Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.5% → 96.9%

Time: 13.5s

Alternatives: 10

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{m + n}{2} - M\\ \frac{\cos M}{e^{\left(t\_0 \cdot t\_0 + \ell\right) - \left|m - n\right|}} \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (/ (+ m n) 2.0) M)))
   (/ (cos M) (exp (- (+ (* t_0 t_0) l) (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = ((m + n) / 2.0) - M;
	return cos(M) / exp((((t_0 * t_0) + l) - fabs((m - n))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = ((m + n) / 2.0d0) - m_1
    code = cos(m_1) / exp((((t_0 * t_0) + l) - abs((m - n))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = ((m + n) / 2.0) - M;
	return Math.cos(M) / Math.exp((((t_0 * t_0) + l) - Math.abs((m - n))));
}

Python

def code(K, m, n, M, l):
	t_0 = ((m + n) / 2.0) - M
	return math.cos(M) / math.exp((((t_0 * t_0) + l) - math.fabs((m - n))))

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(Float64(m + n) / 2.0) - M)
	return Float64(cos(M) / exp(Float64(Float64(Float64(t_0 * t_0) + l) - abs(Float64(m - n)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	t_0 = ((m + n) / 2.0) - M;
	tmp = cos(M) / exp((((t_0 * t_0) + l) - abs((m - n))));
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]}, N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + l), $MachinePrecision] - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. associate--l- N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

   3. exp-diff N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

   5. exp-0 N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

   6. *-rgt-identity N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

3. Simplified 74.8%

   \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
4. Add Preprocessing

5. Taylor expanded in K around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. cos-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   2. cos-lowering-cos.f64 97.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

7. Simplified 97.4%

   \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]
8. Add Preprocessing

Alternative 2: 65.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3 \cdot 10^{-132}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 6:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 3e-132)
   (/ (cos M) (exp (* 0.25 (* m m))))
   (if (<= n 6.0)
     (/ (cos M) (exp (* M M)))
     (/ (cos M) (exp (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 3e-132) {
		tmp = cos(M) / exp((0.25 * (m * m)));
	} else if (n <= 6.0) {
		tmp = cos(M) / exp((M * M));
	} else {
		tmp = cos(M) / exp((0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 3d-132) then
        tmp = cos(m_1) / exp((0.25d0 * (m * m)))
    else if (n <= 6.0d0) then
        tmp = cos(m_1) / exp((m_1 * m_1))
    else
        tmp = cos(m_1) / exp((0.25d0 * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 3e-132) {
		tmp = Math.cos(M) / Math.exp((0.25 * (m * m)));
	} else if (n <= 6.0) {
		tmp = Math.cos(M) / Math.exp((M * M));
	} else {
		tmp = Math.cos(M) / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 3e-132:
		tmp = math.cos(M) / math.exp((0.25 * (m * m)))
	elif n <= 6.0:
		tmp = math.cos(M) / math.exp((M * M))
	else:
		tmp = math.cos(M) / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 3e-132)
		tmp = Float64(cos(M) / exp(Float64(0.25 * Float64(m * m))));
	elseif (n <= 6.0)
		tmp = Float64(cos(M) / exp(Float64(M * M)));
	else
		tmp = Float64(cos(M) / exp(Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 3e-132)
		tmp = cos(M) / exp((0.25 * (m * m)));
	elseif (n <= 6.0)
		tmp = cos(M) / exp((M * M));
	else
		tmp = cos(M) / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 3e-132], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6.0], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 3e-132

   1. Initial program 74.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 74.5%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 98.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 98.4%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 61.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 61.4%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   if 3e-132 < n < 6

   1. Initial program 80.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 88.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 88.4%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in M around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left({M}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\left(M \cdot M\right)\right)\right) \]

      2. *-lowering-*.f64 65.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(M, M\right)\right)\right) \]

   10. Simplified 65.6%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{M \cdot M}}} \]

   if 6 < n

   1. Initial program 71.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {n}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({n}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(n \cdot n\right)\right)\right)\right) \]

      3. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(n, n\right)\right)\right)\right) \]

   10. Simplified 98.3%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{0.25 \cdot \left(n \cdot n\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{if}\;m \leq -14500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* (* m m) -0.25))))
   (if (<= m -14500.0) t_0 (if (<= m 8.5e-9) (/ (cos M) (exp (* M M))) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((m * m) * -0.25));
	double tmp;
	if (m <= -14500.0) {
		tmp = t_0;
	} else if (m <= 8.5e-9) {
		tmp = cos(M) / exp((M * M));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((m * m) * (-0.25d0)))
    if (m <= (-14500.0d0)) then
        tmp = t_0
    else if (m <= 8.5d-9) then
        tmp = cos(m_1) / exp((m_1 * m_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((m * m) * -0.25));
	double tmp;
	if (m <= -14500.0) {
		tmp = t_0;
	} else if (m <= 8.5e-9) {
		tmp = Math.cos(M) / Math.exp((M * M));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((m * m) * -0.25))
	tmp = 0
	if m <= -14500.0:
		tmp = t_0
	elif m <= 8.5e-9:
		tmp = math.cos(M) / math.exp((M * M))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(m * m) * -0.25))
	tmp = 0.0
	if (m <= -14500.0)
		tmp = t_0;
	elseif (m <= 8.5e-9)
		tmp = Float64(cos(M) / exp(Float64(M * M)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((m * m) * -0.25));
	tmp = 0.0;
	if (m <= -14500.0)
		tmp = t_0;
	elseif (m <= 8.5e-9)
		tmp = cos(M) / exp((M * M));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -14500.0], t$95$0, If[LessEqual[m, 8.5e-9], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -14500 or 8.5e-9 < m

   1. Initial program 70.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 70.3%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 96.6%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)} \]

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

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left({m}^{2} \cdot \frac{1}{4}\right)\right)\right) \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \frac{-1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({m}^{2}\right), \frac{-1}{4}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(m \cdot m\right), \frac{-1}{4}\right)\right) \]

       8. *-lowering-*.f64 96.6%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \frac{-1}{4}\right)\right) \]

   13. Simplified 96.6%

       \[\leadsto \color{blue}{e^{\left(m \cdot m\right) \cdot -0.25}} \]

   if -14500 < m < 8.5e-9

   1. Initial program 80.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 80.5%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 95.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 95.0%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in M around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left({M}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\left(M \cdot M\right)\right)\right) \]

      2. *-lowering-*.f64 58.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(M, M\right)\right)\right) \]

   10. Simplified 58.3%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{M \cdot M}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 72.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+87}:\\ \;\;\;\;\frac{\cos M}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-278}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos \left(0.5 \cdot \left(n \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{0 - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* (* m m) -0.25))))
   (if (<= l -1.65e+87)
     (/ (cos M) (+ 1.0 (* l (+ 1.0 (* l (+ 0.5 (* l 0.16666666666666666)))))))
     (if (<= l -1e-78)
       t_0
       (if (<= l 9e-278)
         (* -0.16666666666666666 (* (* l (* l l)) (cos (* 0.5 (* n K)))))
         (if (<= l 720.0) t_0 (exp (- 0.0 l))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((m * m) * -0.25));
	double tmp;
	if (l <= -1.65e+87) {
		tmp = cos(M) / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))));
	} else if (l <= -1e-78) {
		tmp = t_0;
	} else if (l <= 9e-278) {
		tmp = -0.16666666666666666 * ((l * (l * l)) * cos((0.5 * (n * K))));
	} else if (l <= 720.0) {
		tmp = t_0;
	} else {
		tmp = exp((0.0 - l));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((m * m) * (-0.25d0)))
    if (l <= (-1.65d+87)) then
        tmp = cos(m_1) / (1.0d0 + (l * (1.0d0 + (l * (0.5d0 + (l * 0.16666666666666666d0))))))
    else if (l <= (-1d-78)) then
        tmp = t_0
    else if (l <= 9d-278) then
        tmp = (-0.16666666666666666d0) * ((l * (l * l)) * cos((0.5d0 * (n * k))))
    else if (l <= 720.0d0) then
        tmp = t_0
    else
        tmp = exp((0.0d0 - l))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((m * m) * -0.25));
	double tmp;
	if (l <= -1.65e+87) {
		tmp = Math.cos(M) / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))));
	} else if (l <= -1e-78) {
		tmp = t_0;
	} else if (l <= 9e-278) {
		tmp = -0.16666666666666666 * ((l * (l * l)) * Math.cos((0.5 * (n * K))));
	} else if (l <= 720.0) {
		tmp = t_0;
	} else {
		tmp = Math.exp((0.0 - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((m * m) * -0.25))
	tmp = 0
	if l <= -1.65e+87:
		tmp = math.cos(M) / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))))
	elif l <= -1e-78:
		tmp = t_0
	elif l <= 9e-278:
		tmp = -0.16666666666666666 * ((l * (l * l)) * math.cos((0.5 * (n * K))))
	elif l <= 720.0:
		tmp = t_0
	else:
		tmp = math.exp((0.0 - l))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(m * m) * -0.25))
	tmp = 0.0
	if (l <= -1.65e+87)
		tmp = Float64(cos(M) / Float64(1.0 + Float64(l * Float64(1.0 + Float64(l * Float64(0.5 + Float64(l * 0.16666666666666666)))))));
	elseif (l <= -1e-78)
		tmp = t_0;
	elseif (l <= 9e-278)
		tmp = Float64(-0.16666666666666666 * Float64(Float64(l * Float64(l * l)) * cos(Float64(0.5 * Float64(n * K)))));
	elseif (l <= 720.0)
		tmp = t_0;
	else
		tmp = exp(Float64(0.0 - l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((m * m) * -0.25));
	tmp = 0.0;
	if (l <= -1.65e+87)
		tmp = cos(M) / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))));
	elseif (l <= -1e-78)
		tmp = t_0;
	elseif (l <= 9e-278)
		tmp = -0.16666666666666666 * ((l * (l * l)) * cos((0.5 * (n * K))));
	elseif (l <= 720.0)
		tmp = t_0;
	else
		tmp = exp((0.0 - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.65e+87], N[(N[Cos[M], $MachinePrecision] / N[(1.0 + N[(l * N[(1.0 + N[(l * N[(0.5 + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-78], t$95$0, If[LessEqual[l, 9e-278], N[(-0.16666666666666666 * N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 720.0], t$95$0, N[Exp[N[(0.0 - l), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.6500000000000001e87

   1. Initial program 65.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 65.1%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 95.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 95.3%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in l around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 12.9%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{\ell}}} \]

   11. Taylor expanded in l around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \color{blue}{\left(1 + \ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \color{blue}{\left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot \ell\right)}\right)\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 81.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 81.8%

       \[\leadsto \frac{\cos M}{\color{blue}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)}} \]

   if -1.6500000000000001e87 < l < -9.99999999999999999e-79 or 8.9999999999999996e-278 < l < 720

   1. Initial program 76.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 76.0%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 96.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 96.2%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 62.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 62.5%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)} \]

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

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left({m}^{2} \cdot \frac{1}{4}\right)\right)\right) \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \frac{-1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({m}^{2}\right), \frac{-1}{4}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(m \cdot m\right), \frac{-1}{4}\right)\right) \]

       8. *-lowering-*.f64 62.5%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \frac{-1}{4}\right)\right) \]

   13. Simplified 62.5%

       \[\leadsto \color{blue}{e^{\left(m \cdot m\right) \cdot -0.25}} \]

   if -9.99999999999999999e-79 < l < 8.9999999999999996e-278

   1. Initial program 80.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), 2\right), M\right)\right), \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \ell\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), 2\right), M\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), 2\right), M\right)\right), \mathsf{exp.f64}\left(\left(0 - \ell\right)\right)\right) \]

      3. --lowering--.f64 7.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), 2\right), M\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

   5. Simplified 7.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{0 - \ell}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)}\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(K \cdot n\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, \ell\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(n \cdot K\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      3. *-lowering-*.f64 7.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

   8. Simplified 7.5%

      \[\leadsto \cos \color{blue}{\left(0.5 \cdot \left(n \cdot K\right)\right)} \cdot e^{0 - \ell} \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right)\right)}\right) \]
   10. Step-by-step derivation

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

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

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

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

       3. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + -1\right)\right)\right)\right) \]

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right)\right), -1\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot \ell\right)\right)\right), -1\right)\right)\right)\right) \]

       8. *-lowering-*.f64 7.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \ell\right)\right)\right), -1\right)\right)\right)\right) \]

   11. Simplified 7.5%

       \[\leadsto \cos \left(0.5 \cdot \left(n \cdot K\right)\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + -0.16666666666666666 \cdot \ell\right) + -1\right)\right)} \]

   12. Taylor expanded in l around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({\ell}^{3} \cdot \cos \left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)\right)} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({\ell}^{3} \cdot \cos \left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({\ell}^{3}\right), \color{blue}{\cos \left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)}\right)\right) \]

       4. unpow2 N/A

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

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

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

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), \cos \left(\frac{1}{2} \cdot \color{blue}{\left(K \cdot n\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos \left(\frac{1}{2} \cdot \color{blue}{\left(K \cdot n\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(K \cdot n\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 80.4%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(K, n\right)\right)\right)\right)\right) \]

   14. Simplified 80.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos \left(0.5 \cdot \left(K \cdot n\right)\right)\right)} \]

   if 720 < l

   1. Initial program 75.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in l around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 100.0%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{\ell}}} \]

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\ell\right)} \]

       2. neg-mul-1 N/A

          \[\leadsto e^{-1 \cdot \ell} \]

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

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \ell\right)\right) \]

       4. neg-mul-1 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(0 - \ell\right)\right) \]

       6. --lowering--.f64 100.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{e^{0 - \ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+87}:\\ \;\;\;\;\frac{\cos M}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-78}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-278}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos \left(0.5 \cdot \left(n \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{+134}:\\ \;\;\;\;\frac{\cos M}{1 + \ell \cdot \left(1 + \ell \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-278}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos \left(0.5 \cdot \left(n \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 740:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{0 - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* (* m m) -0.25))))
   (if (<= l -8e+134)
     (/ (cos M) (+ 1.0 (* l (+ 1.0 (* l 0.5)))))
     (if (<= l -1e-78)
       t_0
       (if (<= l 9e-278)
         (* -0.16666666666666666 (* (* l (* l l)) (cos (* 0.5 (* n K)))))
         (if (<= l 740.0) t_0 (exp (- 0.0 l))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((m * m) * -0.25));
	double tmp;
	if (l <= -8e+134) {
		tmp = cos(M) / (1.0 + (l * (1.0 + (l * 0.5))));
	} else if (l <= -1e-78) {
		tmp = t_0;
	} else if (l <= 9e-278) {
		tmp = -0.16666666666666666 * ((l * (l * l)) * cos((0.5 * (n * K))));
	} else if (l <= 740.0) {
		tmp = t_0;
	} else {
		tmp = exp((0.0 - l));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((m * m) * (-0.25d0)))
    if (l <= (-8d+134)) then
        tmp = cos(m_1) / (1.0d0 + (l * (1.0d0 + (l * 0.5d0))))
    else if (l <= (-1d-78)) then
        tmp = t_0
    else if (l <= 9d-278) then
        tmp = (-0.16666666666666666d0) * ((l * (l * l)) * cos((0.5d0 * (n * k))))
    else if (l <= 740.0d0) then
        tmp = t_0
    else
        tmp = exp((0.0d0 - l))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((m * m) * -0.25));
	double tmp;
	if (l <= -8e+134) {
		tmp = Math.cos(M) / (1.0 + (l * (1.0 + (l * 0.5))));
	} else if (l <= -1e-78) {
		tmp = t_0;
	} else if (l <= 9e-278) {
		tmp = -0.16666666666666666 * ((l * (l * l)) * Math.cos((0.5 * (n * K))));
	} else if (l <= 740.0) {
		tmp = t_0;
	} else {
		tmp = Math.exp((0.0 - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((m * m) * -0.25))
	tmp = 0
	if l <= -8e+134:
		tmp = math.cos(M) / (1.0 + (l * (1.0 + (l * 0.5))))
	elif l <= -1e-78:
		tmp = t_0
	elif l <= 9e-278:
		tmp = -0.16666666666666666 * ((l * (l * l)) * math.cos((0.5 * (n * K))))
	elif l <= 740.0:
		tmp = t_0
	else:
		tmp = math.exp((0.0 - l))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(m * m) * -0.25))
	tmp = 0.0
	if (l <= -8e+134)
		tmp = Float64(cos(M) / Float64(1.0 + Float64(l * Float64(1.0 + Float64(l * 0.5)))));
	elseif (l <= -1e-78)
		tmp = t_0;
	elseif (l <= 9e-278)
		tmp = Float64(-0.16666666666666666 * Float64(Float64(l * Float64(l * l)) * cos(Float64(0.5 * Float64(n * K)))));
	elseif (l <= 740.0)
		tmp = t_0;
	else
		tmp = exp(Float64(0.0 - l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((m * m) * -0.25));
	tmp = 0.0;
	if (l <= -8e+134)
		tmp = cos(M) / (1.0 + (l * (1.0 + (l * 0.5))));
	elseif (l <= -1e-78)
		tmp = t_0;
	elseif (l <= 9e-278)
		tmp = -0.16666666666666666 * ((l * (l * l)) * cos((0.5 * (n * K))));
	elseif (l <= 740.0)
		tmp = t_0;
	else
		tmp = exp((0.0 - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -8e+134], N[(N[Cos[M], $MachinePrecision] / N[(1.0 + N[(l * N[(1.0 + N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-78], t$95$0, If[LessEqual[l, 9e-278], N[(-0.16666666666666666 * N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 740.0], t$95$0, N[Exp[N[(0.0 - l), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -7.99999999999999937e134

   1. Initial program 62.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 94.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 94.3%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in l around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 12.7%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{\ell}}} \]

   11. Taylor expanded in l around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \color{blue}{\left(1 + \ell \cdot \left(1 + \frac{1}{2} \cdot \ell\right)\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(1 + \frac{1}{2} \cdot \ell\right)\right)}\right)\right) \]

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

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

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

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

       4. *-lowering-*.f64 72.6%

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

   13. Simplified 72.6%

       \[\leadsto \frac{\cos M}{\color{blue}{1 + \ell \cdot \left(1 + 0.5 \cdot \ell\right)}} \]

   if -7.99999999999999937e134 < l < -9.99999999999999999e-79 or 8.9999999999999996e-278 < l < 740

   1. Initial program 75.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 75.9%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 96.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 96.5%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 63.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 63.5%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)} \]

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

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left({m}^{2} \cdot \frac{1}{4}\right)\right)\right) \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \frac{-1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({m}^{2}\right), \frac{-1}{4}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(m \cdot m\right), \frac{-1}{4}\right)\right) \]

       8. *-lowering-*.f64 63.5%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \frac{-1}{4}\right)\right) \]

   13. Simplified 63.5%

       \[\leadsto \color{blue}{e^{\left(m \cdot m\right) \cdot -0.25}} \]

   if -9.99999999999999999e-79 < l < 8.9999999999999996e-278

   1. Initial program 80.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), 2\right), M\right)\right), \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \ell\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), 2\right), M\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), 2\right), M\right)\right), \mathsf{exp.f64}\left(\left(0 - \ell\right)\right)\right) \]

      3. --lowering--.f64 7.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), 2\right), M\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

   5. Simplified 7.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{0 - \ell}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)}\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(K \cdot n\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, \ell\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(n \cdot K\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      3. *-lowering-*.f64 7.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

   8. Simplified 7.5%

      \[\leadsto \cos \color{blue}{\left(0.5 \cdot \left(n \cdot K\right)\right)} \cdot e^{0 - \ell} \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right)\right)}\right) \]
   10. Step-by-step derivation

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

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

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

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

       3. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + -1\right)\right)\right)\right) \]

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right)\right), -1\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot \ell\right)\right)\right), -1\right)\right)\right)\right) \]

       8. *-lowering-*.f64 7.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \ell\right)\right)\right), -1\right)\right)\right)\right) \]

   11. Simplified 7.5%

       \[\leadsto \cos \left(0.5 \cdot \left(n \cdot K\right)\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + -0.16666666666666666 \cdot \ell\right) + -1\right)\right)} \]

   12. Taylor expanded in l around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({\ell}^{3} \cdot \cos \left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)\right)} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({\ell}^{3} \cdot \cos \left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({\ell}^{3}\right), \color{blue}{\cos \left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)}\right)\right) \]

       4. unpow2 N/A

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

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

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

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), \cos \left(\frac{1}{2} \cdot \color{blue}{\left(K \cdot n\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos \left(\frac{1}{2} \cdot \color{blue}{\left(K \cdot n\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(K \cdot n\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 80.4%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(K, n\right)\right)\right)\right)\right) \]

   14. Simplified 80.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos \left(0.5 \cdot \left(K \cdot n\right)\right)\right)} \]

   if 740 < l

   1. Initial program 75.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in l around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 100.0%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{\ell}}} \]

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\ell\right)} \]

       2. neg-mul-1 N/A

          \[\leadsto e^{-1 \cdot \ell} \]

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

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \ell\right)\right) \]

       4. neg-mul-1 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(0 - \ell\right)\right) \]

       6. --lowering--.f64 100.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{e^{0 - \ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+134}:\\ \;\;\;\;\frac{\cos M}{1 + \ell \cdot \left(1 + \ell \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-78}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-278}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos \left(0.5 \cdot \left(n \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 740:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+134}:\\ \;\;\;\;\frac{\cos M}{1 + \ell \cdot \left(1 + \ell \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 740:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -8e+134)
   (/ (cos M) (+ 1.0 (* l (+ 1.0 (* l 0.5)))))
   (if (<= l 740.0) (exp (* (* m m) -0.25)) (exp (- 0.0 l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -8e+134) {
		tmp = cos(M) / (1.0 + (l * (1.0 + (l * 0.5))));
	} else if (l <= 740.0) {
		tmp = exp(((m * m) * -0.25));
	} else {
		tmp = exp((0.0 - l));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-8d+134)) then
        tmp = cos(m_1) / (1.0d0 + (l * (1.0d0 + (l * 0.5d0))))
    else if (l <= 740.0d0) then
        tmp = exp(((m * m) * (-0.25d0)))
    else
        tmp = exp((0.0d0 - l))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -8e+134) {
		tmp = Math.cos(M) / (1.0 + (l * (1.0 + (l * 0.5))));
	} else if (l <= 740.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else {
		tmp = Math.exp((0.0 - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -8e+134:
		tmp = math.cos(M) / (1.0 + (l * (1.0 + (l * 0.5))))
	elif l <= 740.0:
		tmp = math.exp(((m * m) * -0.25))
	else:
		tmp = math.exp((0.0 - l))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -8e+134)
		tmp = Float64(cos(M) / Float64(1.0 + Float64(l * Float64(1.0 + Float64(l * 0.5)))));
	elseif (l <= 740.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	else
		tmp = exp(Float64(0.0 - l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -8e+134)
		tmp = cos(M) / (1.0 + (l * (1.0 + (l * 0.5))));
	elseif (l <= 740.0)
		tmp = exp(((m * m) * -0.25));
	else
		tmp = exp((0.0 - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -8e+134], N[(N[Cos[M], $MachinePrecision] / N[(1.0 + N[(l * N[(1.0 + N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 740.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[Exp[N[(0.0 - l), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.99999999999999937e134

   1. Initial program 62.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 94.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 94.3%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in l around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 12.7%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{\ell}}} \]

   11. Taylor expanded in l around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \color{blue}{\left(1 + \ell \cdot \left(1 + \frac{1}{2} \cdot \ell\right)\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(1 + \frac{1}{2} \cdot \ell\right)\right)}\right)\right) \]

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

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

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

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

       4. *-lowering-*.f64 72.6%

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

   13. Simplified 72.6%

       \[\leadsto \frac{\cos M}{\color{blue}{1 + \ell \cdot \left(1 + 0.5 \cdot \ell\right)}} \]

   if -7.99999999999999937e134 < l < 740

   1. Initial program 77.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 96.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 96.9%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 61.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 61.8%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)} \]

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

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left({m}^{2} \cdot \frac{1}{4}\right)\right)\right) \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \frac{-1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({m}^{2}\right), \frac{-1}{4}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(m \cdot m\right), \frac{-1}{4}\right)\right) \]

       8. *-lowering-*.f64 61.8%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \frac{-1}{4}\right)\right) \]

   13. Simplified 61.8%

       \[\leadsto \color{blue}{e^{\left(m \cdot m\right) \cdot -0.25}} \]

   if 740 < l

   1. Initial program 75.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in l around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 100.0%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{\ell}}} \]

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\ell\right)} \]

       2. neg-mul-1 N/A

          \[\leadsto e^{-1 \cdot \ell} \]

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

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \ell\right)\right) \]

       4. neg-mul-1 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(0 - \ell\right)\right) \]

       6. --lowering--.f64 100.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{e^{0 - \ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+134}:\\ \;\;\;\;\frac{\cos M}{1 + \ell \cdot \left(1 + \ell \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 740:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{if}\;m \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 55:\\ \;\;\;\;e^{0 - \ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* (* m m) -0.25))))
   (if (<= m -4.2e-5) t_0 (if (<= m 55.0) (exp (- 0.0 l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((m * m) * -0.25));
	double tmp;
	if (m <= -4.2e-5) {
		tmp = t_0;
	} else if (m <= 55.0) {
		tmp = exp((0.0 - l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((m * m) * (-0.25d0)))
    if (m <= (-4.2d-5)) then
        tmp = t_0
    else if (m <= 55.0d0) then
        tmp = exp((0.0d0 - l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((m * m) * -0.25));
	double tmp;
	if (m <= -4.2e-5) {
		tmp = t_0;
	} else if (m <= 55.0) {
		tmp = Math.exp((0.0 - l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((m * m) * -0.25))
	tmp = 0
	if m <= -4.2e-5:
		tmp = t_0
	elif m <= 55.0:
		tmp = math.exp((0.0 - l))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(m * m) * -0.25))
	tmp = 0.0
	if (m <= -4.2e-5)
		tmp = t_0;
	elseif (m <= 55.0)
		tmp = exp(Float64(0.0 - l));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((m * m) * -0.25));
	tmp = 0.0;
	if (m <= -4.2e-5)
		tmp = t_0;
	elseif (m <= 55.0)
		tmp = exp((0.0 - l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -4.2e-5], t$95$0, If[LessEqual[m, 55.0], N[Exp[N[(0.0 - l), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -4.19999999999999977e-5 or 55 < m

   1. Initial program 69.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 69.9%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 97.9%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)} \]

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

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left({m}^{2} \cdot \frac{1}{4}\right)\right)\right) \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \frac{-1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({m}^{2}\right), \frac{-1}{4}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(m \cdot m\right), \frac{-1}{4}\right)\right) \]

       8. *-lowering-*.f64 97.9%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \frac{-1}{4}\right)\right) \]

   13. Simplified 97.9%

       \[\leadsto \color{blue}{e^{\left(m \cdot m\right) \cdot -0.25}} \]

   if -4.19999999999999977e-5 < m < 55

   1. Initial program 80.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--l- N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      3. exp-diff N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

      5. exp-0 N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 95.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   7. Simplified 95.1%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

   8. Taylor expanded in l around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 42.5%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{\ell}}} \]

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\ell\right)} \]

       2. neg-mul-1 N/A

          \[\leadsto e^{-1 \cdot \ell} \]

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

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \ell\right)\right) \]

       4. neg-mul-1 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(0 - \ell\right)\right) \]

       6. --lowering--.f64 41.6%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right) \]

   13. Simplified 41.6%

       \[\leadsto \color{blue}{e^{0 - \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 35.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ e^{0 - \ell} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (exp (- 0.0 l)))

C

double code(double K, double m, double n, double M, double l) {
	return exp((0.0 - l));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp((0.0d0 - l))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((0.0 - l));
}

Python

def code(K, m, n, M, l):
	return math.exp((0.0 - l))

Julia

function code(K, m, n, M, l)
	return exp(Float64(0.0 - l))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp((0.0 - l));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(0.0 - l), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 74.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. associate--l- N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

   3. exp-diff N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

   5. exp-0 N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

   6. *-rgt-identity N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

3. Simplified 74.8%

   \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
4. Add Preprocessing

5. Taylor expanded in K around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. cos-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   2. cos-lowering-cos.f64 97.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

7. Simplified 97.4%

   \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

8. Taylor expanded in l around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
9. Step-by-step derivation

10. Simplified 34.3%

    \[\leadsto \frac{\cos M}{e^{\color{blue}{\ell}}} \]

11. Taylor expanded in M around 0

    \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
12. Step-by-step derivation

    1. rec-exp N/A

       \[\leadsto e^{\mathsf{neg}\left(\ell\right)} \]

    2. neg-mul-1 N/A

       \[\leadsto e^{-1 \cdot \ell} \]

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

       \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \ell\right)\right) \]

    4. neg-mul-1 N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right) \]

    5. neg-sub0 N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(0 - \ell\right)\right) \]

    6. --lowering--.f64 33.9%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right) \]

13. Simplified 33.9%

    \[\leadsto \color{blue}{e^{0 - \ell}} \]
14. Add Preprocessing

Alternative 9: 7.5% accurate, 85.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\ell + 1} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (/ 1.0 (+ l 1.0)))

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 / (l + 1.0);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0 / (l + 1.0d0)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 / (l + 1.0);
}

Python

def code(K, m, n, M, l):
	return 1.0 / (l + 1.0)

Julia

function code(K, m, n, M, l)
	return Float64(1.0 / Float64(l + 1.0))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 / (l + 1.0);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 / N[(l + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. associate--l- N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

   3. exp-diff N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

   5. exp-0 N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

   6. *-rgt-identity N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

3. Simplified 74.8%

   \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
4. Add Preprocessing

5. Taylor expanded in K around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. cos-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   2. cos-lowering-cos.f64 97.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

7. Simplified 97.4%

   \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

8. Taylor expanded in l around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
9. Step-by-step derivation

10. Simplified 34.3%

    \[\leadsto \frac{\cos M}{e^{\color{blue}{\ell}}} \]

11. Taylor expanded in l around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \color{blue}{\left(1 + \ell\right)}\right) \]
12. Step-by-step derivation

    1. +-lowering-+.f64 6.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \color{blue}{\ell}\right)\right) \]

13. Simplified 6.0%

    \[\leadsto \frac{\cos M}{\color{blue}{1 + \ell}} \]

14. Taylor expanded in M around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \ell\right)\right) \]
15. Step-by-step derivation

16. Simplified 6.0%

    \[\leadsto \frac{\color{blue}{1}}{1 + \ell} \]

17. Final simplification 6.0%

    \[\leadsto \frac{1}{\ell + 1} \]
18. Add Preprocessing

Alternative 10: 6.6% accurate, 425.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 1.0)

C

double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

Python

def code(K, m, n, M, l):
	return 1.0

Julia

function code(K, m, n, M, l)
	return 1.0
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0;
end

Wolfram

code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 74.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(0 - {\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. associate--l- N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0 - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

   3. exp-diff N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \frac{e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{0}}{\color{blue}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

   5. exp-0 N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot 1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \color{blue}{\left(\ell - \left|m - n\right|\right)}}} \]

   6. *-rgt-identity N/A

      \[\leadsto \frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

3. Simplified 74.8%

   \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
4. Add Preprocessing

5. Taylor expanded in K around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. cos-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

   2. cos-lowering-cos.f64 97.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)}\right)\right) \]

7. Simplified 97.4%

   \[\leadsto \frac{\color{blue}{\cos M}}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}} \]

8. Taylor expanded in m around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

   3. *-lowering-*.f64 57.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

10. Simplified 57.7%

    \[\leadsto \frac{\cos M}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

11. Taylor expanded in M around 0

    \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
12. Step-by-step derivation

    1. rec-exp N/A

       \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)} \]

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

       \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left({m}^{2} \cdot \frac{1}{4}\right)\right)\right) \]

    4. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{exp.f64}\left(\left({m}^{2} \cdot \frac{-1}{4}\right)\right) \]

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

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({m}^{2}\right), \frac{-1}{4}\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(m \cdot m\right), \frac{-1}{4}\right)\right) \]

    8. *-lowering-*.f64 57.7%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \frac{-1}{4}\right)\right) \]

13. Simplified 57.7%

    \[\leadsto \color{blue}{e^{\left(m \cdot m\right) \cdot -0.25}} \]

14. Taylor expanded in m around 0

    \[\leadsto \color{blue}{1} \]
15. Step-by-step derivation

16. Simplified 4.7%

    \[\leadsto \color{blue}{1} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024147 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))