Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.7%

Time: 25.1s

Alternatives: 9

Speedup: 2.0×

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: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

3. Taylor expanded in K around 0 97.2%

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

   1. cos-neg 97.2%

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

5. Simplified 97.2%

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

6. Final simplification 97.2%

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

Alternative 2: 96.2% accurate, 1.4× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

3. Taylor expanded in m around inf 81.4%

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

   1. *-commutative 81.4%

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

   2. associate-*l* 81.4%

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

5. Simplified 81.4%

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

6. Taylor expanded in K around 0 97.0%

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

7. Final simplification 97.0%

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

Alternative 3: 85.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 2 \cdot 10^{+20}:\\ \;\;\;\;e^{t\_0 + \left(\left(\left(n + m \cdot 0.5\right) - M\right) \cdot \left(M - m \cdot 0.5\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 + \left(\left(n \cdot 0.5 - M\right) \cdot \left(M - n \cdot 0.5\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 2e+20)
     (exp (+ t_0 (- (* (- (+ n (* m 0.5)) M) (- M (* m 0.5))) l)))
     (exp (+ t_0 (- (* (- (* n 0.5) M) (- M (* n 0.5))) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= 2e+20) {
		tmp = exp((t_0 + ((((n + (m * 0.5)) - M) * (M - (m * 0.5))) - l)));
	} else {
		tmp = exp((t_0 + ((((n * 0.5) - M) * (M - (n * 0.5))) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (n <= 2d+20) then
        tmp = exp((t_0 + ((((n + (m * 0.5d0)) - m_1) * (m_1 - (m * 0.5d0))) - l)))
    else
        tmp = exp((t_0 + ((((n * 0.5d0) - m_1) * (m_1 - (n * 0.5d0))) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (n <= 2e+20) {
		tmp = Math.exp((t_0 + ((((n + (m * 0.5)) - M) * (M - (m * 0.5))) - l)));
	} else {
		tmp = Math.exp((t_0 + ((((n * 0.5) - M) * (M - (n * 0.5))) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 2e+20:
		tmp = math.exp((t_0 + ((((n + (m * 0.5)) - M) * (M - (m * 0.5))) - l)))
	else:
		tmp = math.exp((t_0 + ((((n * 0.5) - M) * (M - (n * 0.5))) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 2e+20)
		tmp = exp(Float64(t_0 + Float64(Float64(Float64(Float64(n + Float64(m * 0.5)) - M) * Float64(M - Float64(m * 0.5))) - l)));
	else
		tmp = exp(Float64(t_0 + Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(M - Float64(n * 0.5))) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (n <= 2e+20)
		tmp = exp((t_0 + ((((n + (m * 0.5)) - M) * (M - (m * 0.5))) - l)));
	else
		tmp = exp((t_0 + ((((n * 0.5) - M) * (M - (n * 0.5))) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 2e+20], N[Exp[N[(t$95$0 + N[(N[(N[(N[(n + N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision] * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 + N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 2e20

   1. Initial program 71.1%

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

   3. Taylor expanded in n around 0 63.0%

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

      1. +-commutative 63.0%

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

      2. unpow2 63.0%

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

      3. distribute-rgt-out 65.6%

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

      4. *-commutative 65.6%

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

      5. *-commutative 65.6%

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

   5. Simplified 65.6%

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

   6. Taylor expanded in n around inf 81.5%

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

      1. *-commutative 81.5%

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

      2. associate-*l* 81.5%

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

   8. Simplified 81.5%

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

   9. Taylor expanded in K around 0 86.5%

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

   if 2e20 < n

   1. Initial program 72.1%

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

   3. Taylor expanded in m around inf 95.1%

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

      1. *-commutative 95.1%

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

      2. associate-*l* 95.1%

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

   5. Simplified 95.1%

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

   6. Taylor expanded in m around 0 95.2%

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

      1. unpow2 95.2%

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

   8. Applied egg-rr 95.2%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{+20}:\\ \;\;\;\;e^{\left|n - m\right| + \left(\left(\left(n + m \cdot 0.5\right) - M\right) \cdot \left(M - m \cdot 0.5\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| + \left(\left(n \cdot 0.5 - M\right) \cdot \left(M - n \cdot 0.5\right) - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;M \leq -0.27 \lor \neg \left(M \leq 2 \cdot 10^{+85}\right):\\ \;\;\;\;e^{t\_0 + \left(M \cdot \left(n - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 + \left(n \cdot \left(M - n \cdot 0.25\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (or (<= M -0.27) (not (<= M 2e+85)))
     (exp (+ t_0 (- (* M (- n M)) l)))
     (exp (+ t_0 (- (* n (- M (* n 0.25))) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if ((M <= -0.27) || !(M <= 2e+85)) {
		tmp = exp((t_0 + ((M * (n - M)) - l)));
	} else {
		tmp = exp((t_0 + ((n * (M - (n * 0.25))) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if ((m_1 <= (-0.27d0)) .or. (.not. (m_1 <= 2d+85))) then
        tmp = exp((t_0 + ((m_1 * (n - m_1)) - l)))
    else
        tmp = exp((t_0 + ((n * (m_1 - (n * 0.25d0))) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if ((M <= -0.27) || !(M <= 2e+85)) {
		tmp = Math.exp((t_0 + ((M * (n - M)) - l)));
	} else {
		tmp = Math.exp((t_0 + ((n * (M - (n * 0.25))) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if (M <= -0.27) or not (M <= 2e+85):
		tmp = math.exp((t_0 + ((M * (n - M)) - l)))
	else:
		tmp = math.exp((t_0 + ((n * (M - (n * 0.25))) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if ((M <= -0.27) || !(M <= 2e+85))
		tmp = exp(Float64(t_0 + Float64(Float64(M * Float64(n - M)) - l)));
	else
		tmp = exp(Float64(t_0 + Float64(Float64(n * Float64(M - Float64(n * 0.25))) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if ((M <= -0.27) || ~((M <= 2e+85)))
		tmp = exp((t_0 + ((M * (n - M)) - l)));
	else
		tmp = exp((t_0 + ((n * (M - (n * 0.25))) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[M, -0.27], N[Not[LessEqual[M, 2e+85]], $MachinePrecision]], N[Exp[N[(t$95$0 + N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 + N[(N[(n * N[(M - N[(n * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -0.27000000000000002 or 2e85 < M

   1. Initial program 74.3%

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

   3. Taylor expanded in m around inf 84.1%

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

      1. *-commutative 84.1%

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

      2. associate-*l* 84.1%

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

   5. Simplified 84.1%

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

   6. Taylor expanded in m around 0 95.6%

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

   7. Taylor expanded in n around 0 82.5%

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

      1. mul-1-neg 82.5%

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

      2. distribute-rgt-neg-in 82.5%

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

      3. mul-1-neg 82.5%

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

      4. unpow2 82.5%

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

      5. distribute-lft-in 86.9%

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

      6. +-commutative 86.9%

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

      7. mul-1-neg 86.9%

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

      8. unsub-neg 86.9%

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

   9. Simplified 86.9%

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

   if -0.27000000000000002 < M < 2e85

   1. Initial program 69.0%

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

   3. Taylor expanded in m around inf 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-*l* 79.3%

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

   5. Simplified 79.3%

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

   6. Taylor expanded in m around 0 69.9%

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

      1. unpow2 69.9%

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

   8. Applied egg-rr 69.9%

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

   9. Taylor expanded in M around 0 69.9%

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

       1. associate-+r+ 69.9%

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

       2. unpow2 69.9%

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

       3. associate-*r* 69.9%

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

       4. *-commutative 69.9%

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

       5. associate-+r+ 69.9%

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

       6. mul-1-neg 69.9%

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

       7. *-commutative 69.9%

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

       8. distribute-rgt-neg-in 69.9%

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

       9. distribute-lft-in 69.9%

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

       10. +-commutative 69.9%

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

       11. unsub-neg 69.9%

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

       12. *-commutative 69.9%

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

   11. Simplified 69.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -0.27 \lor \neg \left(M \leq 2 \cdot 10^{+85}\right):\\ \;\;\;\;e^{\left|n - m\right| + \left(M \cdot \left(n - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| + \left(n \cdot \left(M - n \cdot 0.25\right) - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq -8 \cdot 10^{-95}:\\ \;\;\;\;\cos \left(\left(n \cdot 0.5\right) \cdot K\right) \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{+47}:\\ \;\;\;\;e^{t\_0 + \left(M \cdot \left(n - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 + \left(n \cdot \left(M - n \cdot 0.25\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n -8e-95)
     (* (cos (* (* n 0.5) K)) (exp (* n (- M (* m 0.5)))))
     (if (<= n 4.8e+47)
       (exp (+ t_0 (- (* M (- n M)) l)))
       (exp (+ t_0 (- (* n (- M (* n 0.25))) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= -8e-95) {
		tmp = cos(((n * 0.5) * K)) * exp((n * (M - (m * 0.5))));
	} else if (n <= 4.8e+47) {
		tmp = exp((t_0 + ((M * (n - M)) - l)));
	} else {
		tmp = exp((t_0 + ((n * (M - (n * 0.25))) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (n <= (-8d-95)) then
        tmp = cos(((n * 0.5d0) * k)) * exp((n * (m_1 - (m * 0.5d0))))
    else if (n <= 4.8d+47) then
        tmp = exp((t_0 + ((m_1 * (n - m_1)) - l)))
    else
        tmp = exp((t_0 + ((n * (m_1 - (n * 0.25d0))) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (n <= -8e-95) {
		tmp = Math.cos(((n * 0.5) * K)) * Math.exp((n * (M - (m * 0.5))));
	} else if (n <= 4.8e+47) {
		tmp = Math.exp((t_0 + ((M * (n - M)) - l)));
	} else {
		tmp = Math.exp((t_0 + ((n * (M - (n * 0.25))) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= -8e-95:
		tmp = math.cos(((n * 0.5) * K)) * math.exp((n * (M - (m * 0.5))))
	elif n <= 4.8e+47:
		tmp = math.exp((t_0 + ((M * (n - M)) - l)))
	else:
		tmp = math.exp((t_0 + ((n * (M - (n * 0.25))) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= -8e-95)
		tmp = Float64(cos(Float64(Float64(n * 0.5) * K)) * exp(Float64(n * Float64(M - Float64(m * 0.5)))));
	elseif (n <= 4.8e+47)
		tmp = exp(Float64(t_0 + Float64(Float64(M * Float64(n - M)) - l)));
	else
		tmp = exp(Float64(t_0 + Float64(Float64(n * Float64(M - Float64(n * 0.25))) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (n <= -8e-95)
		tmp = cos(((n * 0.5) * K)) * exp((n * (M - (m * 0.5))));
	elseif (n <= 4.8e+47)
		tmp = exp((t_0 + ((M * (n - M)) - l)));
	else
		tmp = exp((t_0 + ((n * (M - (n * 0.25))) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -8e-95], N[(N[Cos[N[(N[(n * 0.5), $MachinePrecision] * K), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.8e+47], N[Exp[N[(t$95$0 + N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 + N[(N[(n * N[(M - N[(n * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.99999999999999992e-95

   1. Initial program 67.4%

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

   3. Taylor expanded in n around 0 49.0%

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

      1. +-commutative 49.0%

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

      2. unpow2 49.0%

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

      3. distribute-rgt-out 54.9%

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

      4. *-commutative 54.9%

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

      5. *-commutative 54.9%

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

   5. Simplified 54.9%

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

   6. Taylor expanded in n around inf 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-*l* 67.6%

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

   8. Simplified 67.6%

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

   9. Taylor expanded in n around inf 36.9%

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

   if -7.99999999999999992e-95 < n < 4.80000000000000037e47

   1. Initial program 75.1%

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

   3. Taylor expanded in m around inf 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-*l* 75.8%

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

   5. Simplified 75.8%

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

   6. Taylor expanded in m around 0 71.8%

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

   7. Taylor expanded in n around 0 70.9%

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

      1. mul-1-neg 70.9%

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

      2. distribute-rgt-neg-in 70.9%

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

      3. mul-1-neg 70.9%

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

      4. unpow2 70.9%

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

      5. distribute-lft-in 70.9%

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

      6. +-commutative 70.9%

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

      7. mul-1-neg 70.9%

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

      8. unsub-neg 70.9%

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

   9. Simplified 70.9%

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

   if 4.80000000000000037e47 < n

   1. Initial program 69.6%

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

   3. Taylor expanded in m around inf 94.6%

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

      1. *-commutative 94.6%

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

      2. associate-*l* 94.6%

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

   5. Simplified 94.6%

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

   6. Taylor expanded in m around 0 94.7%

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

      1. unpow2 94.7%

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

   8. Applied egg-rr 94.7%

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

   9. Taylor expanded in M around 0 78.8%

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

       1. associate-+r+ 78.8%

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

       2. unpow2 78.8%

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

       3. associate-*r* 78.8%

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

       4. *-commutative 78.8%

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

       5. associate-+r+ 78.8%

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

       6. mul-1-neg 78.8%

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

       7. *-commutative 78.8%

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

       8. distribute-rgt-neg-in 78.8%

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

       9. distribute-lft-in 82.4%

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

       10. +-commutative 82.4%

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

       11. unsub-neg 82.4%

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

       12. *-commutative 82.4%

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

   11. Simplified 82.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-95}:\\ \;\;\;\;\cos \left(\left(n \cdot 0.5\right) \cdot K\right) \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{+47}:\\ \;\;\;\;e^{\left|n - m\right| + \left(M \cdot \left(n - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| + \left(n \cdot \left(M - n \cdot 0.25\right) - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1300000000 \lor \neg \left(\ell \leq 7 \cdot 10^{+180}\right):\\ \;\;\;\;e^{\left|n - m\right| + \left(M \cdot \left(n - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(n \cdot 0.5\right) \cdot K\right) \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= l 1300000000.0) (not (<= l 7e+180)))
   (exp (+ (fabs (- n m)) (- (* M (- n M)) l)))
   (* (cos (* (* n 0.5) K)) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((l <= 1300000000.0) || !(l <= 7e+180)) {
		tmp = exp((fabs((n - m)) + ((M * (n - M)) - l)));
	} else {
		tmp = cos(((n * 0.5) * K)) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l <= 1300000000.0d0) .or. (.not. (l <= 7d+180))) then
        tmp = exp((abs((n - m)) + ((m_1 * (n - m_1)) - l)))
    else
        tmp = cos(((n * 0.5d0) * k)) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((l <= 1300000000.0) || !(l <= 7e+180)) {
		tmp = Math.exp((Math.abs((n - m)) + ((M * (n - M)) - l)));
	} else {
		tmp = Math.cos(((n * 0.5) * K)) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (l <= 1300000000.0) or not (l <= 7e+180):
		tmp = math.exp((math.fabs((n - m)) + ((M * (n - M)) - l)))
	else:
		tmp = math.cos(((n * 0.5) * K)) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((l <= 1300000000.0) || !(l <= 7e+180))
		tmp = exp(Float64(abs(Float64(n - m)) + Float64(Float64(M * Float64(n - M)) - l)));
	else
		tmp = Float64(cos(Float64(Float64(n * 0.5) * K)) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((l <= 1300000000.0) || ~((l <= 7e+180)))
		tmp = exp((abs((n - m)) + ((M * (n - M)) - l)));
	else
		tmp = cos(((n * 0.5) * K)) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[l, 1300000000.0], N[Not[LessEqual[l, 7e+180]], $MachinePrecision]], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] + N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(N[(n * 0.5), $MachinePrecision] * K), $MachinePrecision]], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e9 or 6.9999999999999996e180 < l

   1. Initial program 70.2%

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

   3. Taylor expanded in m around inf 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-*l* 80.4%

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

   5. Simplified 80.4%

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

   6. Taylor expanded in m around 0 79.4%

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

   7. Taylor expanded in n around 0 52.9%

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

      1. mul-1-neg 52.9%

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

      2. distribute-rgt-neg-in 52.9%

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

      3. mul-1-neg 52.9%

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

      4. unpow2 52.9%

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

      5. distribute-lft-in 54.7%

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

      6. +-commutative 54.7%

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

      7. mul-1-neg 54.7%

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

      8. unsub-neg 54.7%

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

   9. Simplified 54.7%

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

   if 1.3e9 < l < 6.9999999999999996e180

   1. Initial program 78.8%

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

   3. Taylor expanded in n around 0 72.8%

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

      1. +-commutative 72.8%

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

      2. unpow2 72.8%

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

      3. distribute-rgt-out 75.8%

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

      4. *-commutative 75.8%

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

      5. *-commutative 75.8%

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

   5. Simplified 75.8%

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

   6. Taylor expanded in n around inf 84.9%

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

      1. *-commutative 84.9%

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

      2. associate-*l* 84.9%

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

   8. Simplified 84.9%

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

   9. Taylor expanded in l around inf 87.9%

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

       1. neg-mul-1 87.9%

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

   11. Simplified 87.9%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1300000000 \lor \neg \left(\ell \leq 7 \cdot 10^{+180}\right):\\ \;\;\;\;e^{\left|n - m\right| + \left(M \cdot \left(n - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(n \cdot 0.5\right) \cdot K\right) \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.8% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (+ (fabs (- n m)) (- (* (- (* n 0.5) M) (- M (* n 0.5))) l))))

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) + ((((n * 0.5) - M) * (M - (n * 0.5))) - l)));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp((abs((n - m)) + ((((n * 0.5d0) - m_1) * (m_1 - (n * 0.5d0))) - l)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((Math.abs((n - m)) + ((((n * 0.5) - M) * (M - (n * 0.5))) - l)));
}

Python

def code(K, m, n, M, l):
	return math.exp((math.fabs((n - m)) + ((((n * 0.5) - M) * (M - (n * 0.5))) - l)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

3. Taylor expanded in m around inf 81.4%

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

   1. *-commutative 81.4%

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

   2. associate-*l* 81.4%

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

5. Simplified 81.4%

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

6. Taylor expanded in m around 0 81.3%

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

   1. unpow2 81.3%

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

8. Applied egg-rr 81.3%

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

9. Final simplification 81.3%

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

Alternative 8: 31.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(n \cdot 0.5\right) \cdot K\right) \cdot e^{-\ell} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (* (cos (* (* n 0.5) K)) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(((n * 0.5) * K)) * exp(-l);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(((n * 0.5d0) * k)) * exp(-l)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(((n * 0.5) * K)) * Math.exp(-l);
}

Python

def code(K, m, n, M, l):
	return math.cos(((n * 0.5) * K)) * math.exp(-l)

Julia

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

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(n * 0.5), $MachinePrecision] * K), $MachinePrecision]], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

3. Taylor expanded in n around 0 58.2%

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

   1. +-commutative 58.2%

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

   2. unpow2 58.2%

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

   3. distribute-rgt-out 61.4%

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

   4. *-commutative 61.4%

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

   5. *-commutative 61.4%

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

5. Simplified 61.4%

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

6. Taylor expanded in n around inf 73.5%

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

   1. *-commutative 73.5%

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

   2. associate-*l* 73.5%

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

8. Simplified 73.5%

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

9. Taylor expanded in l around inf 31.3%

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

    1. neg-mul-1 31.3%

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

11. Simplified 31.3%

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

12. Final simplification 31.3%

    \[\leadsto \cos \left(\left(n \cdot 0.5\right) \cdot K\right) \cdot e^{-\ell} \]
13. Add Preprocessing

Alternative 9: 24.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\left|n - m\right| - \ell} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

3. Taylor expanded in m around inf 81.4%

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

   1. *-commutative 81.4%

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

   2. associate-*l* 81.4%

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

5. Simplified 81.4%

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

6. Taylor expanded in K around 0 97.0%

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

7. Taylor expanded in l around inf 27.1%

   \[\leadsto e^{\left|m - n\right| - \color{blue}{\ell}} \]

8. Final simplification 27.1%

   \[\leadsto e^{\left|n - m\right| - \ell} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024094 
(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)))))))