Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.8%

Time: 28.8s

Alternatives: 12

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: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot e^{\left|n - m\right| - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (fabs (- n m)) (+ l (pow (- (* (+ m n) 0.5) 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) * 0.5) - 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) * 0.5d0) - 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) * 0.5) - 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) * 0.5) - M), 2.0))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(abs(Float64(n - m)) - Float64(l + (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp((abs((n - m)) - (l + ((((m + n) * 0.5) - M) ^ 2.0))));
end

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in K around 0 97.7%

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

   1. cos-neg 97.7%

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

   2. sub-neg 97.7%

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

   3. sub-neg 97.7%

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

   4. associate--r+ 97.7%

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

   5. *-commutative 97.7%

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

   6. associate--r+ 97.7%

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

5. Simplified 97.7%

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

6. Final simplification 97.7%

   \[\leadsto \cos M \cdot e^{\left|n - m\right| - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)} \]
7. Add Preprocessing

Alternative 2: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -3.5 \cdot 10^{+46}:\\ \;\;\;\;\cos M \cdot e^{t\_0 - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(t\_0 - \ell\right) + \left(n \cdot 0.5 - M\right) \cdot \left(\left(M - m\right) - n \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -3.5e+46)
     (* (cos M) (exp (- t_0 (pow (- (* (+ m n) 0.5) M) 2.0))))
     (*
      (cos M)
      (exp (+ (- t_0 l) (* (- (* n 0.5) M) (- (- M m) (* n 0.5)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -3.5e+46) {
		tmp = cos(M) * exp((t_0 - pow((((m + n) * 0.5) - M), 2.0)));
	} else {
		tmp = cos(M) * exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))));
	}
	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 (m <= (-3.5d+46)) then
        tmp = cos(m_1) * exp((t_0 - ((((m + n) * 0.5d0) - m_1) ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(((t_0 - l) + (((n * 0.5d0) - m_1) * ((m_1 - m) - (n * 0.5d0)))))
    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 (m <= -3.5e+46) {
		tmp = Math.cos(M) * Math.exp((t_0 - Math.pow((((m + n) * 0.5) - M), 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if m <= -3.5e+46:
		tmp = math.cos(M) * math.exp((t_0 - math.pow((((m + n) * 0.5) - M), 2.0)))
	else:
		tmp = math.cos(M) * math.exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -3.5e+46)
		tmp = Float64(cos(M) * exp(Float64(t_0 - (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(t_0 - l) + Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - m) - Float64(n * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (m <= -3.5e+46)
		tmp = cos(M) * exp((t_0 - ((((m + n) * 0.5) - M) ^ 2.0)));
	else
		tmp = cos(M) * exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))));
	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[m, -3.5e+46], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] + N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - m), $MachinePrecision] - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -3.49999999999999985e46

   1. Initial program 67.3%

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

   3. Taylor expanded in l around 0 67.3%

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

      1. *-commutative 67.3%

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

      2. *-commutative 67.3%

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

      3. *-commutative 67.3%

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

      4. associate-*r* 67.3%

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

   5. Simplified 67.3%

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

   6. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   8. Simplified 100.0%

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

   if -3.49999999999999985e46 < m

   1. Initial program 85.7%

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

   3. Taylor expanded in m around 0 70.0%

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

      1. +-commutative 70.0%

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

      2. unpow2 70.0%

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

      3. distribute-rgt-out 73.0%

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

      4. *-commutative 73.0%

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

      5. *-commutative 73.0%

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

   5. Simplified 73.0%

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

   6. Taylor expanded in K around 0 82.9%

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

      1. cos-neg 82.9%

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

      2. associate--r+ 82.9%

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

      3. sub-neg 82.9%

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

      4. +-commutative 82.9%

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

      5. neg-mul-1 82.9%

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

      6. associate--l+ 82.9%

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

      7. sub-neg 82.9%

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

      8. +-commutative 82.9%

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

      9. neg-mul-1 82.9%

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

      10. *-commutative 82.9%

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

      11. *-commutative 82.9%

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

      12. neg-mul-1 82.9%

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

      13. +-commutative 82.9%

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

      14. sub-neg 82.9%

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

   8. Simplified 82.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{+46}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) + \left(n \cdot 0.5 - M\right) \cdot \left(\left(M - m\right) - n \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -8.5 \cdot 10^{+139}:\\ \;\;\;\;\cos M \cdot e^{t\_0 - \left(\ell + 0.25 \cdot {m}^{2}\right)}\\ \mathbf{elif}\;m \leq -3.5 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \left(\left(m \cdot K\right) \cdot \left(M \cdot e^{t\_0 - \left(0.25 \cdot {\left(m + n\right)}^{2} + \ell\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(t\_0 - \ell\right) + \left(n \cdot 0.5 - M\right) \cdot \left(\left(M - m\right) - n \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -8.5e+139)
     (* (cos M) (exp (- t_0 (+ l (* 0.25 (pow m 2.0))))))
     (if (<= m -3.5e+46)
       (* 0.5 (* (* m K) (* M (exp (- t_0 (+ (* 0.25 (pow (+ m n) 2.0)) l))))))
       (*
        (cos M)
        (exp (+ (- t_0 l) (* (- (* n 0.5) M) (- (- M m) (* n 0.5))))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -8.5e+139) {
		tmp = cos(M) * exp((t_0 - (l + (0.25 * pow(m, 2.0)))));
	} else if (m <= -3.5e+46) {
		tmp = 0.5 * ((m * K) * (M * exp((t_0 - ((0.25 * pow((m + n), 2.0)) + l)))));
	} else {
		tmp = cos(M) * exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))));
	}
	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 (m <= (-8.5d+139)) then
        tmp = cos(m_1) * exp((t_0 - (l + (0.25d0 * (m ** 2.0d0)))))
    else if (m <= (-3.5d+46)) then
        tmp = 0.5d0 * ((m * k) * (m_1 * exp((t_0 - ((0.25d0 * ((m + n) ** 2.0d0)) + l)))))
    else
        tmp = cos(m_1) * exp(((t_0 - l) + (((n * 0.5d0) - m_1) * ((m_1 - m) - (n * 0.5d0)))))
    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 (m <= -8.5e+139) {
		tmp = Math.cos(M) * Math.exp((t_0 - (l + (0.25 * Math.pow(m, 2.0)))));
	} else if (m <= -3.5e+46) {
		tmp = 0.5 * ((m * K) * (M * Math.exp((t_0 - ((0.25 * Math.pow((m + n), 2.0)) + l)))));
	} else {
		tmp = Math.cos(M) * Math.exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if m <= -8.5e+139:
		tmp = math.cos(M) * math.exp((t_0 - (l + (0.25 * math.pow(m, 2.0)))))
	elif m <= -3.5e+46:
		tmp = 0.5 * ((m * K) * (M * math.exp((t_0 - ((0.25 * math.pow((m + n), 2.0)) + l)))))
	else:
		tmp = math.cos(M) * math.exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -8.5e+139)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(l + Float64(0.25 * (m ^ 2.0))))));
	elseif (m <= -3.5e+46)
		tmp = Float64(0.5 * Float64(Float64(m * K) * Float64(M * exp(Float64(t_0 - Float64(Float64(0.25 * (Float64(m + n) ^ 2.0)) + l))))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(t_0 - l) + Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - m) - Float64(n * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (m <= -8.5e+139)
		tmp = cos(M) * exp((t_0 - (l + (0.25 * (m ^ 2.0)))));
	elseif (m <= -3.5e+46)
		tmp = 0.5 * ((m * K) * (M * exp((t_0 - ((0.25 * ((m + n) ^ 2.0)) + l)))));
	else
		tmp = cos(M) * exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))));
	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[m, -8.5e+139], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -3.5e+46], N[(0.5 * N[(N[(m * K), $MachinePrecision] * N[(M * N[Exp[N[(t$95$0 - N[(N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] + N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - m), $MachinePrecision] - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.5e139

   1. Initial program 56.4%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--r+ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate--r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around inf 97.5%

      \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(\ell + \color{blue}{0.25 \cdot {m}^{2}}\right)} \]

   if -8.5e139 < m < -3.49999999999999985e46

   1. Initial program 93.8%

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

   3. Taylor expanded in K around 0 93.8%

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

      1. cos-neg 93.8%

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

      2. sin-neg 93.8%

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

   5. Simplified 93.8%

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

   6. Taylor expanded in m around inf 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in M around 0 93.8%

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

       1. +-commutative 87.6%

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

   11. Simplified 93.8%

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

   if -3.49999999999999985e46 < m

   1. Initial program 85.7%

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

   3. Taylor expanded in m around 0 70.0%

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

      1. +-commutative 70.0%

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

      2. unpow2 70.0%

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

      3. distribute-rgt-out 73.0%

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

      4. *-commutative 73.0%

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

      5. *-commutative 73.0%

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

   5. Simplified 73.0%

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

   6. Taylor expanded in K around 0 82.9%

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

      1. cos-neg 82.9%

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

      2. associate--r+ 82.9%

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

      3. sub-neg 82.9%

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

      4. +-commutative 82.9%

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

      5. neg-mul-1 82.9%

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

      6. associate--l+ 82.9%

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

      7. sub-neg 82.9%

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

      8. +-commutative 82.9%

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

      9. neg-mul-1 82.9%

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

      10. *-commutative 82.9%

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

      11. *-commutative 82.9%

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

      12. neg-mul-1 82.9%

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

      13. +-commutative 82.9%

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

      14. sub-neg 82.9%

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

   8. Simplified 82.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{+139}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\ell + 0.25 \cdot {m}^{2}\right)}\\ \mathbf{elif}\;m \leq -3.5 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \left(\left(m \cdot K\right) \cdot \left(M \cdot e^{\left|n - m\right| - \left(0.25 \cdot {\left(m + n\right)}^{2} + \ell\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) + \left(n \cdot 0.5 - M\right) \cdot \left(\left(M - m\right) - n \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -3.5 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \left(\left(M \cdot e^{t\_0 - \left(0.25 \cdot {\left(m + n\right)}^{2} + \ell\right)}\right) \cdot \left(n \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(t\_0 - \ell\right) + \left(n \cdot 0.5 - M\right) \cdot \left(\left(M - m\right) - n \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -3.5e+46)
     (* 0.5 (* (* M (exp (- t_0 (+ (* 0.25 (pow (+ m n) 2.0)) l)))) (* n K)))
     (*
      (cos M)
      (exp (+ (- t_0 l) (* (- (* n 0.5) M) (- (- M m) (* n 0.5)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -3.5e+46) {
		tmp = 0.5 * ((M * exp((t_0 - ((0.25 * pow((m + n), 2.0)) + l)))) * (n * K));
	} else {
		tmp = cos(M) * exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))));
	}
	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 (m <= (-3.5d+46)) then
        tmp = 0.5d0 * ((m_1 * exp((t_0 - ((0.25d0 * ((m + n) ** 2.0d0)) + l)))) * (n * k))
    else
        tmp = cos(m_1) * exp(((t_0 - l) + (((n * 0.5d0) - m_1) * ((m_1 - m) - (n * 0.5d0)))))
    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 (m <= -3.5e+46) {
		tmp = 0.5 * ((M * Math.exp((t_0 - ((0.25 * Math.pow((m + n), 2.0)) + l)))) * (n * K));
	} else {
		tmp = Math.cos(M) * Math.exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if m <= -3.5e+46:
		tmp = 0.5 * ((M * math.exp((t_0 - ((0.25 * math.pow((m + n), 2.0)) + l)))) * (n * K))
	else:
		tmp = math.cos(M) * math.exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -3.5e+46)
		tmp = Float64(0.5 * Float64(Float64(M * exp(Float64(t_0 - Float64(Float64(0.25 * (Float64(m + n) ^ 2.0)) + l)))) * Float64(n * K)));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(t_0 - l) + Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - m) - Float64(n * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (m <= -3.5e+46)
		tmp = 0.5 * ((M * exp((t_0 - ((0.25 * ((m + n) ^ 2.0)) + l)))) * (n * K));
	else
		tmp = cos(M) * exp(((t_0 - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))));
	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[m, -3.5e+46], N[(0.5 * N[(N[(M * N[Exp[N[(t$95$0 - N[(N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(n * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] + N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - m), $MachinePrecision] - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -3.49999999999999985e46

   1. Initial program 67.3%

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

   3. Taylor expanded in K around 0 83.6%

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

      1. cos-neg 83.6%

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

      2. sin-neg 83.6%

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

   5. Simplified 83.6%

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

   6. Taylor expanded in n around inf 100.0%

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

   8. Simplified 89.1%

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

   9. Taylor expanded in M around 0 87.3%

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

       1. +-commutative 87.3%

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

   11. Simplified 87.3%

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

   if -3.49999999999999985e46 < m

   1. Initial program 85.7%

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

   3. Taylor expanded in m around 0 70.0%

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

      1. +-commutative 70.0%

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

      2. unpow2 70.0%

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

      3. distribute-rgt-out 73.0%

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

      4. *-commutative 73.0%

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

      5. *-commutative 73.0%

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

   5. Simplified 73.0%

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

   6. Taylor expanded in K around 0 82.9%

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

      1. cos-neg 82.9%

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

      2. associate--r+ 82.9%

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

      3. sub-neg 82.9%

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

      4. +-commutative 82.9%

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

      5. neg-mul-1 82.9%

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

      6. associate--l+ 82.9%

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

      7. sub-neg 82.9%

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

      8. +-commutative 82.9%

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

      9. neg-mul-1 82.9%

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

      10. *-commutative 82.9%

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

      11. *-commutative 82.9%

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

      12. neg-mul-1 82.9%

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

      13. +-commutative 82.9%

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

      14. sub-neg 82.9%

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

   8. Simplified 82.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \left(\left(M \cdot e^{\left|n - m\right| - \left(0.25 \cdot {\left(m + n\right)}^{2} + \ell\right)}\right) \cdot \left(n \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) + \left(n \cdot 0.5 - M\right) \cdot \left(\left(M - m\right) - n \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -3800000000000 \lor \neg \left(M \leq 34\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n \cdot 0.5\right) + \left(\left(m - \ell\right) - n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -3800000000000.0) (not (<= M 34.0)))
   (* (cos M) (exp (- (pow M 2.0))))
   (* (cos M) (exp (+ (* M (- M (* n 0.5))) (- (- m l) n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -3800000000000.0) || !(M <= 34.0)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp(((M * (M - (n * 0.5))) + ((m - l) - 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 ((m_1 <= (-3800000000000.0d0)) .or. (.not. (m_1 <= 34.0d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(((m_1 * (m_1 - (n * 0.5d0))) + ((m - l) - 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 ((M <= -3800000000000.0) || !(M <= 34.0)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp(((M * (M - (n * 0.5))) + ((m - l) - n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -3800000000000.0) or not (M <= 34.0):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp(((M * (M - (n * 0.5))) + ((m - l) - n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -3800000000000.0) || !(M <= 34.0))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(M - Float64(n * 0.5))) + Float64(Float64(m - l) - n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -3800000000000.0) || ~((M <= 34.0)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp(((M * (M - (n * 0.5))) + ((m - l) - n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -3800000000000.0], N[Not[LessEqual[M, 34.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(m - l), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -3.8e12 or 34 < M

   1. Initial program 86.4%

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

   3. Taylor expanded in l around 0 84.8%

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

      1. *-commutative 84.8%

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

      2. *-commutative 84.8%

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

      3. *-commutative 84.8%

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

      4. associate-*r* 84.8%

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

   5. Simplified 84.8%

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

   6. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in M around inf 98.4%

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

       1. mul-1-neg 98.4%

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

   11. Simplified 98.4%

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

   if -3.8e12 < M < 34

   1. Initial program 77.3%

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

      1. exp-diff 23.8%

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

      2. associate-*r/ 23.8%

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

   4. Applied egg-rr 21.3%

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

      1. associate-/l* 21.3%

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

      2. fma-neg 21.3%

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

      3. associate-*r* 21.3%

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

      4. *-commutative 21.3%

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

      5. *-commutative 21.3%

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

      6. associate-*r* 21.3%

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

      7. div-exp 24.7%

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

      8. associate--r- 24.7%

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

      9. +-commutative 24.7%

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

   6. Simplified 24.7%

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

   7. Taylor expanded in m around 0 35.9%

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

      1. *-commutative 35.9%

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

      2. *-commutative 35.9%

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

      3. *-commutative 35.9%

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

      4. unpow2 35.9%

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

      5. distribute-lft-in 37.4%

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

      6. sub-neg 37.4%

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

      7. *-commutative 37.4%

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

      8. +-commutative 37.4%

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

      9. associate-+r+ 37.4%

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

      10. sub-neg 37.4%

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

      11. *-commutative 37.4%

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

   9. Simplified 37.4%

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

   10. Taylor expanded in K around 0 46.0%

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

       1. cos-neg 82.2%

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

   12. Simplified 46.0%

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

   13. Taylor expanded in M around inf 61.8%

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

       1. neg-mul-1 61.8%

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

   15. Simplified 61.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3800000000000 \lor \neg \left(M \leq 34\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n \cdot 0.5\right) + \left(\left(m - \ell\right) - n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(m \cdot K\right) - M\right) \cdot e^{M \cdot \left(m - M\right)}\\ \mathbf{if}\;M \leq -1.55 \cdot 10^{+192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq -1.55 \cdot 10^{+126}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{elif}\;M \leq -1.35 \cdot 10^{+47} \lor \neg \left(M \leq 1.2 \cdot 10^{+17}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n \cdot 0.5\right) + \left(\left(m - \ell\right) - n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos (- (* 0.5 (* m K)) M)) (exp (* M (- m M))))))
   (if (<= M -1.55e+192)
     t_0
     (if (<= M -1.55e+126)
       (* (cos M) (exp (* m (- (+ (* n 0.5) 1.0) M))))
       (if (or (<= M -1.35e+47) (not (<= M 1.2e+17)))
         t_0
         (* (cos M) (exp (+ (* M (- M (* n 0.5))) (- (- m l) n)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(((0.5 * (m * K)) - M)) * exp((M * (m - M)));
	double tmp;
	if (M <= -1.55e+192) {
		tmp = t_0;
	} else if (M <= -1.55e+126) {
		tmp = cos(M) * exp((m * (((n * 0.5) + 1.0) - M)));
	} else if ((M <= -1.35e+47) || !(M <= 1.2e+17)) {
		tmp = t_0;
	} else {
		tmp = cos(M) * exp(((M * (M - (n * 0.5))) + ((m - l) - 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 = cos(((0.5d0 * (m * k)) - m_1)) * exp((m_1 * (m - m_1)))
    if (m_1 <= (-1.55d+192)) then
        tmp = t_0
    else if (m_1 <= (-1.55d+126)) then
        tmp = cos(m_1) * exp((m * (((n * 0.5d0) + 1.0d0) - m_1)))
    else if ((m_1 <= (-1.35d+47)) .or. (.not. (m_1 <= 1.2d+17))) then
        tmp = t_0
    else
        tmp = cos(m_1) * exp(((m_1 * (m_1 - (n * 0.5d0))) + ((m - l) - 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(((0.5 * (m * K)) - M)) * Math.exp((M * (m - M)));
	double tmp;
	if (M <= -1.55e+192) {
		tmp = t_0;
	} else if (M <= -1.55e+126) {
		tmp = Math.cos(M) * Math.exp((m * (((n * 0.5) + 1.0) - M)));
	} else if ((M <= -1.35e+47) || !(M <= 1.2e+17)) {
		tmp = t_0;
	} else {
		tmp = Math.cos(M) * Math.exp(((M * (M - (n * 0.5))) + ((m - l) - n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(((0.5 * (m * K)) - M)) * math.exp((M * (m - M)))
	tmp = 0
	if M <= -1.55e+192:
		tmp = t_0
	elif M <= -1.55e+126:
		tmp = math.cos(M) * math.exp((m * (((n * 0.5) + 1.0) - M)))
	elif (M <= -1.35e+47) or not (M <= 1.2e+17):
		tmp = t_0
	else:
		tmp = math.cos(M) * math.exp(((M * (M - (n * 0.5))) + ((m - l) - n)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(Float64(Float64(0.5 * Float64(m * K)) - M)) * exp(Float64(M * Float64(m - M))))
	tmp = 0.0
	if (M <= -1.55e+192)
		tmp = t_0;
	elseif (M <= -1.55e+126)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(Float64(Float64(n * 0.5) + 1.0) - M))));
	elseif ((M <= -1.35e+47) || !(M <= 1.2e+17))
		tmp = t_0;
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(M - Float64(n * 0.5))) + Float64(Float64(m - l) - n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(((0.5 * (m * K)) - M)) * exp((M * (m - M)));
	tmp = 0.0;
	if (M <= -1.55e+192)
		tmp = t_0;
	elseif (M <= -1.55e+126)
		tmp = cos(M) * exp((m * (((n * 0.5) + 1.0) - M)));
	elseif ((M <= -1.35e+47) || ~((M <= 1.2e+17)))
		tmp = t_0;
	else
		tmp = cos(M) * exp(((M * (M - (n * 0.5))) + ((m - l) - n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[N[(N[(0.5 * N[(m * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -1.55e+192], t$95$0, If[LessEqual[M, -1.55e+126], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(N[(N[(n * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[M, -1.35e+47], N[Not[LessEqual[M, 1.2e+17]], $MachinePrecision]], t$95$0, N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(m - l), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -1.5499999999999999e192 or -1.55e126 < M < -1.34999999999999998e47 or 1.2e17 < M

   1. Initial program 92.0%

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

   3. Taylor expanded in m around 0 77.0%

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

      1. +-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. distribute-rgt-out 87.1%

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

      4. *-commutative 87.1%

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

      5. *-commutative 87.1%

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

   5. Simplified 87.1%

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

   6. Taylor expanded in n around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. mul-1-neg 79.2%

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

      3. unsub-neg 79.2%

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

   8. Simplified 79.2%

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

   9. Taylor expanded in M around inf 78.1%

      \[\leadsto \cos \left(0.5 \cdot \left(m \cdot K\right) - M\right) \cdot e^{\color{blue}{-1 \cdot {M}^{2} + M \cdot m}} \]
   10. Step-by-step derivation

       1. +-commutative 78.1%

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

       2. mul-1-neg 78.1%

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

       3. unpow2 78.1%

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

       4. distribute-rgt-neg-in 78.1%

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

       5. distribute-lft-in 84.1%

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

       6. sub-neg 84.1%

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

   11. Simplified 84.1%

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

   if -1.5499999999999999e192 < M < -1.55e126

   1. Initial program 62.5%

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

      1. exp-diff 12.5%

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

      2. associate-*r/ 12.5%

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

   4. Applied egg-rr 0.5%

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

      1. associate-/l* 0.5%

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

      2. fma-neg 0.5%

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

      3. associate-*r* 0.5%

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

      4. *-commutative 0.5%

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

      5. *-commutative 0.5%

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

      6. associate-*r* 0.5%

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

      7. div-exp 1.0%

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

      8. associate--r- 1.0%

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

      9. +-commutative 1.0%

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

   6. Simplified 1.0%

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

   7. Taylor expanded in m around 0 7.0%

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

      1. *-commutative 7.0%

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

      2. *-commutative 7.0%

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

      3. *-commutative 7.0%

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

      4. unpow2 7.0%

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

      5. distribute-lft-in 13.3%

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

      6. sub-neg 13.3%

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

      7. *-commutative 13.3%

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

      8. +-commutative 13.3%

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

      9. associate-+r+ 13.3%

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

      10. sub-neg 13.3%

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

      11. *-commutative 13.3%

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

   9. Simplified 13.3%

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

   10. Taylor expanded in K around 0 20.0%

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

       1. cos-neg 100.0%

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

   12. Simplified 20.0%

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

   13. Taylor expanded in m around inf 45.0%

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

       1. *-commutative 45.0%

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

   15. Simplified 45.0%

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

   if -1.34999999999999998e47 < M < 1.2e17

   1. Initial program 76.6%

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

      1. exp-diff 23.0%

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

      2. associate-*r/ 23.0%

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

   4. Applied egg-rr 20.7%

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

      1. associate-/l* 20.7%

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

      2. fma-neg 20.7%

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

      3. associate-*r* 20.7%

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

      4. *-commutative 20.7%

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

      5. *-commutative 20.7%

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

      6. associate-*r* 20.7%

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

      7. div-exp 23.8%

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

      8. associate--r- 23.8%

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

      9. +-commutative 23.8%

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

   6. Simplified 23.8%

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

   7. Taylor expanded in m around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. *-commutative 35.7%

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

      3. *-commutative 35.7%

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

      4. unpow2 35.7%

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

      5. distribute-lft-in 37.2%

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

      6. sub-neg 37.2%

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

      7. *-commutative 37.2%

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

      8. +-commutative 37.2%

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

      9. associate-+r+ 37.2%

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

      10. sub-neg 37.2%

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

      11. *-commutative 37.2%

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

   9. Simplified 37.2%

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

   10. Taylor expanded in K around 0 45.2%

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

       1. cos-neg 82.7%

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

   12. Simplified 45.2%

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

   13. Taylor expanded in M around inf 59.3%

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

       1. neg-mul-1 59.3%

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

   15. Simplified 59.3%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.55 \cdot 10^{+192}:\\ \;\;\;\;\cos \left(0.5 \cdot \left(m \cdot K\right) - M\right) \cdot e^{M \cdot \left(m - M\right)}\\ \mathbf{elif}\;M \leq -1.55 \cdot 10^{+126}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{elif}\;M \leq -1.35 \cdot 10^{+47} \lor \neg \left(M \leq 1.2 \cdot 10^{+17}\right):\\ \;\;\;\;\cos \left(0.5 \cdot \left(m \cdot K\right) - M\right) \cdot e^{M \cdot \left(m - M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n \cdot 0.5\right) + \left(\left(m - \ell\right) - n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot {n}^{2}}\\ \mathbf{if}\;n \leq -2.9 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-263}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-28}:\\ \;\;\;\;\cos \left(0.5 \cdot \left(m \cdot K\right) - M\right) \cdot e^{M \cdot \left(m - M\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (pow n 2.0)))))
   (if (<= n -2.9e-19)
     t_0
     (if (<= n 2.2e-263)
       (* (cos M) (exp (* m (- (+ (* n 0.5) 1.0) M))))
       (if (<= n 1.3e-28)
         (* (cos (- (* 0.5 (* m K)) M)) (exp (* M (- m M))))
         (if (<= n 54.0) (* (cos M) (exp (- l))) t_0))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * pow(n, 2.0)));
	double tmp;
	if (n <= -2.9e-19) {
		tmp = t_0;
	} else if (n <= 2.2e-263) {
		tmp = cos(M) * exp((m * (((n * 0.5) + 1.0) - M)));
	} else if (n <= 1.3e-28) {
		tmp = cos(((0.5 * (m * K)) - M)) * exp((M * (m - M)));
	} else if (n <= 54.0) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((-0.25d0) * (n ** 2.0d0)))
    if (n <= (-2.9d-19)) then
        tmp = t_0
    else if (n <= 2.2d-263) then
        tmp = cos(m_1) * exp((m * (((n * 0.5d0) + 1.0d0) - m_1)))
    else if (n <= 1.3d-28) then
        tmp = cos(((0.5d0 * (m * k)) - m_1)) * exp((m_1 * (m - m_1)))
    else if (n <= 54.0d0) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((-0.25 * Math.pow(n, 2.0)));
	double tmp;
	if (n <= -2.9e-19) {
		tmp = t_0;
	} else if (n <= 2.2e-263) {
		tmp = Math.cos(M) * Math.exp((m * (((n * 0.5) + 1.0) - M)));
	} else if (n <= 1.3e-28) {
		tmp = Math.cos(((0.5 * (m * K)) - M)) * Math.exp((M * (m - M)));
	} else if (n <= 54.0) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * math.pow(n, 2.0)))
	tmp = 0
	if n <= -2.9e-19:
		tmp = t_0
	elif n <= 2.2e-263:
		tmp = math.cos(M) * math.exp((m * (((n * 0.5) + 1.0) - M)))
	elif n <= 1.3e-28:
		tmp = math.cos(((0.5 * (m * K)) - M)) * math.exp((M * (m - M)))
	elif n <= 54.0:
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * (n ^ 2.0)))
	tmp = 0.0
	if (n <= -2.9e-19)
		tmp = t_0;
	elseif (n <= 2.2e-263)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(Float64(Float64(n * 0.5) + 1.0) - M))));
	elseif (n <= 1.3e-28)
		tmp = Float64(cos(Float64(Float64(0.5 * Float64(m * K)) - M)) * exp(Float64(M * Float64(m - M))));
	elseif (n <= 54.0)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (n ^ 2.0)));
	tmp = 0.0;
	if (n <= -2.9e-19)
		tmp = t_0;
	elseif (n <= 2.2e-263)
		tmp = cos(M) * exp((m * (((n * 0.5) + 1.0) - M)));
	elseif (n <= 1.3e-28)
		tmp = cos(((0.5 * (m * K)) - M)) * exp((M * (m - M)));
	elseif (n <= 54.0)
		tmp = cos(M) * exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -2.9e-19], t$95$0, If[LessEqual[n, 2.2e-263], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(N[(N[(n * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.3e-28], N[(N[Cos[N[(N[(0.5 * N[(m * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.9e-19 or 54 < n

   1. Initial program 77.9%

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

   3. Taylor expanded in l around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. *-commutative 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-*r* 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in K around 0 95.2%

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

      1. cos-neg 95.2%

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

   8. Simplified 95.2%

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

   9. Taylor expanded in n around inf 91.0%

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

   10. Taylor expanded in M around 0 91.0%

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

   if -2.9e-19 < n < 2.2e-263

   1. Initial program 89.2%

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

      1. exp-diff 47.9%

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

      2. associate-*r/ 47.9%

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

   4. Applied egg-rr 26.4%

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

      1. associate-/l* 26.4%

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

      2. fma-neg 26.4%

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

      3. associate-*r* 26.4%

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

      4. *-commutative 26.4%

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

      5. *-commutative 26.4%

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

      6. associate-*r* 26.4%

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

      7. div-exp 26.7%

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

      8. associate--r- 26.7%

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

      9. +-commutative 26.7%

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

   6. Simplified 26.7%

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

   7. Taylor expanded in m around 0 32.8%

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

      1. *-commutative 32.8%

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

      2. *-commutative 32.8%

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

      3. *-commutative 32.8%

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

      4. unpow2 32.8%

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

      5. distribute-lft-in 36.0%

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

      6. sub-neg 36.0%

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

      7. *-commutative 36.0%

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

      8. +-commutative 36.0%

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

      9. associate-+r+ 36.0%

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

      10. sub-neg 36.0%

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

      11. *-commutative 36.0%

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

   9. Simplified 36.0%

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

   10. Taylor expanded in K around 0 42.2%

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

       1. cos-neg 84.5%

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

   12. Simplified 42.2%

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

   13. Taylor expanded in m around inf 45.7%

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

       1. *-commutative 45.7%

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

   15. Simplified 45.7%

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

   if 2.2e-263 < n < 1.3e-28

   1. Initial program 85.7%

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

   3. Taylor expanded in m around 0 59.6%

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

      1. +-commutative 59.6%

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

      2. unpow2 59.6%

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

      3. distribute-rgt-out 59.6%

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

      4. *-commutative 59.6%

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

      5. *-commutative 59.6%

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

   5. Simplified 59.6%

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

   6. Taylor expanded in n around 0 59.6%

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

      1. *-commutative 59.6%

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

      2. mul-1-neg 59.6%

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

      3. unsub-neg 59.6%

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

   8. Simplified 59.6%

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

   9. Taylor expanded in M around inf 57.9%

      \[\leadsto \cos \left(0.5 \cdot \left(m \cdot K\right) - M\right) \cdot e^{\color{blue}{-1 \cdot {M}^{2} + M \cdot m}} \]
   10. Step-by-step derivation

       1. +-commutative 57.9%

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

       2. mul-1-neg 57.9%

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

       3. unpow2 57.9%

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

       4. distribute-rgt-neg-in 57.9%

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

       5. distribute-lft-in 57.9%

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

       6. sub-neg 57.9%

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

   11. Simplified 57.9%

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

   if 1.3e-28 < n < 54

   1. Initial program 50.4%

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

   3. Taylor expanded in m around 0 25.8%

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

      1. +-commutative 25.8%

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

      2. unpow2 25.8%

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

      3. distribute-rgt-out 25.8%

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

      4. *-commutative 25.8%

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

      5. *-commutative 25.8%

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

   5. Simplified 25.8%

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

   6. Taylor expanded in n around 0 25.8%

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

      1. *-commutative 25.8%

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

      2. mul-1-neg 25.8%

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

      3. unsub-neg 25.8%

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

   8. Simplified 25.8%

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

   9. Taylor expanded in l around inf 26.2%

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

       1. mul-1-neg 26.2%

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

   11. Simplified 26.2%

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

   12. Taylor expanded in m around 0 27.0%

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

       1. cos-neg 27.0%

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

   14. Simplified 27.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-19}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-263}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-28}:\\ \;\;\;\;\cos \left(0.5 \cdot \left(m \cdot K\right) - M\right) \cdot e^{M \cdot \left(m - M\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -53 \lor \neg \left(n \leq 56\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m + 0.5 \cdot \left(n \cdot \left(m + n \cdot 0.5\right)\right)\right) - \left(n + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -53.0) (not (<= n 56.0)))
   (exp (* -0.25 (pow n 2.0)))
   (* (cos M) (exp (- (+ m (* 0.5 (* n (+ m (* n 0.5))))) (+ n l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -53.0) || !(n <= 56.0)) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = cos(M) * exp(((m + (0.5 * (n * (m + (n * 0.5))))) - (n + 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 ((n <= (-53.0d0)) .or. (.not. (n <= 56.0d0))) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(((m + (0.5d0 * (n * (m + (n * 0.5d0))))) - (n + 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 ((n <= -53.0) || !(n <= 56.0)) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(((m + (0.5 * (n * (m + (n * 0.5))))) - (n + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -53.0) or not (n <= 56.0):
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(((m + (0.5 * (n * (m + (n * 0.5))))) - (n + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -53.0) || !(n <= 56.0))
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(m + Float64(0.5 * Float64(n * Float64(m + Float64(n * 0.5))))) - Float64(n + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -53.0) || ~((n <= 56.0)))
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = cos(M) * exp(((m + (0.5 * (n * (m + (n * 0.5))))) - (n + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -53.0], N[Not[LessEqual[n, 56.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m + N[(0.5 * N[(n * N[(m + N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -53 or 56 < n

   1. Initial program 77.4%

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

   3. Taylor expanded in l around 0 74.5%

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

      1. *-commutative 74.5%

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

      2. *-commutative 74.5%

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

      3. *-commutative 74.5%

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

      4. associate-*r* 74.5%

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

   5. Simplified 74.5%

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

   6. Taylor expanded in K around 0 95.6%

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

      1. cos-neg 95.6%

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

   8. Simplified 95.6%

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

   9. Taylor expanded in n around inf 95.6%

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

   10. Taylor expanded in M around 0 95.6%

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

   if -53 < n < 56

   1. Initial program 86.4%

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

      1. exp-diff 43.3%

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

      2. associate-*r/ 43.3%

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

   4. Applied egg-rr 21.1%

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

      1. associate-/l* 21.1%

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

      2. fma-neg 21.1%

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

      3. associate-*r* 21.1%

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

      4. *-commutative 21.1%

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

      5. *-commutative 21.1%

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

      6. associate-*r* 21.1%

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

      7. div-exp 23.1%

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

      8. associate--r- 23.1%

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

      9. +-commutative 23.1%

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

   6. Simplified 23.1%

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

   7. Taylor expanded in m around 0 34.3%

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

      1. *-commutative 34.3%

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

      2. *-commutative 34.3%

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

      3. *-commutative 34.3%

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

      4. unpow2 34.3%

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

      5. distribute-lft-in 35.9%

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

      6. sub-neg 35.9%

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

      7. *-commutative 35.9%

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

      8. +-commutative 35.9%

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

      9. associate-+r+ 35.9%

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

      10. sub-neg 35.9%

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

      11. *-commutative 35.9%

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

   9. Simplified 35.9%

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

   10. Taylor expanded in K around 0 42.5%

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

       1. cos-neg 84.3%

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

   12. Simplified 42.5%

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

   13. Taylor expanded in M around 0 54.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -53 \lor \neg \left(n \leq 56\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m + 0.5 \cdot \left(n \cdot \left(m + n \cdot 0.5\right)\right)\right) - \left(n + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-19} \lor \neg \left(n \leq 54\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -2.9e-19) (not (<= n 54.0)))
   (exp (* -0.25 (pow n 2.0)))
   (* (cos M) (exp (* m (- (+ (* n 0.5) 1.0) M))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -2.9e-19) || !(n <= 54.0)) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = cos(M) * exp((m * (((n * 0.5) + 1.0) - M)));
	}
	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 <= (-2.9d-19)) .or. (.not. (n <= 54.0d0))) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = cos(m_1) * exp((m * (((n * 0.5d0) + 1.0d0) - m_1)))
    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 <= -2.9e-19) || !(n <= 54.0)) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp((m * (((n * 0.5) + 1.0) - M)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -2.9e-19) or not (n <= 54.0):
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.cos(M) * math.exp((m * (((n * 0.5) + 1.0) - M)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -2.9e-19) || !(n <= 54.0))
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = Float64(cos(M) * exp(Float64(m * Float64(Float64(Float64(n * 0.5) + 1.0) - M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -2.9e-19) || ~((n <= 54.0)))
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = cos(M) * exp((m * (((n * 0.5) + 1.0) - M)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -2.9e-19], N[Not[LessEqual[n, 54.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(N[(N[(n * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.9e-19 or 54 < n

   1. Initial program 77.9%

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

   3. Taylor expanded in l around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. *-commutative 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-*r* 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in K around 0 95.2%

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

      1. cos-neg 95.2%

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

   8. Simplified 95.2%

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

   9. Taylor expanded in n around inf 91.0%

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

   10. Taylor expanded in M around 0 91.0%

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

   if -2.9e-19 < n < 54

   1. Initial program 86.4%

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

      1. exp-diff 45.0%

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

      2. associate-*r/ 45.0%

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

   4. Applied egg-rr 22.4%

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

      1. associate-/l* 22.4%

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

      2. fma-neg 22.4%

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

      3. associate-*r* 22.4%

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

      4. *-commutative 22.4%

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

      5. *-commutative 22.4%

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

      6. associate-*r* 22.4%

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

      7. div-exp 24.4%

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

      8. associate--r- 24.4%

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

      9. +-commutative 24.4%

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

   6. Simplified 24.4%

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

   7. Taylor expanded in m around 0 35.4%

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

      1. *-commutative 35.4%

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

      2. *-commutative 35.4%

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

      3. *-commutative 35.4%

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

      4. unpow2 35.4%

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

      5. distribute-lft-in 37.2%

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

      6. sub-neg 37.2%

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

      7. *-commutative 37.2%

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

      8. +-commutative 37.2%

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

      9. associate-+r+ 37.2%

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

      10. sub-neg 37.2%

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

      11. *-commutative 37.2%

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

   9. Simplified 37.2%

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

   10. Taylor expanded in K around 0 44.0%

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

       1. cos-neg 84.0%

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

   12. Simplified 44.0%

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

   13. Taylor expanded in m around inf 46.0%

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

       1. *-commutative 46.0%

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

   15. Simplified 46.0%

       \[\leadsto \cos M \cdot e^{\color{blue}{m \cdot \left(\left(1 + n \cdot 0.5\right) - M\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-19} \lor \neg \left(n \leq 54\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{-47} \lor \neg \left(n \leq 54\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -2.4e-47) (not (<= n 54.0)))
   (exp (* -0.25 (pow n 2.0)))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -2.4e-47) || !(n <= 54.0)) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} 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 ((n <= (-2.4d-47)) .or. (.not. (n <= 54.0d0))) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    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 ((n <= -2.4e-47) || !(n <= 54.0)) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -2.4e-47) or not (n <= 54.0):
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -2.4e-47) || !(n <= 54.0))
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	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 ((n <= -2.4e-47) || ~((n <= 54.0)))
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -2.4e-47], N[Not[LessEqual[n, 54.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.3999999999999999e-47 or 54 < n

   1. Initial program 78.6%

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

   3. Taylor expanded in l around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. *-commutative 75.9%

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

      4. associate-*r* 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in K around 0 95.4%

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

      1. cos-neg 95.4%

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

   8. Simplified 95.4%

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

   9. Taylor expanded in n around inf 88.7%

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

   10. Taylor expanded in M around 0 88.7%

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

   if -2.3999999999999999e-47 < n < 54

   1. Initial program 85.8%

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

   3. Taylor expanded in m around 0 59.1%

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

      1. +-commutative 59.1%

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

      2. unpow2 59.1%

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

      3. distribute-rgt-out 61.0%

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

      4. *-commutative 61.0%

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

      5. *-commutative 61.0%

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

   5. Simplified 61.0%

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

   6. Taylor expanded in n around 0 61.0%

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

      1. *-commutative 61.0%

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

      2. mul-1-neg 61.0%

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

      3. unsub-neg 61.0%

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

   8. Simplified 61.0%

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

   9. Taylor expanded in l around inf 36.7%

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

       1. mul-1-neg 36.7%

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

   11. Simplified 36.7%

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

   12. Taylor expanded in m around 0 41.2%

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

       1. cos-neg 41.2%

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

   14. Simplified 41.2%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{-47} \lor \neg \left(n \leq 54\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{-0.25 \cdot {n}^{2}} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (exp (* -0.25 (pow n 2.0))))

C

double code(double K, double m, double n, double M, double l) {
	return exp((-0.25 * pow(n, 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 = exp(((-0.25d0) * (n ** 2.0d0)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((-0.25 * Math.pow(n, 2.0)));
}

Python

def code(K, m, n, M, l):
	return math.exp((-0.25 * math.pow(n, 2.0)))

Julia

function code(K, m, n, M, l)
	return exp(Float64(-0.25 * (n ^ 2.0)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp((-0.25 * (n ^ 2.0)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in l around 0 74.9%

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

   1. *-commutative 74.9%

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

   2. *-commutative 74.9%

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

   3. *-commutative 74.9%

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

   4. associate-*r* 74.9%

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

5. Simplified 74.9%

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

6. Taylor expanded in K around 0 90.1%

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

   1. cos-neg 90.1%

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

8. Simplified 90.1%

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

9. Taylor expanded in n around inf 55.3%

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

10. Taylor expanded in M around 0 55.3%

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

11. Final simplification 55.3%

    \[\leadsto e^{-0.25 \cdot {n}^{2}} \]
12. Add Preprocessing

Alternative 12: 7.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \cos M \end{array} \]

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[Cos[M], $MachinePrecision]

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in l around 0 74.9%

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

   1. *-commutative 74.9%

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

   2. *-commutative 74.9%

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

   3. *-commutative 74.9%

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

   4. associate-*r* 74.9%

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

5. Simplified 74.9%

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

6. Taylor expanded in K around 0 90.1%

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

   1. cos-neg 90.1%

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

8. Simplified 90.1%

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

9. Taylor expanded in n around inf 55.3%

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

10. Taylor expanded in n around 0 7.3%

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

11. Final simplification 7.3%

    \[\leadsto \cos M \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(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)))))))