Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.3% → 96.6%

Time: 18.9s

Alternatives: 10

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% 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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. associate-/l* 80.0%

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

   2. +-commutative 80.0%

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

   3. fabs-sub 80.0%

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

   4. +-commutative 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

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

Alternative 2: 75.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 1.25 \cdot 10^{-146}:\\ \;\;\;\;-0.5 \cdot \left(\left(n \cdot K\right) \cdot \left(e^{t_0 + \left(\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) - \ell\right)} \cdot \left(-\sin M\right)\right)\right)\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{n}} - M\right) \cdot e^{M \cdot \left(n - M\right) + \left(t_0 - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 1.25e-146)
     (*
      -0.5
      (*
       (* n K)
       (*
        (exp (+ t_0 (- (* (+ n (- (* m 0.5) M)) (- M (* m 0.5))) l)))
        (- (sin M)))))
     (if (<= n 54.0)
       (* (cos (- (/ K (/ 2.0 n)) M)) (exp (+ (* M (- n M)) (- t_0 l))))
       (* (cos M) (exp (* -0.25 (* n n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= 1.25e-146) {
		tmp = -0.5 * ((n * K) * (exp((t_0 + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - l))) * -sin(M)));
	} else if (n <= 54.0) {
		tmp = cos(((K / (2.0 / n)) - M)) * exp(((M * (n - M)) + (t_0 - l)));
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (n <= 1.25d-146) then
        tmp = (-0.5d0) * ((n * k) * (exp((t_0 + (((n + ((m * 0.5d0) - m_1)) * (m_1 - (m * 0.5d0))) - l))) * -sin(m_1)))
    else if (n <= 54.0d0) then
        tmp = cos(((k / (2.0d0 / n)) - m_1)) * exp(((m_1 * (n - m_1)) + (t_0 - l)))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (n <= 1.25e-146) {
		tmp = -0.5 * ((n * K) * (Math.exp((t_0 + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - l))) * -Math.sin(M)));
	} else if (n <= 54.0) {
		tmp = Math.cos(((K / (2.0 / n)) - M)) * Math.exp(((M * (n - M)) + (t_0 - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 1.25e-146:
		tmp = -0.5 * ((n * K) * (math.exp((t_0 + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - l))) * -math.sin(M)))
	elif n <= 54.0:
		tmp = math.cos(((K / (2.0 / n)) - M)) * math.exp(((M * (n - M)) + (t_0 - l)))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 1.25e-146)
		tmp = Float64(-0.5 * Float64(Float64(n * K) * Float64(exp(Float64(t_0 + Float64(Float64(Float64(n + Float64(Float64(m * 0.5) - M)) * Float64(M - Float64(m * 0.5))) - l))) * Float64(-sin(M)))));
	elseif (n <= 54.0)
		tmp = Float64(cos(Float64(Float64(K / Float64(2.0 / n)) - M)) * exp(Float64(Float64(M * Float64(n - M)) + Float64(t_0 - l))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (n <= 1.25e-146)
		tmp = -0.5 * ((n * K) * (exp((t_0 + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - l))) * -sin(M)));
	elseif (n <= 54.0)
		tmp = cos(((K / (2.0 / n)) - M)) * exp(((M * (n - M)) + (t_0 - l)));
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 1.25e-146], N[(-0.5 * N[(N[(n * K), $MachinePrecision] * N[(N[Exp[N[(t$95$0 + N[(N[(N[(n + N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sin[M], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[(N[Cos[N[(N[(K / N[(2.0 / n), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.24999999999999989e-146

   1. Initial program 85.3%

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

      1. associate-/l* 85.9%

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

      2. +-commutative 85.9%

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

      3. fabs-sub 85.9%

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

      4. +-commutative 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in n around 0 72.2%

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

      1. +-commutative 72.2%

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

      2. unpow2 72.2%

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

      3. distribute-rgt-out 73.5%

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

      4. *-commutative 73.5%

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

      5. *-commutative 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in m around 0 80.4%

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

   9. Taylor expanded in K around 0 79.5%

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

       1. cos-neg 79.5%

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

   11. Simplified 79.5%

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

   12. Taylor expanded in K around inf 76.4%

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

   14. Simplified 73.9%

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

   if 1.24999999999999989e-146 < n < 54

   1. Initial program 83.3%

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

      1. associate-/l* 83.3%

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

      2. +-commutative 83.3%

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

      3. fabs-sub 83.3%

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

      4. +-commutative 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in n around 0 83.3%

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

      1. +-commutative 83.3%

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

      2. unpow2 83.3%

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

      3. distribute-rgt-out 83.3%

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

      4. *-commutative 83.3%

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

      5. *-commutative 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in m around 0 93.3%

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

   9. Taylor expanded in m around 0 67.1%

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

       1. mul-1-neg 67.1%

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

       2. distribute-rgt-neg-in 67.1%

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

   11. Simplified 67.1%

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

   if 54 < n

   1. Initial program 64.7%

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

      1. associate-/l* 64.7%

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

      2. +-commutative 64.7%

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

      3. fabs-sub 64.7%

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

      4. +-commutative 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in n around inf 94.2%

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

      1. unpow2 94.2%

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

   10. Applied egg-rr 94.2%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.25 \cdot 10^{-146}:\\ \;\;\;\;-0.5 \cdot \left(\left(n \cdot K\right) \cdot \left(e^{\left|n - m\right| + \left(\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) - \ell\right)} \cdot \left(-\sin M\right)\right)\right)\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{n}} - M\right) \cdot e^{M \cdot \left(n - M\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 290:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{n}} - M\right) \cdot e^{\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 290.0)
   (*
    (cos (- (/ K (/ 2.0 n)) M))
    (exp (+ (* (+ n (- (* m 0.5) M)) (- M (* m 0.5))) (- (fabs (- n m)) l))))
   (* (cos M) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 290.0) {
		tmp = cos(((K / (2.0 / n)) - M)) * exp((((n + ((m * 0.5) - M)) * (M - (m * 0.5))) + (fabs((n - m)) - l)));
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 290.0d0) then
        tmp = cos(((k / (2.0d0 / n)) - m_1)) * exp((((n + ((m * 0.5d0) - m_1)) * (m_1 - (m * 0.5d0))) + (abs((n - m)) - l)))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 290.0) {
		tmp = Math.cos(((K / (2.0 / n)) - M)) * Math.exp((((n + ((m * 0.5) - M)) * (M - (m * 0.5))) + (Math.abs((n - m)) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 290.0:
		tmp = math.cos(((K / (2.0 / n)) - M)) * math.exp((((n + ((m * 0.5) - M)) * (M - (m * 0.5))) + (math.fabs((n - m)) - l)))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 290.0)
		tmp = Float64(cos(Float64(Float64(K / Float64(2.0 / n)) - M)) * exp(Float64(Float64(Float64(n + Float64(Float64(m * 0.5) - M)) * Float64(M - Float64(m * 0.5))) + Float64(abs(Float64(n - m)) - l))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 290.0)
		tmp = cos(((K / (2.0 / n)) - M)) * exp((((n + ((m * 0.5) - M)) * (M - (m * 0.5))) + (abs((n - m)) - l)));
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 290.0], N[(N[Cos[N[(N[(K / N[(2.0 / n), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(n + N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 290

   1. Initial program 85.0%

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

      1. associate-/l* 85.5%

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

      2. +-commutative 85.5%

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

      3. fabs-sub 85.5%

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

      4. +-commutative 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in n around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. unpow2 73.9%

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

      3. distribute-rgt-out 75.0%

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

      4. *-commutative 75.0%

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

      5. *-commutative 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in m around 0 82.5%

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

   if 290 < n

   1. Initial program 64.7%

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

      1. associate-/l* 64.7%

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

      2. +-commutative 64.7%

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

      3. fabs-sub 64.7%

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

      4. +-commutative 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in n around inf 94.2%

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

      1. unpow2 94.2%

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

   10. Applied egg-rr 94.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 290:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{n}} - M\right) \cdot e^{\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-{M}^{2}}\\ t_1 := e^{\left(\left|n - m\right| - \ell\right) - \left(m \cdot 0.5\right) \cdot \left(n + m \cdot 0.5\right)}\\ \mathbf{if}\;n \leq 4.1 \cdot 10^{-235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-201}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- (pow M 2.0)))))
        (t_1 (exp (- (- (fabs (- n m)) l) (* (* m 0.5) (+ n (* m 0.5)))))))
   (if (<= n 4.1e-235)
     t_1
     (if (<= n 3.5e-201)
       t_0
       (if (<= n 1.05e-49)
         t_1
         (if (<= n 54.0) t_0 (* (cos M) (exp (* -0.25 (* n n))))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp(-pow(M, 2.0));
	double t_1 = exp(((fabs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	double tmp;
	if (n <= 4.1e-235) {
		tmp = t_1;
	} else if (n <= 3.5e-201) {
		tmp = t_0;
	} else if (n <= 1.05e-49) {
		tmp = t_1;
	} else if (n <= 54.0) {
		tmp = t_0;
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(m_1) * exp(-(m_1 ** 2.0d0))
    t_1 = exp(((abs((n - m)) - l) - ((m * 0.5d0) * (n + (m * 0.5d0)))))
    if (n <= 4.1d-235) then
        tmp = t_1
    else if (n <= 3.5d-201) then
        tmp = t_0
    else if (n <= 1.05d-49) then
        tmp = t_1
    else if (n <= 54.0d0) then
        tmp = t_0
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	double t_1 = Math.exp(((Math.abs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	double tmp;
	if (n <= 4.1e-235) {
		tmp = t_1;
	} else if (n <= 3.5e-201) {
		tmp = t_0;
	} else if (n <= 1.05e-49) {
		tmp = t_1;
	} else if (n <= 54.0) {
		tmp = t_0;
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp(-math.pow(M, 2.0))
	t_1 = math.exp(((math.fabs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))))
	tmp = 0
	if n <= 4.1e-235:
		tmp = t_1
	elif n <= 3.5e-201:
		tmp = t_0
	elif n <= 1.05e-49:
		tmp = t_1
	elif n <= 54.0:
		tmp = t_0
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-(M ^ 2.0))))
	t_1 = exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(Float64(m * 0.5) * Float64(n + Float64(m * 0.5)))))
	tmp = 0.0
	if (n <= 4.1e-235)
		tmp = t_1;
	elseif (n <= 3.5e-201)
		tmp = t_0;
	elseif (n <= 1.05e-49)
		tmp = t_1;
	elseif (n <= 54.0)
		tmp = t_0;
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp(-(M ^ 2.0));
	t_1 = exp(((abs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	tmp = 0.0;
	if (n <= 4.1e-235)
		tmp = t_1;
	elseif (n <= 3.5e-201)
		tmp = t_0;
	elseif (n <= 1.05e-49)
		tmp = t_1;
	elseif (n <= 54.0)
		tmp = t_0;
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[(m * 0.5), $MachinePrecision] * N[(n + N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 4.1e-235], t$95$1, If[LessEqual[n, 3.5e-201], t$95$0, If[LessEqual[n, 1.05e-49], t$95$1, If[LessEqual[n, 54.0], t$95$0, N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < 4.09999999999999997e-235 or 3.50000000000000008e-201 < n < 1.0499999999999999e-49

   1. Initial program 84.7%

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

      1. associate-/l* 85.2%

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

      2. +-commutative 85.2%

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

      3. fabs-sub 85.2%

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

      4. +-commutative 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in n around 0 72.5%

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

      1. +-commutative 72.5%

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

      2. unpow2 72.5%

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

      3. distribute-rgt-out 73.7%

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

      4. *-commutative 73.7%

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

      5. *-commutative 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in m around 0 80.8%

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

   9. Taylor expanded in K around 0 81.1%

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

       1. cos-neg 81.1%

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

   11. Simplified 81.1%

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

   12. Taylor expanded in M around 0 72.2%

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

       1. associate--r+ 72.2%

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

       2. associate-*r* 72.2%

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

       3. *-commutative 72.2%

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

       4. *-commutative 72.2%

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

   14. Simplified 72.2%

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

   if 4.09999999999999997e-235 < n < 3.50000000000000008e-201 or 1.0499999999999999e-49 < n < 54

   1. Initial program 88.2%

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

      1. associate-/l* 88.2%

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

      2. +-commutative 88.2%

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

      3. fabs-sub 88.2%

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

      4. +-commutative 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around inf 77.2%

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

      1. mul-1-neg 77.2%

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

   10. Simplified 77.2%

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

   if 54 < n

   1. Initial program 64.7%

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

      1. associate-/l* 64.7%

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

      2. +-commutative 64.7%

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

      3. fabs-sub 64.7%

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

      4. +-commutative 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in n around inf 94.2%

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

      1. unpow2 94.2%

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

   10. Applied egg-rr 94.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.1 \cdot 10^{-235}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \left(m \cdot 0.5\right) \cdot \left(n + m \cdot 0.5\right)}\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-201}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-49}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \left(m \cdot 0.5\right) \cdot \left(n + m \cdot 0.5\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(\left|n - m\right| - \ell\right) - \left(m \cdot 0.5\right) \cdot \left(n + m \cdot 0.5\right)}\\ t_1 := e^{-{M}^{2}}\\ \mathbf{if}\;n \leq 9 \cdot 10^{-235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-161}:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{n}} - M\right) \cdot t_1\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (- (fabs (- n m)) l) (* (* m 0.5) (+ n (* m 0.5))))))
        (t_1 (exp (- (pow M 2.0)))))
   (if (<= n 9e-235)
     t_0
     (if (<= n 3.2e-161)
       (* (cos (- (/ K (/ 2.0 n)) M)) t_1)
       (if (<= n 3.2e-49)
         t_0
         (if (<= n 54.0)
           (* (cos M) t_1)
           (* (cos M) (exp (* -0.25 (* n n))))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((fabs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	double t_1 = exp(-pow(M, 2.0));
	double tmp;
	if (n <= 9e-235) {
		tmp = t_0;
	} else if (n <= 3.2e-161) {
		tmp = cos(((K / (2.0 / n)) - M)) * t_1;
	} else if (n <= 3.2e-49) {
		tmp = t_0;
	} else if (n <= 54.0) {
		tmp = cos(M) * t_1;
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((abs((n - m)) - l) - ((m * 0.5d0) * (n + (m * 0.5d0)))))
    t_1 = exp(-(m_1 ** 2.0d0))
    if (n <= 9d-235) then
        tmp = t_0
    else if (n <= 3.2d-161) then
        tmp = cos(((k / (2.0d0 / n)) - m_1)) * t_1
    else if (n <= 3.2d-49) then
        tmp = t_0
    else if (n <= 54.0d0) then
        tmp = cos(m_1) * t_1
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((Math.abs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	double t_1 = Math.exp(-Math.pow(M, 2.0));
	double tmp;
	if (n <= 9e-235) {
		tmp = t_0;
	} else if (n <= 3.2e-161) {
		tmp = Math.cos(((K / (2.0 / n)) - M)) * t_1;
	} else if (n <= 3.2e-49) {
		tmp = t_0;
	} else if (n <= 54.0) {
		tmp = Math.cos(M) * t_1;
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((math.fabs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))))
	t_1 = math.exp(-math.pow(M, 2.0))
	tmp = 0
	if n <= 9e-235:
		tmp = t_0
	elif n <= 3.2e-161:
		tmp = math.cos(((K / (2.0 / n)) - M)) * t_1
	elif n <= 3.2e-49:
		tmp = t_0
	elif n <= 54.0:
		tmp = math.cos(M) * t_1
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(Float64(m * 0.5) * Float64(n + Float64(m * 0.5)))))
	t_1 = exp(Float64(-(M ^ 2.0)))
	tmp = 0.0
	if (n <= 9e-235)
		tmp = t_0;
	elseif (n <= 3.2e-161)
		tmp = Float64(cos(Float64(Float64(K / Float64(2.0 / n)) - M)) * t_1);
	elseif (n <= 3.2e-49)
		tmp = t_0;
	elseif (n <= 54.0)
		tmp = Float64(cos(M) * t_1);
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((abs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	t_1 = exp(-(M ^ 2.0));
	tmp = 0.0;
	if (n <= 9e-235)
		tmp = t_0;
	elseif (n <= 3.2e-161)
		tmp = cos(((K / (2.0 / n)) - M)) * t_1;
	elseif (n <= 3.2e-49)
		tmp = t_0;
	elseif (n <= 54.0)
		tmp = cos(M) * t_1;
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[(m * 0.5), $MachinePrecision] * N[(n + N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]}, If[LessEqual[n, 9e-235], t$95$0, If[LessEqual[n, 3.2e-161], N[(N[Cos[N[(N[(K / N[(2.0 / n), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[n, 3.2e-49], t$95$0, If[LessEqual[n, 54.0], N[(N[Cos[M], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < 8.9999999999999996e-235 or 3.19999999999999985e-161 < n < 3.20000000000000002e-49

   1. Initial program 85.6%

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

      1. associate-/l* 86.2%

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

      2. +-commutative 86.2%

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

      3. fabs-sub 86.2%

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

      4. +-commutative 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in n around 0 72.9%

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

      1. +-commutative 72.9%

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

      2. unpow2 72.9%

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

      3. distribute-rgt-out 74.2%

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

      4. *-commutative 74.2%

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

      5. *-commutative 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in m around 0 80.3%

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

   9. Taylor expanded in K around 0 80.7%

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

       1. cos-neg 80.7%

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

   11. Simplified 80.7%

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

   12. Taylor expanded in M around 0 72.5%

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

       1. associate--r+ 72.5%

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

       2. associate-*r* 72.5%

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

       3. *-commutative 72.5%

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

       4. *-commutative 72.5%

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

   14. Simplified 72.5%

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

   if 8.9999999999999996e-235 < n < 3.19999999999999985e-161

   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* 77.6%

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

      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. Add Preprocessing

   5. Taylor expanded in m around 0 95.2%

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

   6. Taylor expanded in M around inf 72.4%

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

      1. mul-1-neg 72.4%

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

   8. Simplified 72.4%

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

   9. Taylor expanded in m around 0 72.4%

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

   if 3.20000000000000002e-49 < n < 54

   1. Initial program 87.5%

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

      1. associate-/l* 87.5%

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

      2. +-commutative 87.5%

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

      3. fabs-sub 87.5%

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

      4. +-commutative 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around inf 87.9%

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

      1. mul-1-neg 87.9%

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

   10. Simplified 87.9%

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

   if 54 < n

   1. Initial program 64.7%

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

      1. associate-/l* 64.7%

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

      2. +-commutative 64.7%

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

      3. fabs-sub 64.7%

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

      4. +-commutative 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in n around inf 94.2%

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

      1. unpow2 94.2%

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

   10. Applied egg-rr 94.2%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 9 \cdot 10^{-235}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \left(m \cdot 0.5\right) \cdot \left(n + m \cdot 0.5\right)}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-161}:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{n}} - M\right) \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-49}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \left(m \cdot 0.5\right) \cdot \left(n + m \cdot 0.5\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 54:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \left(m \cdot 0.5\right) \cdot \left(n + m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 54.0)
   (exp (- (- (fabs (- n m)) l) (* (* m 0.5) (+ n (* m 0.5)))))
   (* (cos M) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 54.0) {
		tmp = exp(((fabs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 54.0d0) then
        tmp = exp(((abs((n - m)) - l) - ((m * 0.5d0) * (n + (m * 0.5d0)))))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 54.0) {
		tmp = Math.exp(((Math.abs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 54.0:
		tmp = math.exp(((math.fabs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 54.0)
		tmp = exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(Float64(m * 0.5) * Float64(n + Float64(m * 0.5)))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 54.0)
		tmp = exp(((abs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 54.0], N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[(m * 0.5), $MachinePrecision] * N[(n + N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 54

   1. Initial program 85.0%

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

      1. associate-/l* 85.5%

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

      2. +-commutative 85.5%

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

      3. fabs-sub 85.5%

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

      4. +-commutative 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in n around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. unpow2 73.9%

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

      3. distribute-rgt-out 75.0%

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

      4. *-commutative 75.0%

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

      5. *-commutative 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in m around 0 82.5%

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

   9. Taylor expanded in K around 0 82.8%

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

       1. cos-neg 82.8%

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

   11. Simplified 82.8%

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

   12. Taylor expanded in M around 0 71.6%

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

       1. associate--r+ 71.6%

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

       2. associate-*r* 71.6%

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

       3. *-commutative 71.6%

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

       4. *-commutative 71.6%

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

   14. Simplified 71.6%

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

   if 54 < n

   1. Initial program 64.7%

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

      1. associate-/l* 64.7%

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

      2. +-commutative 64.7%

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

      3. fabs-sub 64.7%

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

      4. +-commutative 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in n around inf 94.2%

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

      1. unpow2 94.2%

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

   10. Applied egg-rr 94.2%

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

4. Final simplification 77.6%

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

Alternative 7: 72.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+19}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 1.3:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -4.8e+19)
   (* (cos M) (exp l))
   (if (<= l 1.3)
     (* (cos M) (exp (* -0.25 (* n n))))
     (* (cos M) (exp (- l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -4.8e+19) {
		tmp = cos(M) * exp(l);
	} else if (l <= 1.3) {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-4.8d+19)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 1.3d0) then
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -4.8e+19) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 1.3) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -4.8e+19:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 1.3:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -4.8e+19)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 1.3)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -4.8e+19)
		tmp = cos(M) * exp(l);
	elseif (l <= 1.3)
		tmp = cos(M) * exp((-0.25 * (n * n)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -4.8e+19], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.8e19

   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* 80.0%

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

      2. +-commutative 80.0%

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

      3. fabs-sub 80.0%

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

      4. +-commutative 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in K around 0 96.7%

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

      1. cos-neg 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in l around inf 24.5%

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

      1. mul-1-neg 24.5%

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

   10. Simplified 24.5%

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

       1. expm1-log1p-u 24.2%

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

       2. expm1-udef 24.2%

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

       3. add-sqr-sqrt 24.2%

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

       4. sqrt-unprod 24.2%

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

       5. sqr-neg 24.2%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 73.8%

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

   12. Applied egg-rr 73.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 73.8%

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

       2. expm1-log1p 73.8%

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

   14. Simplified 73.8%

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

   if -4.8e19 < l < 1.30000000000000004

   1. Initial program 79.1%

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

      1. associate-/l* 79.1%

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

      2. +-commutative 79.1%

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

      3. fabs-sub 79.1%

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

      4. +-commutative 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in K around 0 96.6%

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

      1. cos-neg 96.6%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in n around inf 53.4%

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

      1. unpow2 53.4%

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

   10. Applied egg-rr 53.4%

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

   if 1.30000000000000004 < l

   1. Initial program 81.8%

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

      1. associate-/l* 81.8%

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

      2. +-commutative 81.8%

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

      3. fabs-sub 81.8%

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

      4. +-commutative 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 98.5%

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

      1. mul-1-neg 98.5%

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

   10. Simplified 98.5%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+19}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 1.3:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-172}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -1e-172) (* (cos M) (exp l)) (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -1e-172) {
		tmp = cos(M) * exp(l);
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-1d-172)) then
        tmp = cos(m_1) * exp(l)
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -1e-172) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -1e-172:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -1e-172)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -1e-172)
		tmp = cos(M) * exp(l);
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -1e-172], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1e-172

   1. Initial program 78.9%

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

      1. associate-/l* 80.0%

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

      2. +-commutative 80.0%

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

      3. fabs-sub 80.0%

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

      4. +-commutative 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in K around 0 95.0%

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

      1. cos-neg 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in l around inf 17.2%

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

      1. mul-1-neg 17.2%

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

   10. Simplified 17.2%

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

       1. expm1-log1p-u 16.9%

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

       2. expm1-udef 16.9%

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

       3. add-sqr-sqrt 16.9%

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

       4. sqrt-unprod 16.9%

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

       5. sqr-neg 16.9%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 51.1%

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

   12. Applied egg-rr 51.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 51.1%

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

       2. expm1-log1p 51.1%

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

   14. Simplified 51.1%

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

   if -1e-172 < l

   1. Initial program 80.0%

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

      1. associate-/l* 80.0%

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

      2. +-commutative 80.0%

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

      3. fabs-sub 80.0%

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

      4. +-commutative 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in K around 0 98.9%

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

      1. cos-neg 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in l around inf 49.0%

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

      1. mul-1-neg 49.0%

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

   10. Simplified 49.0%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-172}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 1.4e-8) (* (cos M) (exp l)) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1.4e-8) {
		tmp = cos(M) * exp(l);
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 1.4d-8) then
        tmp = cos(m_1) * exp(l)
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1.4e-8) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 1.4e-8:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 1.4e-8)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 1.4e-8)
		tmp = cos(M) * exp(l);
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 1.4e-8], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.4e-8

   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* 79.2%

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

      2. +-commutative 79.2%

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

      3. fabs-sub 79.2%

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

      4. +-commutative 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in K around 0 96.6%

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

      1. cos-neg 96.6%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in l around inf 16.3%

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

      1. mul-1-neg 16.3%

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

   10. Simplified 16.3%

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

       1. expm1-log1p-u 16.1%

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

       2. expm1-udef 16.1%

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

       3. add-sqr-sqrt 11.0%

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

       4. sqrt-unprod 16.1%

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

       5. sqr-neg 16.1%

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

       6. sqrt-unprod 5.1%

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

       7. add-sqr-sqrt 32.9%

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

   12. Applied egg-rr 32.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 32.9%

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

       2. expm1-log1p 32.9%

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

   14. Simplified 32.9%

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

   if 1.4e-8 < l

   1. Initial program 82.1%

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

      1. associate-/l* 82.1%

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

      2. +-commutative 82.1%

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

      3. fabs-sub 82.1%

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

      4. +-commutative 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 97.1%

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

      1. mul-1-neg 97.1%

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

   10. Simplified 97.1%

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

   11. Taylor expanded in M around 0 97.1%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. associate-/l* 80.0%

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

   2. +-commutative 80.0%

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

   3. fabs-sub 80.0%

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

   4. +-commutative 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in l around inf 37.4%

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

   1. mul-1-neg 37.4%

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

10. Simplified 37.4%

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

11. Taylor expanded in M around 0 37.4%

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

12. Final simplification 37.4%

    \[\leadsto e^{-\ell} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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)))))))