-rgt1-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       4. lft-mult-inverse N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + \frac{1}{{im}^{2}} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{{im}^{2}} + \frac{1}{2}\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{im}^{2}} \cdot {im}^{2} + \frac{1}{2} \cdot {im}^{2}\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       9. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right)\right) \]

       18. *-lowering-*.f64 25.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right)\right) \]

   11. Simplified 25.0%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)} \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\frac{-1}{2}} \]

       2. associate-*r* N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) \cdot \frac{-1}{2}\right)} \]

       3. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\color{blue}{\frac{-1}{2}} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)}\right)\right) \]

       8. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} \cdot 1 + \color{blue}{\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{2} + \color{blue}{\frac{-1}{2}} \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)\right)\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left({im}^{2} \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{-1}{2}\right)}\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{-1}{4}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{4}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 25.0%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{4}\right)\right)\right)\right) \]

   14. Simplified 25.0%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(-0.5 + \left(im \cdot im\right) \cdot -0.25\right)\right)} \]

   if 2.10000000000000013e198 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 87.6%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 11.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 11.6%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right) \]

       5. *-lowering-*.f64 11.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right) \]

   11. Simplified 11.6%

       \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot -0.5} \]

   13. Applied egg-rr 33.4%

       \[\leadsto \color{blue}{1 - re \cdot \left(re \cdot -0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 48.1% accurate, 25.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 6.8 \cdot 10^{+172}:\\ \;\;\;\;1 + 0.5 \cdot \left(im\_m \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - re \cdot \left(re \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 6.8e+172)
   (+ 1.0 (* 0.5 (* im_m im_m)))
   (- 1.0 (* re (* re -0.5)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 6.8e+172) {
		tmp = 1.0 + (0.5 * (im_m * im_m));
	} else {
		tmp = 1.0 - (re * (re * -0.5));
	}
	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 <= 6.8d+172) then
        tmp = 1.0d0 + (0.5d0 * (im_m * im_m))
    else
        tmp = 1.0d0 - (re * (re * (-0.5d0)))
    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 <= 6.8e+172) {
		tmp = 1.0 + (0.5 * (im_m * im_m));
	} else {
		tmp = 1.0 - (re * (re * -0.5));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 6.8e+172:
		tmp = 1.0 + (0.5 * (im_m * im_m))
	else:
		tmp = 1.0 - (re * (re * -0.5))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 6.8e+172)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im_m * im_m)));
	else
		tmp = Float64(1.0 - Float64(re * Float64(re * -0.5)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 6.8e+172)
		tmp = 1.0 + (0.5 * (im_m * im_m));
	else
		tmp = 1.0 - (re * (re * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 6.8e+172], N[(1.0 + N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.7999999999999996e172

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 83.9%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 63.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 63.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{2} \cdot {re}^{2} + \frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       3. distribute-rgt1-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       4. lft-mult-inverse N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + \frac{1}{{im}^{2}} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{{im}^{2}} + \frac{1}{2}\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{im}^{2}} \cdot {im}^{2} + \frac{1}{2} \cdot {im}^{2}\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       9. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right)\right) \]

       18. *-lowering-*.f64 53.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right)\right) \]

   11. Simplified 53.5%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       4. *-lowering-*.f64 50.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   14. Simplified 50.2%

       \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   if 6.7999999999999996e172 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 78.9%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 21.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 21.3%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right) \]

       5. *-lowering-*.f64 21.3%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right) \]

   11. Simplified 21.3%

       \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot -0.5} \]

   13. Applied egg-rr 34.3%

       \[\leadsto \color{blue}{1 - re \cdot \left(re \cdot -0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 48.2% accurate, 25.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 4.6 \cdot 10^{+169}:\\ \;\;\;\;1 + 0.5 \cdot \left(im\_m \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 4.6e+169)
   (+ 1.0 (* 0.5 (* im_m im_m)))
   (+ 1.0 (* (* re re) -0.5))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 4.6e+169) {
		tmp = 1.0 + (0.5 * (im_m * im_m));
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	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 <= 4.6d+169) then
        tmp = 1.0d0 + (0.5d0 * (im_m * im_m))
    else
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    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 <= 4.6e+169) {
		tmp = 1.0 + (0.5 * (im_m * im_m));
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 4.6e+169:
		tmp = 1.0 + (0.5 * (im_m * im_m))
	else:
		tmp = 1.0 + ((re * re) * -0.5)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 4.6e+169)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im_m * im_m)));
	else
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 4.6e+169)
		tmp = 1.0 + (0.5 * (im_m * im_m));
	else
		tmp = 1.0 + ((re * re) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 4.6e+169], N[(1.0 + N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.5999999999999999e169

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 83.9%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 63.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 63.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{2} \cdot {re}^{2} + \frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) + \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       3. distribute-rgt1-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]

       4. lft-mult-inverse N/A

          \[\leadsto \left(\frac{1}{2} \cdot {im}^{2} + \frac{1}{{im}^{2}} \cdot {im}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right) \cdot \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{im}^{2}}\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{1}{{im}^{2}} + \frac{1}{2}\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{im}^{2}} \cdot {im}^{2} + \frac{1}{2} \cdot {im}^{2}\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       9. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {im}^{2}\right)\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({im}^{2}\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(1 + \frac{-1}{2} \cdot {re}^{2}\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right)\right) \]

       18. *-lowering-*.f64 53.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right)\right) \]

   11. Simplified 53.5%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       4. *-lowering-*.f64 50.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   14. Simplified 50.2%

       \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im\right)} \]

   if 4.5999999999999999e169 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 45.8%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 45.8%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({re}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot \color{blue}{re}\right)\right)\right) \]

      4. *-lowering-*.f64 21.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   8. Simplified 21.3%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \left(re \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.6 \cdot 10^{+169}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 34.3% accurate, 25.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.25 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.25e+22) 1.0 (+ 1.0 (* (* re re) -0.5))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.25e+22) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	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 <= 1.25d+22) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    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 <= 1.25e+22) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.25e+22:
		tmp = 1.0
	else:
		tmp = 1.0 + ((re * re) * -0.5)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.25e+22)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.25e+22)
		tmp = 1.0;
	else
		tmp = 1.0 + ((re * re) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.25e+22], 1.0, N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.2499999999999999e22

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 58.6%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 58.6%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 33.9%

      \[\leadsto \color{blue}{1} \]

   if 1.2499999999999999e22 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({re}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot \color{blue}{re}\right)\right)\right) \]

      4. *-lowering-*.f64 18.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   8. Simplified 18.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \left(re \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.25 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 33.8% accurate, 30.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 3.2 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 3.2e+24) 1.0 (* (* re re) -0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3.2e+24) {
		tmp = 1.0;
	} else {
		tmp = (re * re) * -0.5;
	}
	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 <= 3.2d+24) then
        tmp = 1.0d0
    else
        tmp = (re * re) * (-0.5d0)
    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 <= 3.2e+24) {
		tmp = 1.0;
	} else {
		tmp = (re * re) * -0.5;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 3.2e+24:
		tmp = 1.0
	else:
		tmp = (re * re) * -0.5
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 3.2e+24)
		tmp = 1.0;
	else
		tmp = Float64(Float64(re * re) * -0.5);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 3.2e+24)
		tmp = 1.0;
	else
		tmp = (re * re) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 3.2e+24], 1.0, N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.1999999999999997e24

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 57.8%

         \[\leadsto \mathsf{cos.f64}\left(re\right) \]

   5. Simplified 57.8%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 33.4%

      \[\leadsto \color{blue}{1} \]

   if 3.1999999999999997e24 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{2} \cdot \cos re\right) + \color{blue}{\cos re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}\right) + \cos \color{blue}{re} \]

      3. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2} + \cos re\right)} \]

      4. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \cos re\right) + \left(\left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{{im}^{2}} + \cos re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot {im}^{2}} + \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(\cos re \cdot {im}^{2}\right) + \cos \color{blue}{re}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \cos re + \cos \color{blue}{re}\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \color{blue}{\cos re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \cos re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \cos re \]

   5. Simplified 75.7%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right), 1\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)}, 1\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({re}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \color{blue}{\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)\right)}\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

      4. *-lowering-*.f64 65.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, re\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(im, im\right)\right)}\right)\right), 1\right)\right) \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right) + 1\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({re}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right)\right) \]

       5. *-lowering-*.f64 18.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right)\right) \]

   11. Simplified 18.9%

       \[\leadsto \color{blue}{1 + \left(re \cdot re\right) \cdot -0.5} \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\frac{-1}{2}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{2}}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{2}\right) \]

       4. *-lowering-*.f64 18.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{2}\right) \]

   14. Simplified 18.1%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 28.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 1 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 1.0)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 1.0;
}

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
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 1.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 1.0

Julia

im_m = abs(im)
function code(re, im_m)
	return 1.0
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 1.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 1.0

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\cos re} \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 46.3%

      \[\leadsto \mathsf{cos.f64}\left(re\right) \]

5. Simplified 46.3%

   \[\leadsto \color{blue}{\cos re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 26.9%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))