Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.7% → 97.0%

Time: 14.8s

Alternatives: 12

Speedup: 1.9×

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: 75.7% 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: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right)\\ e^{\left(n - m\right) + \left(\left(t\_0 - M\right) \cdot \left(M - t\_0\right) - \ell\right)} \cdot \cos M \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ n m))))
   (* (exp (+ (- n m) (- (* (- t_0 M) (- M t_0)) l))) (cos M))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	return exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * cos(M);
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 = 0.5d0 * (n + m)
    code = exp(((n - m) + (((t_0 - m_1) * (m_1 - t_0)) - l))) * cos(m_1)
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	return Math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * Math.cos(M);
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = 0.5 * (n + m)
	return math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * math.cos(M)

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(0.5 * Float64(n + m))
	return Float64(exp(Float64(Float64(n - m) + Float64(Float64(Float64(t_0 - M) * Float64(M - t_0)) - l))) * cos(M))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	t_0 = 0.5 * (n + m);
	tmp = exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * cos(M);
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision]}, N[(N[Exp[N[(N[(n - m), $MachinePrecision] + N[(N[(N[(t$95$0 - M), $MachinePrecision] * N[(M - t$95$0), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.7%

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

3. Taylor expanded in K around 0 97.4%

   \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

5. Simplified 97.4%

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]
6. Step-by-step derivation

   1. *-commutative 97.4%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(\color{blue}{\left(m + n\right) \cdot 0.5} - M\right)}^{2}\right)} \cdot \cos M \]

   2. fmm-undef 97.4%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + {\color{blue}{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}}^{2}\right)} \cdot \cos M \]

   3. unpow2 97.4%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + \color{blue}{\mathsf{fma}\left(m + n, 0.5, -M\right) \cdot \mathsf{fma}\left(m + n, 0.5, -M\right)}\right)} \cdot \cos M \]

   4. fmm-undef 97.4%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + \color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)} \cdot \mathsf{fma}\left(m + n, 0.5, -M\right)\right)} \cdot \cos M \]

   5. *-commutative 97.4%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + \left(\color{blue}{0.5 \cdot \left(m + n\right)} - M\right) \cdot \mathsf{fma}\left(m + n, 0.5, -M\right)\right)} \cdot \cos M \]

   6. +-commutative 97.4%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + \left(0.5 \cdot \color{blue}{\left(n + m\right)} - M\right) \cdot \mathsf{fma}\left(m + n, 0.5, -M\right)\right)} \cdot \cos M \]

   7. fmm-undef 97.4%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + \left(0.5 \cdot \left(n + m\right) - M\right) \cdot \color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)}\right)} \cdot \cos M \]

   8. *-commutative 97.4%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + \left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(\color{blue}{0.5 \cdot \left(m + n\right)} - M\right)\right)} \cdot \cos M \]

   9. +-commutative 97.4%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + \left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(0.5 \cdot \color{blue}{\left(n + m\right)} - M\right)\right)} \cdot \cos M \]

7. Applied egg-rr 97.4%

   \[\leadsto e^{\left|n - m\right| - \left(\ell + \color{blue}{\left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(0.5 \cdot \left(n + m\right) - M\right)}\right)} \cdot \cos M \]
8. Step-by-step derivation

   1. sub-neg 97.4%

      \[\leadsto e^{\color{blue}{\left|n - m\right| + \left(-\left(\ell + \left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(0.5 \cdot \left(n + m\right) - M\right)\right)\right)}} \cdot \cos M \]

   2. add-sqr-sqrt 47.3%

      \[\leadsto e^{\left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right| + \left(-\left(\ell + \left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(0.5 \cdot \left(n + m\right) - M\right)\right)\right)} \cdot \cos M \]

   3. fabs-sqr 47.3%

      \[\leadsto e^{\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}} + \left(-\left(\ell + \left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(0.5 \cdot \left(n + m\right) - M\right)\right)\right)} \cdot \cos M \]

   4. add-sqr-sqrt 97.1%

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

   5. pow2 97.1%

      \[\leadsto e^{\left(n - m\right) + \left(-\left(\ell + \color{blue}{{\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}}\right)\right)} \cdot \cos M \]

   6. fmm-def 97.1%

      \[\leadsto e^{\left(n - m\right) + \left(-\left(\ell + {\color{blue}{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}}^{2}\right)\right)} \cdot \cos M \]

9. Applied egg-rr 97.1%

   \[\leadsto e^{\color{blue}{\left(n - m\right) + \left(-\left(\ell + {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}\right)\right)}} \cdot \cos M \]
10. Step-by-step derivation

    1. unsub-neg 97.1%

       \[\leadsto e^{\color{blue}{\left(n - m\right) - \left(\ell + {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}\right)}} \cdot \cos M \]

    2. +-commutative 97.1%

       \[\leadsto e^{\left(n - m\right) - \left(\ell + {\left(\mathsf{fma}\left(0.5, \color{blue}{m + n}, -M\right)\right)}^{2}\right)} \cdot \cos M \]

    3. fmm-def 97.1%

       \[\leadsto e^{\left(n - m\right) - \left(\ell + {\color{blue}{\left(0.5 \cdot \left(m + n\right) - M\right)}}^{2}\right)} \cdot \cos M \]

11. Simplified 97.1%

    \[\leadsto e^{\color{blue}{\left(n - m\right) - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \cdot \cos M \]
12. Step-by-step derivation

    1. *-commutative 97.4%

       \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(\color{blue}{\left(m + n\right) \cdot 0.5} - M\right)}^{2}\right)} \cdot \cos M \]

    2. fmm-undef 97.4%

       \[\leadsto e^{\left|n - m\right| - \left(\ell + {\color{blue}{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}}^{2}\right)} \cdot \cos M \]

    3. unpow2 97.4%

       \[\leadsto e^{\left|n - m\right| - \left(\ell + \color{blue}{\mathsf{fma}\left(m + n, 0.5, -M\right) \cdot \mathsf{fma}\left(m + n, 0.5, -M\right)}\right)} \cdot \cos M \]

    4. fmm-undef 97.4%

       \[\leadsto e^{\left|n - m\right| - \left(\ell + \color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)} \cdot \mathsf{fma}\left(m + n, 0.5, -M\right)\right)} \cdot \cos M \]

    5. *-commutative 97.4%

       \[\leadsto e^{\left|n - m\right| - \left(\ell + \left(\color{blue}{0.5 \cdot \left(m + n\right)} - M\right) \cdot \mathsf{fma}\left(m + n, 0.5, -M\right)\right)} \cdot \cos M \]

    6. +-commutative 97.4%

       \[\leadsto e^{\left|n - m\right| - \left(\ell + \left(0.5 \cdot \color{blue}{\left(n + m\right)} - M\right) \cdot \mathsf{fma}\left(m + n, 0.5, -M\right)\right)} \cdot \cos M \]

    7. fmm-undef 97.4%

       \[\leadsto e^{\left|n - m\right| - \left(\ell + \left(0.5 \cdot \left(n + m\right) - M\right) \cdot \color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)}\right)} \cdot \cos M \]

    8. *-commutative 97.4%

       \[\leadsto e^{\left|n - m\right| - \left(\ell + \left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(\color{blue}{0.5 \cdot \left(m + n\right)} - M\right)\right)} \cdot \cos M \]

    9. +-commutative 97.4%

       \[\leadsto e^{\left|n - m\right| - \left(\ell + \left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(0.5 \cdot \color{blue}{\left(n + m\right)} - M\right)\right)} \cdot \cos M \]

13. Applied egg-rr 97.1%

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

14. Final simplification 97.1%

    \[\leadsto e^{\left(n - m\right) + \left(\left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(M - 0.5 \cdot \left(n + m\right)\right) - \ell\right)} \cdot \cos M \]
15. Add Preprocessing

Alternative 2: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := e^{-0.25 \cdot {m}^{2}}\\ \mathbf{if}\;m \leq -1.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 6.2 \cdot 10^{-242}:\\ \;\;\;\;\cos M \cdot e^{-M \cdot M}\\ \mathbf{elif}\;m \leq 53:\\ \;\;\;\;e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (pow m 2.0)))))
   (if (<= m -1.8)
     t_0
     (if (<= m 6.2e-242)
       (* (cos M) (exp (- (* M M))))
       (if (<= m 53.0) (exp (* n (+ (* m 0.5) (- -1.0 M)))) t_0)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * pow(m, 2.0)));
	double tmp;
	if (m <= -1.8) {
		tmp = t_0;
	} else if (m <= 6.2e-242) {
		tmp = cos(M) * exp(-(M * M));
	} else if (m <= 53.0) {
		tmp = exp((n * ((m * 0.5) + (-1.0 - M))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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(((-0.25d0) * (m ** 2.0d0)))
    if (m <= (-1.8d0)) then
        tmp = t_0
    else if (m <= 6.2d-242) then
        tmp = cos(m_1) * exp(-(m_1 * m_1))
    else if (m <= 53.0d0) then
        tmp = exp((n * ((m * 0.5d0) + ((-1.0d0) - m_1))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((-0.25 * Math.pow(m, 2.0)));
	double tmp;
	if (m <= -1.8) {
		tmp = t_0;
	} else if (m <= 6.2e-242) {
		tmp = Math.cos(M) * Math.exp(-(M * M));
	} else if (m <= 53.0) {
		tmp = Math.exp((n * ((m * 0.5) + (-1.0 - M))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * math.pow(m, 2.0)))
	tmp = 0
	if m <= -1.8:
		tmp = t_0
	elif m <= 6.2e-242:
		tmp = math.cos(M) * math.exp(-(M * M))
	elif m <= 53.0:
		tmp = math.exp((n * ((m * 0.5) + (-1.0 - M))))
	else:
		tmp = t_0
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * (m ^ 2.0)))
	tmp = 0.0
	if (m <= -1.8)
		tmp = t_0;
	elseif (m <= 6.2e-242)
		tmp = Float64(cos(M) * exp(Float64(-Float64(M * M))));
	elseif (m <= 53.0)
		tmp = exp(Float64(n * Float64(Float64(m * 0.5) + Float64(-1.0 - M))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (m ^ 2.0)));
	tmp = 0.0;
	if (m <= -1.8)
		tmp = t_0;
	elseif (m <= 6.2e-242)
		tmp = cos(M) * exp(-(M * M));
	elseif (m <= 53.0)
		tmp = exp((n * ((m * 0.5) + (-1.0 - M))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -1.8], t$95$0, If[LessEqual[m, 6.2e-242], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[(M * M), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 53.0], N[Exp[N[(n * N[(N[(m * 0.5), $MachinePrecision] + N[(-1.0 - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.80000000000000004 or 53 < m

   1. Initial program 73.0%

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

   3. Taylor expanded in K around 0 99.2%

      \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]

   6. Taylor expanded in m around inf 96.8%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \cdot \cos M \]
   7. Step-by-step derivation

      1. *-commutative 96.8%

         \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \cdot \cos M \]

   8. Simplified 96.8%

      \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \cdot \cos M \]

   9. Taylor expanded in M around 0 96.8%

      \[\leadsto \color{blue}{e^{-0.25 \cdot {m}^{2}}} \]

   if -1.80000000000000004 < m < 6.20000000000000031e-242

   1. Initial program 85.8%

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

   3. Taylor expanded in K around 0 95.4%

      \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   5. Simplified 95.4%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]

   6. Taylor expanded in M around inf 65.5%

      \[\leadsto e^{\color{blue}{-1 \cdot {M}^{2}}} \cdot \cos M \]
   7. Step-by-step derivation

      1. mul-1-neg 65.5%

         \[\leadsto e^{\color{blue}{-{M}^{2}}} \cdot \cos M \]

   8. Simplified 65.5%

      \[\leadsto e^{\color{blue}{-{M}^{2}}} \cdot \cos M \]
   9. Step-by-step derivation

      1. unpow2 65.5%

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

      2. distribute-lft-neg-in 65.5%

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

   10. Applied egg-rr 65.5%

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

   if 6.20000000000000031e-242 < m < 53

   1. Initial program 81.2%

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

      1. add-exp-log 79.3%

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

      2. *-commutative 79.3%

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

      3. log-prod 52.9%

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

   4. Applied egg-rr 28.2%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in n around -inf 18.4%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + -1 \cdot \frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 18.4%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \color{blue}{\left(-\frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}\right)} \]

      2. mul-1-neg 18.4%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \color{blue}{\left(-\left(0.5 \cdot m - M\right)\right)}}{n}\right)\right)} \]

      3. *-commutative 18.4%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(\color{blue}{m \cdot 0.5} - M\right)\right)}{n}\right)\right)} \]

   7. Simplified 18.4%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(m \cdot 0.5 - M\right)\right)}{n}\right)\right)}} \]

   8. Taylor expanded in n around 0 37.2%

      \[\leadsto e^{\color{blue}{n \cdot \left(0.5 \cdot m - \left(1 + M\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.8:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 6.2 \cdot 10^{-242}:\\ \;\;\;\;\cos M \cdot e^{-M \cdot M}\\ \mathbf{elif}\;m \leq 53:\\ \;\;\;\;e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := e^{-0.25 \cdot {m}^{2}}\\ \mathbf{if}\;m \leq -54:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -8.2 \cdot 10^{-119}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;m \leq 53:\\ \;\;\;\;e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (pow m 2.0)))))
   (if (<= m -54.0)
     t_0
     (if (<= m -8.2e-119)
       (* (cos M) (exp (- l)))
       (if (<= m 53.0) (exp (* n (+ (* m 0.5) (- -1.0 M)))) t_0)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * pow(m, 2.0)));
	double tmp;
	if (m <= -54.0) {
		tmp = t_0;
	} else if (m <= -8.2e-119) {
		tmp = cos(M) * exp(-l);
	} else if (m <= 53.0) {
		tmp = exp((n * ((m * 0.5) + (-1.0 - M))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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(((-0.25d0) * (m ** 2.0d0)))
    if (m <= (-54.0d0)) then
        tmp = t_0
    else if (m <= (-8.2d-119)) then
        tmp = cos(m_1) * exp(-l)
    else if (m <= 53.0d0) then
        tmp = exp((n * ((m * 0.5d0) + ((-1.0d0) - m_1))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((-0.25 * Math.pow(m, 2.0)));
	double tmp;
	if (m <= -54.0) {
		tmp = t_0;
	} else if (m <= -8.2e-119) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else if (m <= 53.0) {
		tmp = Math.exp((n * ((m * 0.5) + (-1.0 - M))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * math.pow(m, 2.0)))
	tmp = 0
	if m <= -54.0:
		tmp = t_0
	elif m <= -8.2e-119:
		tmp = math.cos(M) * math.exp(-l)
	elif m <= 53.0:
		tmp = math.exp((n * ((m * 0.5) + (-1.0 - M))))
	else:
		tmp = t_0
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * (m ^ 2.0)))
	tmp = 0.0
	if (m <= -54.0)
		tmp = t_0;
	elseif (m <= -8.2e-119)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	elseif (m <= 53.0)
		tmp = exp(Float64(n * Float64(Float64(m * 0.5) + Float64(-1.0 - M))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (m ^ 2.0)));
	tmp = 0.0;
	if (m <= -54.0)
		tmp = t_0;
	elseif (m <= -8.2e-119)
		tmp = cos(M) * exp(-l);
	elseif (m <= 53.0)
		tmp = exp((n * ((m * 0.5) + (-1.0 - M))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -54.0], t$95$0, If[LessEqual[m, -8.2e-119], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 53.0], N[Exp[N[(n * N[(N[(m * 0.5), $MachinePrecision] + N[(-1.0 - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if m < -54 or 53 < m

   1. Initial program 72.7%

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

   3. Taylor expanded in K around 0 99.2%

      \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]

   6. Taylor expanded in m around inf 97.6%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \cdot \cos M \]
   7. Step-by-step derivation

      1. *-commutative 97.6%

         \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \cdot \cos M \]

   8. Simplified 97.6%

      \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \cdot \cos M \]

   9. Taylor expanded in M around 0 97.6%

      \[\leadsto \color{blue}{e^{-0.25 \cdot {m}^{2}}} \]

   if -54 < m < -8.20000000000000041e-119

   1. Initial program 75.1%

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

   3. Taylor expanded in l around inf 37.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 37.1%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in K around 0 36.7%

      \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{-\ell}} \]
   7. Step-by-step derivation

      1. cos-neg 36.7%

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

   8. Simplified 36.7%

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

   if -8.20000000000000041e-119 < m < 53

   1. Initial program 86.0%

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

      1. add-exp-log 84.2%

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

      2. *-commutative 84.2%

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

      3. log-prod 51.1%

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

   4. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in n around -inf 26.3%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + -1 \cdot \frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 26.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \color{blue}{\left(-\frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}\right)} \]

      2. mul-1-neg 26.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \color{blue}{\left(-\left(0.5 \cdot m - M\right)\right)}}{n}\right)\right)} \]

      3. *-commutative 26.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(\color{blue}{m \cdot 0.5} - M\right)\right)}{n}\right)\right)} \]

   7. Simplified 26.3%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(m \cdot 0.5 - M\right)\right)}{n}\right)\right)}} \]

   8. Taylor expanded in n around 0 41.5%

      \[\leadsto e^{\color{blue}{n \cdot \left(0.5 \cdot m - \left(1 + M\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -8.2 \cdot 10^{-119}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;m \leq 53:\\ \;\;\;\;e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := e^{-0.25 \cdot {m}^{2}}\\ \mathbf{if}\;m \leq -54:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -1.1 \cdot 10^{-118}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;m \leq 54:\\ \;\;\;\;e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (pow m 2.0)))))
   (if (<= m -54.0)
     t_0
     (if (<= m -1.1e-118)
       (exp (- l))
       (if (<= m 54.0) (exp (* n (+ (* m 0.5) (- -1.0 M)))) t_0)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * pow(m, 2.0)));
	double tmp;
	if (m <= -54.0) {
		tmp = t_0;
	} else if (m <= -1.1e-118) {
		tmp = exp(-l);
	} else if (m <= 54.0) {
		tmp = exp((n * ((m * 0.5) + (-1.0 - M))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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(((-0.25d0) * (m ** 2.0d0)))
    if (m <= (-54.0d0)) then
        tmp = t_0
    else if (m <= (-1.1d-118)) then
        tmp = exp(-l)
    else if (m <= 54.0d0) then
        tmp = exp((n * ((m * 0.5d0) + ((-1.0d0) - m_1))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((-0.25 * Math.pow(m, 2.0)));
	double tmp;
	if (m <= -54.0) {
		tmp = t_0;
	} else if (m <= -1.1e-118) {
		tmp = Math.exp(-l);
	} else if (m <= 54.0) {
		tmp = Math.exp((n * ((m * 0.5) + (-1.0 - M))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * math.pow(m, 2.0)))
	tmp = 0
	if m <= -54.0:
		tmp = t_0
	elif m <= -1.1e-118:
		tmp = math.exp(-l)
	elif m <= 54.0:
		tmp = math.exp((n * ((m * 0.5) + (-1.0 - M))))
	else:
		tmp = t_0
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * (m ^ 2.0)))
	tmp = 0.0
	if (m <= -54.0)
		tmp = t_0;
	elseif (m <= -1.1e-118)
		tmp = exp(Float64(-l));
	elseif (m <= 54.0)
		tmp = exp(Float64(n * Float64(Float64(m * 0.5) + Float64(-1.0 - M))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (m ^ 2.0)));
	tmp = 0.0;
	if (m <= -54.0)
		tmp = t_0;
	elseif (m <= -1.1e-118)
		tmp = exp(-l);
	elseif (m <= 54.0)
		tmp = exp((n * ((m * 0.5) + (-1.0 - M))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -54.0], t$95$0, If[LessEqual[m, -1.1e-118], N[Exp[(-l)], $MachinePrecision], If[LessEqual[m, 54.0], N[Exp[N[(n * N[(N[(m * 0.5), $MachinePrecision] + N[(-1.0 - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if m < -54 or 54 < m

   1. Initial program 72.7%

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

   3. Taylor expanded in K around 0 99.2%

      \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]

   6. Taylor expanded in m around inf 97.6%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \cdot \cos M \]
   7. Step-by-step derivation

      1. *-commutative 97.6%

         \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \cdot \cos M \]

   8. Simplified 97.6%

      \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \cdot \cos M \]

   9. Taylor expanded in M around 0 97.6%

      \[\leadsto \color{blue}{e^{-0.25 \cdot {m}^{2}}} \]

   if -54 < m < -1.09999999999999992e-118

   1. Initial program 75.1%

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

      1. add-exp-log 73.9%

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

      2. *-commutative 73.9%

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

      3. log-prod 43.5%

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

   4. Applied egg-rr 22.1%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in l around inf 36.7%

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

      1. neg-mul-1 36.7%

         \[\leadsto e^{\color{blue}{-\ell}} \]

   7. Simplified 36.7%

      \[\leadsto e^{\color{blue}{-\ell}} \]

   if -1.09999999999999992e-118 < m < 54

   1. Initial program 86.0%

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

      1. add-exp-log 84.2%

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

      2. *-commutative 84.2%

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

      3. log-prod 51.1%

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

   4. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in n around -inf 26.3%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + -1 \cdot \frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 26.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \color{blue}{\left(-\frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}\right)} \]

      2. mul-1-neg 26.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \color{blue}{\left(-\left(0.5 \cdot m - M\right)\right)}}{n}\right)\right)} \]

      3. *-commutative 26.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(\color{blue}{m \cdot 0.5} - M\right)\right)}{n}\right)\right)} \]

   7. Simplified 26.3%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(m \cdot 0.5 - M\right)\right)}{n}\right)\right)}} \]

   8. Taylor expanded in n around 0 41.5%

      \[\leadsto e^{\color{blue}{n \cdot \left(0.5 \cdot m - \left(1 + M\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -1.1 \cdot 10^{-118}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;m \leq 54:\\ \;\;\;\;e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;e^{\ell}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-244}:\\ \;\;\;\;e^{n \cdot \left(-M\right)}\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{-5}:\\ \;\;\;\;e^{0.5 \cdot \left(n \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= l -720.0)
   (exp l)
   (if (<= l -4e-244)
     (exp (* n (- M)))
     (if (<= l 5.3e-5) (exp (* 0.5 (* n m))) (exp (- l))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -720.0) {
		tmp = exp(l);
	} else if (l <= -4e-244) {
		tmp = exp((n * -M));
	} else if (l <= 5.3e-5) {
		tmp = exp((0.5 * (n * m)));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 <= (-720.0d0)) then
        tmp = exp(l)
    else if (l <= (-4d-244)) then
        tmp = exp((n * -m_1))
    else if (l <= 5.3d-5) then
        tmp = exp((0.5d0 * (n * m)))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -720.0) {
		tmp = Math.exp(l);
	} else if (l <= -4e-244) {
		tmp = Math.exp((n * -M));
	} else if (l <= 5.3e-5) {
		tmp = Math.exp((0.5 * (n * m)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if l <= -720.0:
		tmp = math.exp(l)
	elif l <= -4e-244:
		tmp = math.exp((n * -M))
	elif l <= 5.3e-5:
		tmp = math.exp((0.5 * (n * m)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -720.0)
		tmp = exp(l);
	elseif (l <= -4e-244)
		tmp = exp(Float64(n * Float64(-M)));
	elseif (l <= 5.3e-5)
		tmp = exp(Float64(0.5 * Float64(n * m)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -720.0)
		tmp = exp(l);
	elseif (l <= -4e-244)
		tmp = exp((n * -M));
	elseif (l <= 5.3e-5)
		tmp = exp((0.5 * (n * m)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[l, -720.0], N[Exp[l], $MachinePrecision], If[LessEqual[l, -4e-244], N[Exp[N[(n * (-M)), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5.3e-5], N[Exp[N[(0.5 * N[(n * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -720

   1. Initial program 73.8%

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

      1. add-exp-log 70.5%

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

      2. *-commutative 70.5%

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

      3. log-prod 45.9%

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

   4. Applied egg-rr 18.5%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in l around inf 22.4%

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

      1. neg-mul-1 22.4%

         \[\leadsto e^{\color{blue}{-\ell}} \]

   7. Simplified 22.4%

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

      1. add-sqr-sqrt 22.4%

         \[\leadsto e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}} \]

      2. sqrt-unprod 22.4%

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

      3. sqr-neg 22.4%

         \[\leadsto e^{\sqrt{\color{blue}{\ell \cdot \ell}}} \]

      4. sqrt-unprod 0.0%

         \[\leadsto e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      5. add-sqr-sqrt 69.3%

         \[\leadsto e^{\color{blue}{\ell}} \]

      6. *-un-lft-identity 69.3%

         \[\leadsto e^{\color{blue}{1 \cdot \ell}} \]

   9. Applied egg-rr 69.3%

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

       1. *-lft-identity 69.3%

          \[\leadsto e^{\color{blue}{\ell}} \]

   11. Simplified 69.3%

       \[\leadsto e^{\color{blue}{\ell}} \]

   if -720 < l < -3.9999999999999997e-244

   1. Initial program 88.9%

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

      1. add-exp-log 88.9%

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

      2. *-commutative 88.9%

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

      3. log-prod 64.8%

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

   4. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in n around -inf 36.2%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + -1 \cdot \frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 36.2%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \color{blue}{\left(-\frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}\right)} \]

      2. mul-1-neg 36.2%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \color{blue}{\left(-\left(0.5 \cdot m - M\right)\right)}}{n}\right)\right)} \]

      3. *-commutative 36.2%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(\color{blue}{m \cdot 0.5} - M\right)\right)}{n}\right)\right)} \]

   7. Simplified 36.2%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(m \cdot 0.5 - M\right)\right)}{n}\right)\right)}} \]

   8. Taylor expanded in M around inf 45.9%

      \[\leadsto e^{\color{blue}{-1 \cdot \left(M \cdot n\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 45.9%

         \[\leadsto e^{\color{blue}{\left(-1 \cdot M\right) \cdot n}} \]

      2. neg-mul-1 45.9%

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

   10. Simplified 45.9%

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

   if -3.9999999999999997e-244 < l < 5.3000000000000001e-5

   1. Initial program 71.8%

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

      1. add-exp-log 71.4%

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

      2. *-commutative 71.4%

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

      3. log-prod 35.2%

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

   4. Applied egg-rr 9.2%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in n around -inf 30.1%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + -1 \cdot \frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 30.1%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \color{blue}{\left(-\frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}\right)} \]

      2. mul-1-neg 30.1%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \color{blue}{\left(-\left(0.5 \cdot m - M\right)\right)}}{n}\right)\right)} \]

      3. *-commutative 30.1%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(\color{blue}{m \cdot 0.5} - M\right)\right)}{n}\right)\right)} \]

   7. Simplified 30.1%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(m \cdot 0.5 - M\right)\right)}{n}\right)\right)}} \]

   8. Taylor expanded in m around inf 36.3%

      \[\leadsto e^{\color{blue}{0.5 \cdot \left(m \cdot n\right)}} \]

   if 5.3000000000000001e-5 < l

   1. Initial program 81.9%

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

      1. add-exp-log 81.9%

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

      2. *-commutative 81.9%

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

      3. log-prod 41.7%

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

   4. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in l around inf 94.6%

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

      1. neg-mul-1 94.6%

         \[\leadsto e^{\color{blue}{-\ell}} \]

   7. Simplified 94.6%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;e^{\ell}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-244}:\\ \;\;\;\;e^{n \cdot \left(-M\right)}\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{-5}:\\ \;\;\;\;e^{0.5 \cdot \left(n \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -740:\\ \;\;\;\;e^{\ell}\\ \mathbf{elif}\;\ell \leq 0.105:\\ \;\;\;\;e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= l -740.0)
   (exp l)
   (if (<= l 0.105) (exp (* n (+ (* m 0.5) (- -1.0 M)))) (exp (- l)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -740.0) {
		tmp = exp(l);
	} else if (l <= 0.105) {
		tmp = exp((n * ((m * 0.5) + (-1.0 - M))));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 <= (-740.0d0)) then
        tmp = exp(l)
    else if (l <= 0.105d0) then
        tmp = exp((n * ((m * 0.5d0) + ((-1.0d0) - m_1))))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -740.0) {
		tmp = Math.exp(l);
	} else if (l <= 0.105) {
		tmp = Math.exp((n * ((m * 0.5) + (-1.0 - M))));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if l <= -740.0:
		tmp = math.exp(l)
	elif l <= 0.105:
		tmp = math.exp((n * ((m * 0.5) + (-1.0 - M))))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -740.0)
		tmp = exp(l);
	elseif (l <= 0.105)
		tmp = exp(Float64(n * Float64(Float64(m * 0.5) + Float64(-1.0 - M))));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -740.0)
		tmp = exp(l);
	elseif (l <= 0.105)
		tmp = exp((n * ((m * 0.5) + (-1.0 - M))));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[l, -740.0], N[Exp[l], $MachinePrecision], If[LessEqual[l, 0.105], N[Exp[N[(n * N[(N[(m * 0.5), $MachinePrecision] + N[(-1.0 - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -740

   1. Initial program 73.8%

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

      1. add-exp-log 70.5%

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

      2. *-commutative 70.5%

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

      3. log-prod 45.9%

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

   4. Applied egg-rr 18.5%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in l around inf 22.4%

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

      1. neg-mul-1 22.4%

         \[\leadsto e^{\color{blue}{-\ell}} \]

   7. Simplified 22.4%

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

      1. add-sqr-sqrt 22.4%

         \[\leadsto e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}} \]

      2. sqrt-unprod 22.4%

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

      3. sqr-neg 22.4%

         \[\leadsto e^{\sqrt{\color{blue}{\ell \cdot \ell}}} \]

      4. sqrt-unprod 0.0%

         \[\leadsto e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      5. add-sqr-sqrt 69.3%

         \[\leadsto e^{\color{blue}{\ell}} \]

      6. *-un-lft-identity 69.3%

         \[\leadsto e^{\color{blue}{1 \cdot \ell}} \]

   9. Applied egg-rr 69.3%

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

       1. *-lft-identity 69.3%

          \[\leadsto e^{\color{blue}{\ell}} \]

   11. Simplified 69.3%

       \[\leadsto e^{\color{blue}{\ell}} \]

   if -740 < l < 0.104999999999999996

   1. Initial program 79.6%

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

      1. add-exp-log 79.4%

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

      2. *-commutative 79.4%

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

      3. log-prod 47.4%

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

   4. Applied egg-rr 12.4%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in n around -inf 32.3%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + -1 \cdot \frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 32.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \color{blue}{\left(-\frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}\right)} \]

      2. mul-1-neg 32.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \color{blue}{\left(-\left(0.5 \cdot m - M\right)\right)}}{n}\right)\right)} \]

      3. *-commutative 32.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(\color{blue}{m \cdot 0.5} - M\right)\right)}{n}\right)\right)} \]

   7. Simplified 32.3%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(m \cdot 0.5 - M\right)\right)}{n}\right)\right)}} \]

   8. Taylor expanded in n around 0 48.3%

      \[\leadsto e^{\color{blue}{n \cdot \left(0.5 \cdot m - \left(1 + M\right)\right)}} \]

   if 0.104999999999999996 < l

   1. Initial program 81.4%

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

      1. add-exp-log 81.4%

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

      2. *-commutative 81.4%

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

      3. log-prod 42.9%

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

   4. Applied egg-rr 21.8%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in l around inf 97.2%

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

      1. neg-mul-1 97.2%

         \[\leadsto e^{\color{blue}{-\ell}} \]

   7. Simplified 97.2%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -740:\\ \;\;\;\;e^{\ell}\\ \mathbf{elif}\;\ell \leq 0.105:\\ \;\;\;\;e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;e^{\ell}\\ \mathbf{elif}\;\ell \leq 0.105:\\ \;\;\;\;e^{n \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= l -720.0) (exp l) (if (<= l 0.105) (exp (* n (- M))) (exp (- l)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -720.0) {
		tmp = exp(l);
	} else if (l <= 0.105) {
		tmp = exp((n * -M));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 <= (-720.0d0)) then
        tmp = exp(l)
    else if (l <= 0.105d0) then
        tmp = exp((n * -m_1))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -720.0) {
		tmp = Math.exp(l);
	} else if (l <= 0.105) {
		tmp = Math.exp((n * -M));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if l <= -720.0:
		tmp = math.exp(l)
	elif l <= 0.105:
		tmp = math.exp((n * -M))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -720.0)
		tmp = exp(l);
	elseif (l <= 0.105)
		tmp = exp(Float64(n * Float64(-M)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -720.0)
		tmp = exp(l);
	elseif (l <= 0.105)
		tmp = exp((n * -M));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[l, -720.0], N[Exp[l], $MachinePrecision], If[LessEqual[l, 0.105], N[Exp[N[(n * (-M)), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -720

   1. Initial program 73.8%

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

      1. add-exp-log 70.5%

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

      2. *-commutative 70.5%

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

      3. log-prod 45.9%

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

   4. Applied egg-rr 18.5%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in l around inf 22.4%

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

      1. neg-mul-1 22.4%

         \[\leadsto e^{\color{blue}{-\ell}} \]

   7. Simplified 22.4%

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

      1. add-sqr-sqrt 22.4%

         \[\leadsto e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}} \]

      2. sqrt-unprod 22.4%

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

      3. sqr-neg 22.4%

         \[\leadsto e^{\sqrt{\color{blue}{\ell \cdot \ell}}} \]

      4. sqrt-unprod 0.0%

         \[\leadsto e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      5. add-sqr-sqrt 69.3%

         \[\leadsto e^{\color{blue}{\ell}} \]

      6. *-un-lft-identity 69.3%

         \[\leadsto e^{\color{blue}{1 \cdot \ell}} \]

   9. Applied egg-rr 69.3%

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

       1. *-lft-identity 69.3%

          \[\leadsto e^{\color{blue}{\ell}} \]

   11. Simplified 69.3%

       \[\leadsto e^{\color{blue}{\ell}} \]

   if -720 < l < 0.104999999999999996

   1. Initial program 79.6%

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

      1. add-exp-log 79.4%

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

      2. *-commutative 79.4%

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

      3. log-prod 47.4%

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

   4. Applied egg-rr 12.4%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in n around -inf 32.3%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + -1 \cdot \frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 32.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \color{blue}{\left(-\frac{1 + -1 \cdot \left(0.5 \cdot m - M\right)}{n}\right)}\right)} \]

      2. mul-1-neg 32.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \color{blue}{\left(-\left(0.5 \cdot m - M\right)\right)}}{n}\right)\right)} \]

      3. *-commutative 32.3%

         \[\leadsto e^{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(\color{blue}{m \cdot 0.5} - M\right)\right)}{n}\right)\right)} \]

   7. Simplified 32.3%

      \[\leadsto e^{\color{blue}{{n}^{2} \cdot \left(0.25 + \left(-\frac{1 + \left(-\left(m \cdot 0.5 - M\right)\right)}{n}\right)\right)}} \]

   8. Taylor expanded in M around inf 39.2%

      \[\leadsto e^{\color{blue}{-1 \cdot \left(M \cdot n\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 39.2%

         \[\leadsto e^{\color{blue}{\left(-1 \cdot M\right) \cdot n}} \]

      2. neg-mul-1 39.2%

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

   10. Simplified 39.2%

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

   if 0.104999999999999996 < l

   1. Initial program 81.4%

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

      1. add-exp-log 81.4%

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

      2. *-commutative 81.4%

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

      3. log-prod 42.9%

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

   4. Applied egg-rr 21.8%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in l around inf 97.2%

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

      1. neg-mul-1 97.2%

         \[\leadsto e^{\color{blue}{-\ell}} \]

   7. Simplified 97.2%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;e^{\ell}\\ \mathbf{elif}\;\ell \leq 0.105:\\ \;\;\;\;e^{n \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -105:\\ \;\;\;\;e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l) :precision binary64 (if (<= l -105.0) (exp l) (exp (- l))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -105.0) {
		tmp = exp(l);
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 <= (-105.0d0)) then
        tmp = exp(l)
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -105.0) {
		tmp = Math.exp(l);
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if l <= -105.0:
		tmp = math.exp(l)
	else:
		tmp = math.exp(-l)
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -105.0)
		tmp = exp(l);
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -105.0)
		tmp = exp(l);
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[l, -105.0], N[Exp[l], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -105

   1. Initial program 72.6%

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

      1. add-exp-log 69.4%

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

      2. *-commutative 69.4%

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

      3. log-prod 45.2%

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

   4. Applied egg-rr 18.2%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in l around inf 22.1%

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

      1. neg-mul-1 22.1%

         \[\leadsto e^{\color{blue}{-\ell}} \]

   7. Simplified 22.1%

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

      1. add-sqr-sqrt 22.1%

         \[\leadsto e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}} \]

      2. sqrt-unprod 22.1%

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

      3. sqr-neg 22.1%

         \[\leadsto e^{\sqrt{\color{blue}{\ell \cdot \ell}}} \]

      4. sqrt-unprod 0.0%

         \[\leadsto e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      5. add-sqr-sqrt 68.3%

         \[\leadsto e^{\color{blue}{\ell}} \]

      6. *-un-lft-identity 68.3%

         \[\leadsto e^{\color{blue}{1 \cdot \ell}} \]

   9. Applied egg-rr 68.3%

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

       1. *-lft-identity 68.3%

          \[\leadsto e^{\color{blue}{\ell}} \]

   11. Simplified 68.3%

       \[\leadsto e^{\color{blue}{\ell}} \]

   if -105 < l

   1. Initial program 80.7%

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

      1. add-exp-log 80.6%

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

      2. *-commutative 80.6%

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

      3. log-prod 46.0%

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

   4. Applied egg-rr 15.8%

      \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

   5. Taylor expanded in l around inf 44.3%

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

      1. neg-mul-1 44.3%

         \[\leadsto e^{\color{blue}{-\ell}} \]

   7. Simplified 44.3%

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

Alternative 9: 24.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ e^{\ell} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l) :precision binary64 (exp l))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return exp(l);
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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(l)
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(l);
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return math.exp(l)

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return exp(l)
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = exp(l);
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[Exp[l], $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. add-exp-log 77.8%

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

   2. *-commutative 77.8%

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

   3. log-prod 45.8%

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

4. Applied egg-rr 16.4%

   \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

5. Taylor expanded in l around inf 38.9%

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

   1. neg-mul-1 38.9%

      \[\leadsto e^{\color{blue}{-\ell}} \]

7. Simplified 38.9%

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

   1. add-sqr-sqrt 10.4%

      \[\leadsto e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}} \]

   2. sqrt-unprod 12.7%

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

   3. sqr-neg 12.7%

      \[\leadsto e^{\sqrt{\color{blue}{\ell \cdot \ell}}} \]

   4. sqrt-unprod 2.3%

      \[\leadsto e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

   5. add-sqr-sqrt 23.8%

      \[\leadsto e^{\color{blue}{\ell}} \]

   6. *-un-lft-identity 23.8%

      \[\leadsto e^{\color{blue}{1 \cdot \ell}} \]

9. Applied egg-rr 23.8%

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

    1. *-lft-identity 23.8%

       \[\leadsto e^{\color{blue}{\ell}} \]

11. Simplified 23.8%

    \[\leadsto e^{\color{blue}{\ell}} \]
12. Add Preprocessing

Alternative 10: 9.7% accurate, 32.7× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ 1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right) \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (+ 1.0 (* l (+ (* l (+ 0.5 (* l -0.16666666666666666))) -1.0))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return 1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0));
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 * ((l * (0.5d0 + (l * (-0.16666666666666666d0)))) + (-1.0d0)))
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	return 1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0));
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return 1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0))

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return Float64(1.0 + Float64(l * Float64(Float64(l * Float64(0.5 + Float64(l * -0.16666666666666666))) + -1.0)))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = 1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0));
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[(1.0 + N[(l * N[(N[(l * N[(0.5 + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. add-exp-log 77.8%

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

   2. *-commutative 77.8%

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

   3. log-prod 45.8%

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

4. Applied egg-rr 16.4%

   \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

5. Taylor expanded in l around inf 38.9%

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

   1. neg-mul-1 38.9%

      \[\leadsto e^{\color{blue}{-\ell}} \]

7. Simplified 38.9%

   \[\leadsto e^{\color{blue}{-\ell}} \]

8. Taylor expanded in l around 0 12.5%

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

9. Final simplification 12.5%

   \[\leadsto 1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right) \]
10. Add Preprocessing

Alternative 11: 9.2% accurate, 47.2× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ 1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right) \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l) :precision binary64 (+ 1.0 (* l (+ (* l 0.5) -1.0))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return 1.0 + (l * ((l * 0.5) + -1.0));
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 * ((l * 0.5d0) + (-1.0d0)))
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	return 1.0 + (l * ((l * 0.5) + -1.0));
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return 1.0 + (l * ((l * 0.5) + -1.0))

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return Float64(1.0 + Float64(l * Float64(Float64(l * 0.5) + -1.0)))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = 1.0 + (l * ((l * 0.5) + -1.0));
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[(1.0 + N[(l * N[(N[(l * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. add-exp-log 77.8%

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

   2. *-commutative 77.8%

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

   3. log-prod 45.8%

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

4. Applied egg-rr 16.4%

   \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

5. Taylor expanded in l around inf 38.9%

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

   1. neg-mul-1 38.9%

      \[\leadsto e^{\color{blue}{-\ell}} \]

7. Simplified 38.9%

   \[\leadsto e^{\color{blue}{-\ell}} \]

8. Taylor expanded in l around 0 11.4%

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

9. Final simplification 11.4%

   \[\leadsto 1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right) \]
10. Add Preprocessing

Alternative 12: 6.7% accurate, 425.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ 1 \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l) :precision binary64 1.0)

C

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

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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

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

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return 1.0

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return 1.0
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = 1.0;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 78.7%

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

   1. add-exp-log 77.8%

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

   2. *-commutative 77.8%

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

   3. log-prod 45.8%

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

4. Applied egg-rr 16.4%

   \[\leadsto \color{blue}{e^{\left(\left(m - n\right) + \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} - \ell\right)\right) + \log \cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}} \]

5. Taylor expanded in M around inf 8.4%

   \[\leadsto e^{\color{blue}{{M}^{2}}} \]

6. Taylor expanded in M around 0 8.3%

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

Reproduce

herbie shell --seed 2024151 
(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)))))))