Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.4% → 96.8%

Time: 23.9s

Alternatives: 12

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \left|m - n\right| - \ell\\ t_1 := e^{t_0 - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t_1 \leq \infty:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\left(K \cdot 0.5\right) \cdot \left(\left(n \cdot e^{t_0 - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\right) \cdot \sin M\right)\\ \end{array} \end{array} \]

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 (- (fabs (- m n)) l))
        (t_1 (exp (- t_0 (pow (- (/ (+ m n) 2.0) M) 2.0)))))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_1) INFINITY)
     (* (cos (- (/ K (/ 2.0 (+ m n))) M)) t_1)
     (*
      (* K 0.5)
      (* (* n (exp (- t_0 (pow (- (* (+ m n) 0.5) M) 2.0)))) (sin 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 = fabs((m - n)) - l;
	double t_1 = exp((t_0 - pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_1) <= ((double) INFINITY)) {
		tmp = cos(((K / (2.0 / (m + n))) - M)) * t_1;
	} else {
		tmp = (K * 0.5) * ((n * exp((t_0 - pow((((m + n) * 0.5) - M), 2.0)))) * sin(M));
	}
	return tmp;
}

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.abs((m - n)) - l;
	double t_1 = Math.exp((t_0 - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * t_1) <= Double.POSITIVE_INFINITY) {
		tmp = Math.cos(((K / (2.0 / (m + n))) - M)) * t_1;
	} else {
		tmp = (K * 0.5) * ((n * Math.exp((t_0 - Math.pow((((m + n) * 0.5) - M), 2.0)))) * Math.sin(M));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.fabs((m - n)) - l
	t_1 = math.exp((t_0 - math.pow((((m + n) / 2.0) - M), 2.0)))
	tmp = 0
	if (math.cos((((K * (m + n)) / 2.0) - M)) * t_1) <= math.inf:
		tmp = math.cos(((K / (2.0 / (m + n))) - M)) * t_1
	else:
		tmp = (K * 0.5) * ((n * math.exp((t_0 - math.pow((((m + n) * 0.5) - M), 2.0)))) * math.sin(M))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(m - n)) - l)
	t_1 = exp(Float64(t_0 - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_1) <= Inf)
		tmp = Float64(cos(Float64(Float64(K / Float64(2.0 / Float64(m + n))) - M)) * t_1);
	else
		tmp = Float64(Float64(K * 0.5) * Float64(Float64(n * exp(Float64(t_0 - (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0)))) * sin(M)));
	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 = abs((m - n)) - l;
	t_1 = exp((t_0 - ((((m + n) / 2.0) - M) ^ 2.0)));
	tmp = 0.0;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_1) <= Inf)
		tmp = cos(((K / (2.0 / (m + n))) - M)) * t_1;
	else
		tmp = (K * 0.5) * ((n * exp((t_0 - ((((m + n) * 0.5) - M) ^ 2.0)))) * sin(M));
	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[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(t$95$0 - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], Infinity], N[(N[Cos[N[(N[(K / N[(2.0 / N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(K * 0.5), $MachinePrecision] * N[(N[(n * N[Exp[N[(t$95$0 - N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 97.3%

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

      1. associate-/l* 98.3%

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

      2. +-commutative 98.3%

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

      3. fabs-sub 98.3%

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

      4. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 0.0%

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

      1. associate-/l* 0.0%

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

      2. +-commutative 0.0%

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

      3. fabs-sub 0.0%

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

      4. +-commutative 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in K around 0 31.5%

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

      1. cos-neg 31.5%

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

      2. sin-neg 31.5%

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

   6. Simplified 31.5%

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

   7. Taylor expanded in n around inf 98.1%

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

   9. Simplified 98.1%

      \[\leadsto \color{blue}{\left(0.5 \cdot K\right) \cdot \left(\left(n \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\right) \cdot \sin M\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \leq \infty:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(K \cdot 0.5\right) \cdot \left(\left(n \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\right) \cdot \sin M\right)\\ \end{array} \]

Alternative 2: 96.8% accurate, 1.0× speedup

Math

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

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 (- (- (fabs (- m n)) l) (pow (- (/ (+ m n) 2.0) M) 2.0))) (cos M)))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return exp(((fabs((m - n)) - l) - pow((((m + n) / 2.0) - M), 2.0))) * 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
    code = exp(((abs((m - n)) - l) - ((((m + n) / 2.0d0) - m_1) ** 2.0d0))) * 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) {
	return Math.exp(((Math.abs((m - n)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0))) * Math.cos(M);
}

Python

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

Julia

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

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = exp(((abs((m - n)) - l) - ((((m + n) / 2.0) - M) ^ 2.0))) * 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_] := N[(N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

   1. associate-/l* 77.6%

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

   2. +-commutative 77.6%

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

   3. fabs-sub 77.6%

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

   4. +-commutative 77.6%

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

3. Simplified 77.6%

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

4. Taylor expanded in K around 0 97.0%

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

   1. cos-neg 97.0%

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

6. Simplified 97.0%

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

7. Final simplification 97.0%

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

Alternative 3: 86.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;m \leq -12800000000000:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -7.4 \cdot 10^{-272}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

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 (<= m -12800000000000.0)
   (* (cos M) (exp (* (pow m 2.0) -0.25)))
   (if (<= m -7.4e-272)
     (*
      (cos (- (/ (* K (+ m n)) 2.0) M))
      (exp (+ (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) (- (fabs (- m n)) l))))
     (* (cos M) (exp (* -0.25 (pow n 2.0)))))))

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 (m <= -12800000000000.0) {
		tmp = cos(M) * exp((pow(m, 2.0) * -0.25));
	} else if (m <= -7.4e-272) {
		tmp = cos((((K * (m + n)) / 2.0) - M)) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (fabs((m - n)) - l)));
	} else {
		tmp = cos(M) * exp((-0.25 * pow(n, 2.0)));
	}
	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) :: tmp
    if (m <= (-12800000000000.0d0)) then
        tmp = cos(m_1) * exp(((m ** 2.0d0) * (-0.25d0)))
    else if (m <= (-7.4d-272)) then
        tmp = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) + (abs((m - n)) - l)))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n ** 2.0d0)))
    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 (m <= -12800000000000.0) {
		tmp = Math.cos(M) * Math.exp((Math.pow(m, 2.0) * -0.25));
	} else if (m <= -7.4e-272) {
		tmp = Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (Math.abs((m - n)) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if m <= -12800000000000.0:
		tmp = math.cos(M) * math.exp((math.pow(m, 2.0) * -0.25))
	elif m <= -7.4e-272:
		tmp = math.cos((((K * (m + n)) / 2.0) - M)) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (math.fabs((m - n)) - l)))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	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 (m <= -12800000000000.0)
		tmp = Float64(cos(M) * exp(Float64((m ^ 2.0) * -0.25)));
	elseif (m <= -7.4e-272)
		tmp = Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) + Float64(abs(Float64(m - n)) - l))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (n ^ 2.0))));
	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)
	tmp = 0.0;
	if (m <= -12800000000000.0)
		tmp = cos(M) * exp(((m ^ 2.0) * -0.25));
	elseif (m <= -7.4e-272)
		tmp = cos((((K * (m + n)) / 2.0) - M)) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (abs((m - n)) - l)));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	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_] := If[LessEqual[m, -12800000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -7.4e-272], N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.28e13

   1. Initial program 70.0%

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

      1. associate-/l* 70.0%

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

      2. +-commutative 70.0%

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

      3. fabs-sub 70.0%

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

      4. +-commutative 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 96.7%

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

      1. *-commutative 96.7%

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

   9. Simplified 96.7%

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

   if -1.28e13 < m < -7.3999999999999994e-272

   1. Initial program 91.9%

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

   2. Taylor expanded in m around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. unpow2 91.9%

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

      3. distribute-rgt-out 91.9%

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

      4. *-commutative 91.9%

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

      5. *-commutative 91.9%

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

   4. Simplified 91.9%

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

   if -7.3999999999999994e-272 < m

   1. Initial program 72.9%

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

      1. associate-/l* 72.8%

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

      2. +-commutative 72.8%

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

      3. fabs-sub 72.8%

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

      4. +-commutative 72.8%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in K around 0 95.7%

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

      1. cos-neg 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in n around inf 57.1%

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

      1. *-commutative 57.1%

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

   9. Simplified 57.1%

      \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -12800000000000:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -7.4 \cdot 10^{-272}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]

Alternative 4: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -2.2 \cdot 10^{-268}:\\ \;\;\;\;\frac{\cos M}{e^{{M}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

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 (<= m -53.0)
   (* (cos M) (exp (* (pow m 2.0) -0.25)))
   (if (<= m -2.2e-268)
     (/ (cos M) (exp (pow M 2.0)))
     (* (cos M) (exp (* -0.25 (pow n 2.0)))))))

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 (m <= -53.0) {
		tmp = cos(M) * exp((pow(m, 2.0) * -0.25));
	} else if (m <= -2.2e-268) {
		tmp = cos(M) / exp(pow(M, 2.0));
	} else {
		tmp = cos(M) * exp((-0.25 * pow(n, 2.0)));
	}
	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) :: tmp
    if (m <= (-53.0d0)) then
        tmp = cos(m_1) * exp(((m ** 2.0d0) * (-0.25d0)))
    else if (m <= (-2.2d-268)) then
        tmp = cos(m_1) / exp((m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n ** 2.0d0)))
    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 (m <= -53.0) {
		tmp = Math.cos(M) * Math.exp((Math.pow(m, 2.0) * -0.25));
	} else if (m <= -2.2e-268) {
		tmp = Math.cos(M) / Math.exp(Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if m <= -53.0:
		tmp = math.cos(M) * math.exp((math.pow(m, 2.0) * -0.25))
	elif m <= -2.2e-268:
		tmp = math.cos(M) / math.exp(math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	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 (m <= -53.0)
		tmp = Float64(cos(M) * exp(Float64((m ^ 2.0) * -0.25)));
	elseif (m <= -2.2e-268)
		tmp = Float64(cos(M) / exp((M ^ 2.0)));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (n ^ 2.0))));
	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)
	tmp = 0.0;
	if (m <= -53.0)
		tmp = cos(M) * exp(((m ^ 2.0) * -0.25));
	elseif (m <= -2.2e-268)
		tmp = cos(M) / exp((M ^ 2.0));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	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_] := If[LessEqual[m, -53.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -2.2e-268], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[Power[M, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -53

   1. Initial program 72.3%

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

      1. associate-/l* 72.3%

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

      2. +-commutative 72.3%

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

      3. fabs-sub 72.3%

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

      4. +-commutative 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 97.0%

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

      1. *-commutative 97.0%

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

   9. Simplified 97.0%

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

   if -53 < m < -2.20000000000000004e-268

   1. Initial program 90.9%

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

      1. associate-/l* 94.5%

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

      2. +-commutative 94.5%

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

      3. fabs-sub 94.5%

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

      4. +-commutative 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in K around 0 96.4%

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

      1. cos-neg 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in M around inf 53.7%

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

      1. mul-1-neg 53.7%

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

   9. Simplified 53.7%

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

       1. exp-neg 53.7%

          \[\leadsto \cos M \cdot \color{blue}{\frac{1}{e^{{M}^{2}}}} \]

       2. un-div-inv 53.7%

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

   11. Applied egg-rr 53.7%

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

   if -2.20000000000000004e-268 < m

   1. Initial program 73.3%

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

      1. associate-/l* 73.2%

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

      2. +-commutative 73.2%

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

      3. fabs-sub 73.2%

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

      4. +-commutative 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in K around 0 95.7%

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

      1. cos-neg 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in n around inf 57.8%

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

      1. *-commutative 57.8%

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

   9. Simplified 57.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -2.2 \cdot 10^{-268}:\\ \;\;\;\;\frac{\cos M}{e^{{M}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]

Alternative 5: 73.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;0.5 \cdot \left(m \cdot \left(e^{\ell} \cdot \left(K \cdot \sin M\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;\frac{\cos M}{e^{{M}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

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)
   (* 0.5 (* m (* (exp l) (* K (sin M)))))
   (if (<= l 720.0) (/ (cos M) (exp (pow M 2.0))) (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 = 0.5 * (m * (exp(l) * (K * sin(M))));
	} else if (l <= 720.0) {
		tmp = cos(M) / exp(pow(M, 2.0));
	} 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 = 0.5d0 * (m * (exp(l) * (k * sin(m_1))))
    else if (l <= 720.0d0) then
        tmp = cos(m_1) / exp((m_1 ** 2.0d0))
    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 = 0.5 * (m * (Math.exp(l) * (K * Math.sin(M))));
	} else if (l <= 720.0) {
		tmp = Math.cos(M) / Math.exp(Math.pow(M, 2.0));
	} 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 = 0.5 * (m * (math.exp(l) * (K * math.sin(M))))
	elif l <= 720.0:
		tmp = math.cos(M) / math.exp(math.pow(M, 2.0))
	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 = Float64(0.5 * Float64(m * Float64(exp(l) * Float64(K * sin(M)))));
	elseif (l <= 720.0)
		tmp = Float64(cos(M) / exp((M ^ 2.0)));
	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 = 0.5 * (m * (exp(l) * (K * sin(M))));
	elseif (l <= 720.0)
		tmp = cos(M) / exp((M ^ 2.0));
	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[(0.5 * N[(m * N[(N[Exp[l], $MachinePrecision] * N[(K * N[Sin[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 720.0], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[Power[M, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -720

   1. Initial program 77.4%

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

      1. associate-/l* 79.8%

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

      2. +-commutative 79.8%

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

      3. fabs-sub 79.8%

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

      4. +-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in K around 0 78.6%

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

      1. cos-neg 78.6%

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

      2. sin-neg 78.6%

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

   6. Simplified 78.6%

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

   7. Taylor expanded in l around inf 19.0%

      \[\leadsto \left(\cos M + -0.5 \cdot \left(K \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. mul-1-neg 23.5%

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

   9. Simplified 19.0%

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

   10. Taylor expanded in m around inf 15.4%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(e^{-\ell} \cdot \sin M\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative 15.4%

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

   12. Simplified 15.4%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot e^{-\ell}\right)\right)\right)} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 11.3%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(K \cdot \left(m \cdot \left(\sin M \cdot e^{-\ell}\right)\right)\right)\right)} \]

       2. expm1-udef 11.3%

          \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(K \cdot \left(m \cdot \left(\sin M \cdot e^{-\ell}\right)\right)\right)} - 1\right)} \]

       3. associate-*r* 8.9%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(K \cdot m\right) \cdot \left(\sin M \cdot e^{-\ell}\right)}\right)} - 1\right) \]

       4. *-commutative 8.9%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(m \cdot K\right)} \cdot \left(\sin M \cdot e^{-\ell}\right)\right)} - 1\right) \]

       5. associate-*l* 11.3%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{m \cdot \left(K \cdot \left(\sin M \cdot e^{-\ell}\right)\right)}\right)} - 1\right) \]

       6. add-sqr-sqrt 11.3%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)\right)\right)} - 1\right) \]

       7. sqrt-unprod 11.3%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)\right)\right)} - 1\right) \]

       8. sqr-neg 11.3%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)\right)\right)} - 1\right) \]

       9. sqrt-unprod 0.0%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)\right)\right)} - 1\right) \]

       10. add-sqr-sqrt 74.2%

           \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\color{blue}{\ell}}\right)\right)\right)} - 1\right) \]

   14. Applied egg-rr 74.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\ell}\right)\right)\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. expm1-def 74.2%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\ell}\right)\right)\right)\right)} \]

       2. expm1-log1p 74.2%

          \[\leadsto 0.5 \cdot \color{blue}{\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\ell}\right)\right)\right)} \]

       3. associate-*r* 74.2%

          \[\leadsto 0.5 \cdot \left(m \cdot \color{blue}{\left(\left(K \cdot \sin M\right) \cdot e^{\ell}\right)}\right) \]

       4. *-commutative 74.2%

          \[\leadsto 0.5 \cdot \left(m \cdot \color{blue}{\left(e^{\ell} \cdot \left(K \cdot \sin M\right)\right)}\right) \]

   16. Simplified 74.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(m \cdot \left(e^{\ell} \cdot \left(K \cdot \sin M\right)\right)\right)} \]

   if -720 < l < 720

   1. Initial program 77.6%

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

      1. associate-/l* 77.5%

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

      2. +-commutative 77.5%

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

      3. fabs-sub 77.5%

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

      4. +-commutative 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in M around inf 60.8%

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

      1. mul-1-neg 60.8%

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

   9. Simplified 60.8%

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

       1. exp-neg 60.8%

          \[\leadsto \cos M \cdot \color{blue}{\frac{1}{e^{{M}^{2}}}} \]

       2. un-div-inv 60.8%

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

   11. Applied egg-rr 60.8%

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

   if 720 < l

   1. Initial program 74.0%

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

      1. associate-/l* 74.0%

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

      2. +-commutative 74.0%

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

      3. fabs-sub 74.0%

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

      4. +-commutative 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in K around 0 84.0%

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

      1. cos-neg 84.0%

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

      2. sin-neg 84.0%

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

   6. Simplified 84.0%

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

   7. Taylor expanded in l around inf 84.0%

      \[\leadsto \left(\cos M + -0.5 \cdot \left(K \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. mul-1-neg 74.0%

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

   9. Simplified 84.0%

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

   10. Taylor expanded in M around 0 100.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;0.5 \cdot \left(m \cdot \left(e^{\ell} \cdot \left(K \cdot \sin M\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;\frac{\cos M}{e^{{M}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 6: 74.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{{M}^{2}}}\\ \end{array} \end{array} \]

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 (<= m -53.0)
   (* (cos M) (exp (* (pow m 2.0) -0.25)))
   (/ (cos M) (exp (pow M 2.0)))))

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 (m <= -53.0) {
		tmp = cos(M) * exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = cos(M) / exp(pow(M, 2.0));
	}
	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) :: tmp
    if (m <= (-53.0d0)) then
        tmp = cos(m_1) * exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = cos(m_1) / exp((m_1 ** 2.0d0))
    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 (m <= -53.0) {
		tmp = Math.cos(M) * Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.cos(M) / Math.exp(Math.pow(M, 2.0));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if m <= -53.0:
		tmp = math.cos(M) * math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.cos(M) / math.exp(math.pow(M, 2.0))
	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 (m <= -53.0)
		tmp = Float64(cos(M) * exp(Float64((m ^ 2.0) * -0.25)));
	else
		tmp = Float64(cos(M) / exp((M ^ 2.0)));
	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)
	tmp = 0.0;
	if (m <= -53.0)
		tmp = cos(M) * exp(((m ^ 2.0) * -0.25));
	else
		tmp = cos(M) / exp((M ^ 2.0));
	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_] := If[LessEqual[m, -53.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[Power[M, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -53

   1. Initial program 72.3%

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

      1. associate-/l* 72.3%

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

      2. +-commutative 72.3%

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

      3. fabs-sub 72.3%

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

      4. +-commutative 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 97.0%

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

      1. *-commutative 97.0%

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

   9. Simplified 97.0%

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

   if -53 < m

   1. Initial program 78.3%

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

      1. associate-/l* 79.4%

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

      2. +-commutative 79.4%

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

      3. fabs-sub 79.4%

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

      4. +-commutative 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in K around 0 95.9%

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

      1. cos-neg 95.9%

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

   6. Simplified 95.9%

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

   7. Taylor expanded in M around inf 60.2%

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

      1. mul-1-neg 60.2%

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

   9. Simplified 60.2%

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

       1. exp-neg 60.2%

          \[\leadsto \cos M \cdot \color{blue}{\frac{1}{e^{{M}^{2}}}} \]

       2. un-div-inv 60.2%

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

   11. Applied egg-rr 60.2%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{{M}^{2}}}\\ \end{array} \]

Alternative 7: 41.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \frac{m}{\frac{e^{\ell}}{\sin M}}\right)\\ \end{array} \end{array} \]

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 (<= n 2.5e-17)
   (* (cos M) (exp (- l)))
   (* 0.5 (* K (/ m (/ (exp l) (sin M)))))))

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 (n <= 2.5e-17) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = 0.5 * (K * (m / (exp(l) / sin(M))));
	}
	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 (n <= 2.5d-17) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = 0.5d0 * (k * (m / (exp(l) / sin(m_1))))
    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 (n <= 2.5e-17) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = 0.5 * (K * (m / (Math.exp(l) / Math.sin(M))));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if n <= 2.5e-17:
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = 0.5 * (K * (m / (math.exp(l) / math.sin(M))))
	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 (n <= 2.5e-17)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = Float64(0.5 * Float64(K * Float64(m / Float64(exp(l) / sin(M)))));
	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 (n <= 2.5e-17)
		tmp = cos(M) * exp(-l);
	else
		tmp = 0.5 * (K * (m / (exp(l) / sin(M))));
	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[n, 2.5e-17], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(K * N[(m / N[(N[Exp[l], $MachinePrecision] / N[Sin[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.4999999999999999e-17

   1. Initial program 78.6%

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

      1. associate-/l* 79.7%

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

      2. +-commutative 79.7%

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

      3. fabs-sub 79.7%

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

      4. +-commutative 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in l around inf 32.8%

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

      1. mul-1-neg 32.8%

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

   6. Simplified 32.8%

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

   7. Taylor expanded in K around 0 35.1%

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

      1. cos-neg 35.1%

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

   9. Simplified 35.1%

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

   if 2.4999999999999999e-17 < n

   1. Initial program 71.6%

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

      1. associate-/l* 71.6%

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

      2. +-commutative 71.6%

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

      3. fabs-sub 71.6%

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

      4. +-commutative 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in K around 0 76.1%

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

      1. cos-neg 76.1%

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

      2. sin-neg 76.1%

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

   6. Simplified 76.1%

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

   7. Taylor expanded in l around inf 18.0%

      \[\leadsto \left(\cos M + -0.5 \cdot \left(K \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. mul-1-neg 17.8%

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

   9. Simplified 18.0%

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

   10. Taylor expanded in m around inf 36.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(e^{-\ell} \cdot \sin M\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative 36.0%

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

   12. Simplified 36.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot e^{-\ell}\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 35.9%

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

       2. exp-neg 35.9%

          \[\leadsto 0.5 \cdot \left(K \cdot \left(\left(m \cdot \sin M\right) \cdot \color{blue}{\frac{1}{e^{\ell}}}\right)\right) \]

       3. un-div-inv 35.9%

          \[\leadsto 0.5 \cdot \left(K \cdot \color{blue}{\frac{m \cdot \sin M}{e^{\ell}}}\right) \]

   14. Applied egg-rr 35.9%

       \[\leadsto 0.5 \cdot \left(K \cdot \color{blue}{\frac{m \cdot \sin M}{e^{\ell}}}\right) \]
   15. Step-by-step derivation

       1. associate-/l* 37.4%

          \[\leadsto 0.5 \cdot \left(K \cdot \color{blue}{\frac{m}{\frac{e^{\ell}}{\sin M}}}\right) \]

   16. Simplified 37.4%

       \[\leadsto 0.5 \cdot \left(K \cdot \color{blue}{\frac{m}{\frac{e^{\ell}}{\sin M}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \frac{m}{\frac{e^{\ell}}{\sin M}}\right)\\ \end{array} \]

Alternative 8: 53.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \left(m \cdot \left(e^{\ell} \cdot \left(K \cdot \sin M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

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 4e-97)
   (* 0.5 (* m (* (exp l) (* K (sin M)))))
   (* (cos 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 <= 4e-97) {
		tmp = 0.5 * (m * (exp(l) * (K * sin(M))));
	} else {
		tmp = cos(M) * 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 <= 4d-97) then
        tmp = 0.5d0 * (m * (exp(l) * (k * sin(m_1))))
    else
        tmp = cos(m_1) * 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 <= 4e-97) {
		tmp = 0.5 * (m * (Math.exp(l) * (K * Math.sin(M))));
	} else {
		tmp = Math.cos(M) * 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 <= 4e-97:
		tmp = 0.5 * (m * (math.exp(l) * (K * math.sin(M))))
	else:
		tmp = math.cos(M) * 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 <= 4e-97)
		tmp = Float64(0.5 * Float64(m * Float64(exp(l) * Float64(K * sin(M)))));
	else
		tmp = Float64(cos(M) * 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 <= 4e-97)
		tmp = 0.5 * (m * (exp(l) * (K * sin(M))));
	else
		tmp = cos(M) * 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, 4e-97], N[(0.5 * N[(m * N[(N[Exp[l], $MachinePrecision] * N[(K * N[Sin[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.00000000000000014e-97

   1. Initial program 77.2%

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

      1. associate-/l* 78.3%

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

      2. +-commutative 78.3%

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

      3. fabs-sub 78.3%

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

      4. +-commutative 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in K around 0 80.8%

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

      1. cos-neg 80.8%

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

      2. sin-neg 80.8%

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

   6. Simplified 80.8%

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

   7. Taylor expanded in l around inf 15.3%

      \[\leadsto \left(\cos M + -0.5 \cdot \left(K \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. mul-1-neg 17.5%

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

   9. Simplified 15.3%

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

   10. Taylor expanded in m around inf 17.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(e^{-\ell} \cdot \sin M\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative 17.2%

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

   12. Simplified 17.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot e^{-\ell}\right)\right)\right)} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 15.1%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(K \cdot \left(m \cdot \left(\sin M \cdot e^{-\ell}\right)\right)\right)\right)} \]

       2. expm1-udef 29.2%

          \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(K \cdot \left(m \cdot \left(\sin M \cdot e^{-\ell}\right)\right)\right)} - 1\right)} \]

       3. associate-*r* 28.1%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(K \cdot m\right) \cdot \left(\sin M \cdot e^{-\ell}\right)}\right)} - 1\right) \]

       4. *-commutative 28.1%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(m \cdot K\right)} \cdot \left(\sin M \cdot e^{-\ell}\right)\right)} - 1\right) \]

       5. associate-*l* 29.2%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{m \cdot \left(K \cdot \left(\sin M \cdot e^{-\ell}\right)\right)}\right)} - 1\right) \]

       6. add-sqr-sqrt 17.7%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)\right)\right)} - 1\right) \]

       7. sqrt-unprod 29.2%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)\right)\right)} - 1\right) \]

       8. sqr-neg 29.2%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)\right)\right)} - 1\right) \]

       9. sqrt-unprod 11.5%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)\right)\right)} - 1\right) \]

       10. add-sqr-sqrt 57.6%

           \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\color{blue}{\ell}}\right)\right)\right)} - 1\right) \]

   14. Applied egg-rr 57.6%

       \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\ell}\right)\right)\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. expm1-def 45.9%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\ell}\right)\right)\right)\right)} \]

       2. expm1-log1p 46.2%

          \[\leadsto 0.5 \cdot \color{blue}{\left(m \cdot \left(K \cdot \left(\sin M \cdot e^{\ell}\right)\right)\right)} \]

       3. associate-*r* 46.2%

          \[\leadsto 0.5 \cdot \left(m \cdot \color{blue}{\left(\left(K \cdot \sin M\right) \cdot e^{\ell}\right)}\right) \]

       4. *-commutative 46.2%

          \[\leadsto 0.5 \cdot \left(m \cdot \color{blue}{\left(e^{\ell} \cdot \left(K \cdot \sin M\right)\right)}\right) \]

   16. Simplified 46.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(m \cdot \left(e^{\ell} \cdot \left(K \cdot \sin M\right)\right)\right)} \]

   if 4.00000000000000014e-97 < l

   1. Initial program 75.7%

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

      1. associate-/l* 75.7%

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

      2. +-commutative 75.7%

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

      3. fabs-sub 75.7%

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

      4. +-commutative 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in l around inf 59.1%

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

      1. mul-1-neg 59.1%

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

   6. Simplified 59.1%

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

   7. Taylor expanded in K around 0 77.9%

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

      1. cos-neg 77.9%

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

   9. Simplified 77.9%

      \[\leadsto \color{blue}{\cos M \cdot e^{-\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \left(m \cdot \left(e^{\ell} \cdot \left(K \cdot \sin M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 9: 40.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \frac{M}{e^{\ell}}\right)\right)\\ \end{array} \end{array} \]

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 (<= n 5.6e-17) (* (cos M) (exp (- l))) (* 0.5 (* K (* m (/ 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 (n <= 5.6e-17) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = 0.5 * (K * (m * (M / 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 (n <= 5.6d-17) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = 0.5d0 * (k * (m * (m_1 / 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 (n <= 5.6e-17) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = 0.5 * (K * (m * (M / 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 n <= 5.6e-17:
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = 0.5 * (K * (m * (M / 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 (n <= 5.6e-17)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = Float64(0.5 * Float64(K * Float64(m * Float64(M / exp(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 (n <= 5.6e-17)
		tmp = cos(M) * exp(-l);
	else
		tmp = 0.5 * (K * (m * (M / 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[n, 5.6e-17], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(K * N[(m * N[(M / N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5.5999999999999998e-17

   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. Step-by-step derivation

      1. associate-/l* 79.8%

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

      2. +-commutative 79.8%

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

      3. fabs-sub 79.8%

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

      4. +-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in l around inf 32.6%

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

      1. mul-1-neg 32.6%

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

   6. Simplified 32.6%

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

   7. Taylor expanded in K around 0 34.9%

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

      1. cos-neg 34.9%

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

   9. Simplified 34.9%

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

   if 5.5999999999999998e-17 < n

   1. Initial program 71.2%

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

      1. associate-/l* 71.2%

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

      2. +-commutative 71.2%

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

      3. fabs-sub 71.2%

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

      4. +-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in K around 0 75.8%

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

      1. cos-neg 75.8%

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

      2. sin-neg 75.8%

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

   6. Simplified 75.8%

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

   7. Taylor expanded in l around inf 18.2%

      \[\leadsto \left(\cos M + -0.5 \cdot \left(K \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. mul-1-neg 18.0%

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

   9. Simplified 18.2%

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

   10. Taylor expanded in m around inf 36.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(e^{-\ell} \cdot \sin M\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative 36.5%

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

   12. Simplified 36.5%

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

   13. Taylor expanded in M around 0 36.4%

       \[\leadsto 0.5 \cdot \left(K \cdot \left(m \cdot \color{blue}{\left(M \cdot e^{-\ell}\right)}\right)\right) \]
   14. Step-by-step derivation

       1. exp-neg 36.4%

          \[\leadsto 0.5 \cdot \left(K \cdot \left(m \cdot \left(M \cdot \color{blue}{\frac{1}{e^{\ell}}}\right)\right)\right) \]

       2. associate-*r/ 36.4%

          \[\leadsto 0.5 \cdot \left(K \cdot \left(m \cdot \color{blue}{\frac{M \cdot 1}{e^{\ell}}}\right)\right) \]

       3. *-rgt-identity 36.4%

          \[\leadsto 0.5 \cdot \left(K \cdot \left(m \cdot \frac{\color{blue}{M}}{e^{\ell}}\right)\right) \]

   15. Simplified 36.4%

       \[\leadsto 0.5 \cdot \left(K \cdot \left(m \cdot \color{blue}{\frac{M}{e^{\ell}}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \frac{M}{e^{\ell}}\right)\right)\\ \end{array} \]

Alternative 10: 39.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \frac{M}{e^{\ell}}\right)\right)\\ \end{array} \end{array} \]

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 (<= n 5.6e-17) (exp (- l)) (* 0.5 (* K (* m (/ 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 (n <= 5.6e-17) {
		tmp = exp(-l);
	} else {
		tmp = 0.5 * (K * (m * (M / 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 (n <= 5.6d-17) then
        tmp = exp(-l)
    else
        tmp = 0.5d0 * (k * (m * (m_1 / 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 (n <= 5.6e-17) {
		tmp = Math.exp(-l);
	} else {
		tmp = 0.5 * (K * (m * (M / 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 n <= 5.6e-17:
		tmp = math.exp(-l)
	else:
		tmp = 0.5 * (K * (m * (M / 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 (n <= 5.6e-17)
		tmp = exp(Float64(-l));
	else
		tmp = Float64(0.5 * Float64(K * Float64(m * Float64(M / exp(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 (n <= 5.6e-17)
		tmp = exp(-l);
	else
		tmp = 0.5 * (K * (m * (M / 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[n, 5.6e-17], N[Exp[(-l)], $MachinePrecision], N[(0.5 * N[(K * N[(m * N[(M / N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5.5999999999999998e-17

   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. Step-by-step derivation

      1. associate-/l* 79.8%

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

      2. +-commutative 79.8%

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

      3. fabs-sub 79.8%

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

      4. +-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in K around 0 84.8%

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

      1. cos-neg 84.8%

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

      2. sin-neg 84.8%

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

   6. Simplified 84.8%

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

   7. Taylor expanded in l around inf 33.1%

      \[\leadsto \left(\cos M + -0.5 \cdot \left(K \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. mul-1-neg 32.6%

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

   9. Simplified 33.1%

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

   10. Taylor expanded in M around 0 34.9%

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

   if 5.5999999999999998e-17 < n

   1. Initial program 71.2%

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

      1. associate-/l* 71.2%

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

      2. +-commutative 71.2%

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

      3. fabs-sub 71.2%

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

      4. +-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in K around 0 75.8%

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

      1. cos-neg 75.8%

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

      2. sin-neg 75.8%

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

   6. Simplified 75.8%

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

   7. Taylor expanded in l around inf 18.2%

      \[\leadsto \left(\cos M + -0.5 \cdot \left(K \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. mul-1-neg 18.0%

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

   9. Simplified 18.2%

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

   10. Taylor expanded in m around inf 36.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(e^{-\ell} \cdot \sin M\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative 36.5%

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

   12. Simplified 36.5%

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

   13. Taylor expanded in M around 0 36.4%

       \[\leadsto 0.5 \cdot \left(K \cdot \left(m \cdot \color{blue}{\left(M \cdot e^{-\ell}\right)}\right)\right) \]
   14. Step-by-step derivation

       1. exp-neg 36.4%

          \[\leadsto 0.5 \cdot \left(K \cdot \left(m \cdot \left(M \cdot \color{blue}{\frac{1}{e^{\ell}}}\right)\right)\right) \]

       2. associate-*r/ 36.4%

          \[\leadsto 0.5 \cdot \left(K \cdot \left(m \cdot \color{blue}{\frac{M \cdot 1}{e^{\ell}}}\right)\right) \]

       3. *-rgt-identity 36.4%

          \[\leadsto 0.5 \cdot \left(K \cdot \left(m \cdot \frac{\color{blue}{M}}{e^{\ell}}\right)\right) \]

   15. Simplified 36.4%

       \[\leadsto 0.5 \cdot \left(K \cdot \left(m \cdot \color{blue}{\frac{M}{e^{\ell}}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \frac{M}{e^{\ell}}\right)\right)\\ \end{array} \]

Alternative 11: 35.4% 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(Float64(-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 76.8%

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

   1. associate-/l* 77.6%

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

   2. +-commutative 77.6%

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

   3. fabs-sub 77.6%

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

   4. +-commutative 77.6%

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

3. Simplified 77.6%

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

4. Taylor expanded in K around 0 82.5%

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

   1. cos-neg 82.5%

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

   2. sin-neg 82.5%

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

6. Simplified 82.5%

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

7. Taylor expanded in l around inf 29.2%

   \[\leadsto \left(\cos M + -0.5 \cdot \left(K \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
8. Step-by-step derivation

   1. mul-1-neg 28.9%

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

9. Simplified 29.2%

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

10. Taylor expanded in M around 0 33.0%

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

11. Final simplification 33.0%

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

Alternative 12: 7.1% accurate, 4.2× speedup

Math

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

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 (cos M))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return 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
    code = 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) {
	return Math.cos(M);
}

Python

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

Julia

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

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = 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_] := N[Cos[M], $MachinePrecision]

Derivation

1. Initial program 76.8%

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

   1. associate-/l* 77.6%

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

   2. +-commutative 77.6%

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

   3. fabs-sub 77.6%

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

   4. +-commutative 77.6%

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

3. Simplified 77.6%

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

4. Taylor expanded in l around inf 28.9%

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

   1. mul-1-neg 28.9%

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

6. Simplified 28.9%

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

7. Taylor expanded in l around 0 7.7%

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

8. Taylor expanded in K around 0 8.2%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \]
9. Step-by-step derivation

   1. cos-neg 8.2%

      \[\leadsto \color{blue}{\cos M} \]

10. Simplified 8.2%

    \[\leadsto \color{blue}{\cos M} \]

11. Final simplification 8.2%

    \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023339 
(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)))))))