Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.2% → 96.7%

Time: 17.9s

Alternatives: 6

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: 76.2% 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

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (- (fabs (- m n)) 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((m - n)) - 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((m - n)) - 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((m - n)) - 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((m - n)) - 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(m - n)) - 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((m - n)) - 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[(m - n), $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 77.8%

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

2. Taylor expanded in K around 0 95.6%

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

   1. cos-neg 95.6%

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

4. Simplified 95.6%

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

5. Final simplification 95.6%

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

Alternative 2: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;M \leq -6 \cdot 10^{-41} \lor \neg \left(M \leq 1.9 \cdot 10^{+63}\right):\\ \;\;\;\;\cos M \cdot e^{t_0 - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (or (<= M -6e-41) (not (<= M 1.9e+63)))
     (* (cos M) (exp (- t_0 (pow (- (* (+ m n) 0.5) M) 2.0))))
     (exp (- t_0 (+ l (* 0.25 (* (+ m n) (+ m n)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if ((M <= -6e-41) || !(M <= 1.9e+63)) {
		tmp = cos(M) * exp((t_0 - pow((((m + n) * 0.5) - M), 2.0)));
	} else {
		tmp = exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((m - n))
    if ((m_1 <= (-6d-41)) .or. (.not. (m_1 <= 1.9d+63))) then
        tmp = cos(m_1) * exp((t_0 - ((((m + n) * 0.5d0) - m_1) ** 2.0d0)))
    else
        tmp = exp((t_0 - (l + (0.25d0 * ((m + n) * (m + n))))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((m - n));
	double tmp;
	if ((M <= -6e-41) || !(M <= 1.9e+63)) {
		tmp = Math.cos(M) * Math.exp((t_0 - Math.pow((((m + n) * 0.5) - M), 2.0)));
	} else {
		tmp = Math.exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	tmp = 0
	if (M <= -6e-41) or not (M <= 1.9e+63):
		tmp = math.cos(M) * math.exp((t_0 - math.pow((((m + n) * 0.5) - M), 2.0)))
	else:
		tmp = math.exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if ((M <= -6e-41) || !(M <= 1.9e+63))
		tmp = Float64(cos(M) * exp(Float64(t_0 - (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0))));
	else
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * Float64(Float64(m + n) * Float64(m + n))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	tmp = 0.0;
	if ((M <= -6e-41) || ~((M <= 1.9e+63)))
		tmp = cos(M) * exp((t_0 - ((((m + n) * 0.5) - M) ^ 2.0)));
	else
		tmp = exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[M, -6e-41], N[Not[LessEqual[M, 1.9e+63]], $MachinePrecision]], 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[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -5.99999999999999978e-41 or 1.9000000000000001e63 < M

   1. Initial program 79.7%

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

   2. Taylor expanded in K around 0 97.7%

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

      1. cos-neg 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in l around 0 96.9%

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

      1. *-commutative 96.9%

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

      2. unpow2 96.9%

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

      3. unpow2 96.9%

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

   7. Simplified 96.9%

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

   if -5.99999999999999978e-41 < M < 1.9000000000000001e63

   1. Initial program 75.9%

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

   2. Taylor expanded in K around 0 93.5%

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

      1. cos-neg 93.5%

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

   4. Simplified 93.5%

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

   5. Taylor expanded in M around 0 93.1%

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

      1. unpow2 93.1%

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

   7. Applied egg-rr 93.1%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6 \cdot 10^{-41} \lor \neg \left(M \leq 1.9 \cdot 10^{+63}\right):\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)}\\ \end{array} \]

Alternative 3: 88.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;n \leq 2.9 \cdot 10^{-251} \lor \neg \left(n \leq 900\right):\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(t_0 - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (or (<= n 2.9e-251) (not (<= n 900.0)))
     (exp (- t_0 (+ l (* 0.25 (* (+ m n) (+ m n))))))
     (*
      (cos (- (/ (* (+ m n) K) 2.0) M))
      (exp (+ (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) (- t_0 l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if ((n <= 2.9e-251) || !(n <= 900.0)) {
		tmp = exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	} else {
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (t_0 - 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((m - n))
    if ((n <= 2.9d-251) .or. (.not. (n <= 900.0d0))) then
        tmp = exp((t_0 - (l + (0.25d0 * ((m + n) * (m + n))))))
    else
        tmp = cos(((((m + n) * k) / 2.0d0) - m_1)) * exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) + (t_0 - 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((m - n));
	double tmp;
	if ((n <= 2.9e-251) || !(n <= 900.0)) {
		tmp = Math.exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	} else {
		tmp = Math.cos(((((m + n) * K) / 2.0) - M)) * Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (t_0 - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	tmp = 0
	if (n <= 2.9e-251) or not (n <= 900.0):
		tmp = math.exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))))
	else:
		tmp = math.cos(((((m + n) * K) / 2.0) - M)) * math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (t_0 - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if ((n <= 2.9e-251) || !(n <= 900.0))
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * Float64(Float64(m + n) * Float64(m + n))))));
	else
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) * K) / 2.0) - M)) * exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) + Float64(t_0 - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	tmp = 0.0;
	if ((n <= 2.9e-251) || ~((n <= 900.0)))
		tmp = exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	else
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (t_0 - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[n, 2.9e-251], N[Not[LessEqual[n, 900.0]], $MachinePrecision]], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 2.9000000000000001e-251 or 900 < n

   1. Initial program 74.6%

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

   2. Taylor expanded in K around 0 95.1%

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

      1. cos-neg 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in M around 0 89.6%

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

      1. unpow2 89.6%

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

   7. Applied egg-rr 89.6%

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

   if 2.9000000000000001e-251 < n < 900

   1. Initial program 91.7%

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

   2. Taylor expanded in n around 0 91.7%

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

      1. +-commutative 91.7%

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

      2. unpow2 91.7%

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

      3. distribute-rgt-out 91.7%

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

      4. *-commutative 91.7%

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

      5. *-commutative 91.7%

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

   4. Simplified 91.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.9 \cdot 10^{-251} \lor \neg \left(n \leq 900\right):\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(\left|m - n\right| - \ell\right)}\\ \end{array} \]

Alternative 4: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;M \leq 6 \cdot 10^{+126}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 + \left(M \cdot \left(m - M\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (<= M 6e+126)
     (exp (- t_0 (+ l (* 0.25 (* (+ m n) (+ m n))))))
     (* (cos M) (exp (+ t_0 (- (* M (- m M)) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if (M <= 6e+126) {
		tmp = exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	} else {
		tmp = cos(M) * exp((t_0 + ((M * (m - M)) - 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((m - n))
    if (m_1 <= 6d+126) then
        tmp = exp((t_0 - (l + (0.25d0 * ((m + n) * (m + n))))))
    else
        tmp = cos(m_1) * exp((t_0 + ((m_1 * (m - m_1)) - 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((m - n));
	double tmp;
	if (M <= 6e+126) {
		tmp = Math.exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 + ((M * (m - M)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	tmp = 0
	if M <= 6e+126:
		tmp = math.exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))))
	else:
		tmp = math.cos(M) * math.exp((t_0 + ((M * (m - M)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if (M <= 6e+126)
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * Float64(Float64(m + n) * Float64(m + n))))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 + Float64(Float64(M * Float64(m - M)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	tmp = 0.0;
	if (M <= 6e+126)
		tmp = exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	else
		tmp = cos(M) * exp((t_0 + ((M * (m - M)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, 6e+126], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 6.0000000000000005e126

   1. Initial program 78.1%

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

   2. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   4. Simplified 94.7%

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

   5. Taylor expanded in M around 0 89.0%

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

      1. unpow2 89.0%

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

   7. Applied egg-rr 89.0%

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

   if 6.0000000000000005e126 < M

   1. Initial program 76.2%

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

   2. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in M around inf 54.8%

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

      1. associate-*r* 54.8%

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

      2. neg-mul-1 54.8%

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

      3. unpow2 54.8%

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

      4. sqr-neg 54.8%

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

      5. distribute-lft-out 93.0%

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

      6. unsub-neg 93.0%

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

   7. Simplified 93.0%

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

   8. Taylor expanded in n around 0 90.6%

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

      1. fabs-sub 90.6%

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

      2. mul-1-neg 90.6%

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

      3. unsub-neg 90.6%

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

   10. Simplified 90.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6 \cdot 10^{+126}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| + \left(M \cdot \left(m - M\right) - \ell\right)}\\ \end{array} \]

Alternative 5: 86.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (fabs (- m n)) (+ l (* 0.25 (* (+ m n) (+ m n)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

2. Taylor expanded in K around 0 95.6%

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

   1. cos-neg 95.6%

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

4. Simplified 95.6%

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

5. Taylor expanded in M around 0 88.9%

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

   1. unpow2 88.9%

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

7. Applied egg-rr 88.9%

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

8. Final simplification 88.9%

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

Alternative 6: 24.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((m - n)) - 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((m - n)) - l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

2. Taylor expanded in K around 0 95.6%

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

   1. cos-neg 95.6%

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

4. Simplified 95.6%

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

5. Taylor expanded in M around inf 46.6%

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

   1. associate-*r* 46.6%

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

   2. neg-mul-1 46.6%

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

   3. unpow2 46.6%

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

   4. sqr-neg 46.6%

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

   5. distribute-lft-out 53.7%

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

   6. unsub-neg 53.7%

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

7. Simplified 53.7%

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

8. Taylor expanded in M around 0 27.0%

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \ell}} \]
9. Step-by-step derivation

   1. fabs-sub 27.0%

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

   2. sub-neg 27.0%

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

   3. mul-1-neg 27.0%

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

   4. fabs-neg 27.0%

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

   5. fabs-neg 27.0%

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

   6. mul-1-neg 27.0%

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

   7. sub-neg 27.0%

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

10. Simplified 27.0%

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

11. Final simplification 27.0%

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

Reproduce

herbie shell --seed 2023314 
(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)))))))