Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.2% → 96.9%

Time: 22.0s

Alternatives: 7

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.2% 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: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp((fabs((m - n)) - (l + pow((((m + n) * 0.5) - M), 2.0))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp((abs((m - n)) - (l + ((((m + n) * 0.5d0) - m_1) ** 2.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.exp((Math.abs((m - n)) - (l + Math.pow((((m + n) * 0.5) - M), 2.0))));
}

Python

def code(K, m, n, M, l):
	return math.cos(M) * math.exp((math.fabs((m - n)) - (l + math.pow((((m + n) * 0.5) - M), 2.0))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

3. Taylor expanded in K around 0 96.8%

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

   1. cos-neg 96.8%

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

   2. sub-neg 96.8%

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

   3. sub-neg 96.8%

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

   4. associate--r+ 96.8%

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

   5. *-commutative 96.8%

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

   6. associate--r+ 96.8%

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

5. Simplified 96.8%

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

Alternative 2: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -9500000000 \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -9500000000.0) (not (<= M 27.0)))
   (* (cos M) (exp (- (pow M 2.0))))
   (* (cos M) (exp (+ (- m n) (- (* M (+ M m)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -9500000000.0) || !(M <= 27.0)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp(((m - n) + ((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) :: tmp
    if ((m_1 <= (-9500000000.0d0)) .or. (.not. (m_1 <= 27.0d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(((m - n) + ((m_1 * (m_1 + m)) - 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 ((M <= -9500000000.0) || !(M <= 27.0)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp(((m - n) + ((M * (M + m)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -9500000000.0) or not (M <= 27.0):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp(((m - n) + ((M * (M + m)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -9500000000.0) || !(M <= 27.0))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(m - n) + Float64(Float64(M * Float64(M + m)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -9500000000.0) || ~((M <= 27.0)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp(((m - n) + ((M * (M + m)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -9500000000.0], N[Not[LessEqual[M, 27.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(M * N[(M + m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -9.5e9 or 27 < M

   1. Initial program 78.4%

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

   3. Taylor expanded in K around 0 99.1%

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

      1. cos-neg 99.1%

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

      2. sub-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. associate--r+ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate--r+ 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in M around inf 97.5%

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

      1. mul-1-neg 97.5%

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

   8. Simplified 97.5%

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

   if -9.5e9 < M < 27

   1. Initial program 71.0%

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

   3. Taylor expanded in m around 0 50.6%

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

      1. +-commutative 50.6%

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

      2. unpow2 50.6%

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

      3. distribute-rgt-out 50.7%

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

      4. *-commutative 50.7%

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

      5. *-commutative 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in n around 0 28.9%

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

      1. associate--r+ 28.9%

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

      2. associate-*r* 28.9%

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

      3. neg-mul-1 28.9%

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

      4. cancel-sign-sub 28.9%

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

   8. Simplified 28.9%

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

      1. associate-+l- 28.9%

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

      2. add-sqr-sqrt 16.2%

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

      3. fabs-sqr 16.2%

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

      4. add-sqr-sqrt 50.0%

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

      5. sub-neg 50.0%

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

      6. add-sqr-sqrt 24.8%

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

      7. sqrt-unprod 50.0%

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

      8. sqr-neg 50.0%

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

      9. sqrt-unprod 25.1%

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

      10. add-sqr-sqrt 50.0%

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

   10. Applied egg-rr 50.0%

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

   11. Taylor expanded in K around 0 63.1%

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

       1. cos-neg 63.1%

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

   13. Simplified 63.1%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -9500000000 \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(m + n\right) \cdot K\\ \mathbf{if}\;M \leq -41000000000:\\ \;\;\;\;\cos \left(0.5 \cdot t\_0\right) \cdot e^{m - \left(n + \ell\right)}\\ \mathbf{elif}\;M \leq 1.08 \cdot 10^{+96}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{t\_0}{2} - M\right) \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (+ m n) K)))
   (if (<= M -41000000000.0)
     (* (cos (* 0.5 t_0)) (exp (- m (+ n l))))
     (if (<= M 1.08e+96)
       (* (cos M) (exp (+ (- m n) (- (* M (+ M m)) l))))
       (* (cos (- (/ t_0 2.0) M)) (exp (* m (- M (* n 0.5)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (m + n) * K;
	double tmp;
	if (M <= -41000000000.0) {
		tmp = cos((0.5 * t_0)) * exp((m - (n + l)));
	} else if (M <= 1.08e+96) {
		tmp = cos(M) * exp(((m - n) + ((M * (M + m)) - l)));
	} else {
		tmp = cos(((t_0 / 2.0) - M)) * exp((m * (M - (n * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m + n) * k
    if (m_1 <= (-41000000000.0d0)) then
        tmp = cos((0.5d0 * t_0)) * exp((m - (n + l)))
    else if (m_1 <= 1.08d+96) then
        tmp = cos(m_1) * exp(((m - n) + ((m_1 * (m_1 + m)) - l)))
    else
        tmp = cos(((t_0 / 2.0d0) - m_1)) * exp((m * (m_1 - (n * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (m + n) * K;
	double tmp;
	if (M <= -41000000000.0) {
		tmp = Math.cos((0.5 * t_0)) * Math.exp((m - (n + l)));
	} else if (M <= 1.08e+96) {
		tmp = Math.cos(M) * Math.exp(((m - n) + ((M * (M + m)) - l)));
	} else {
		tmp = Math.cos(((t_0 / 2.0) - M)) * Math.exp((m * (M - (n * 0.5))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (m + n) * K
	tmp = 0
	if M <= -41000000000.0:
		tmp = math.cos((0.5 * t_0)) * math.exp((m - (n + l)))
	elif M <= 1.08e+96:
		tmp = math.cos(M) * math.exp(((m - n) + ((M * (M + m)) - l)))
	else:
		tmp = math.cos(((t_0 / 2.0) - M)) * math.exp((m * (M - (n * 0.5))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(m + n) * K)
	tmp = 0.0
	if (M <= -41000000000.0)
		tmp = Float64(cos(Float64(0.5 * t_0)) * exp(Float64(m - Float64(n + l))));
	elseif (M <= 1.08e+96)
		tmp = Float64(cos(M) * exp(Float64(Float64(m - n) + Float64(Float64(M * Float64(M + m)) - l))));
	else
		tmp = Float64(cos(Float64(Float64(t_0 / 2.0) - M)) * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (m + n) * K;
	tmp = 0.0;
	if (M <= -41000000000.0)
		tmp = cos((0.5 * t_0)) * exp((m - (n + l)));
	elseif (M <= 1.08e+96)
		tmp = cos(M) * exp(((m - n) + ((M * (M + m)) - l)));
	else
		tmp = cos(((t_0 / 2.0) - M)) * exp((m * (M - (n * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision]}, If[LessEqual[M, -41000000000.0], N[(N[Cos[N[(0.5 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(m - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 1.08e+96], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(M * N[(M + m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[(t$95$0 / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < -4.1e10

   1. Initial program 76.3%

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

   3. Taylor expanded in m around 0 62.8%

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

      1. +-commutative 62.8%

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

      2. unpow2 62.8%

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

      3. distribute-rgt-out 66.3%

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

      4. *-commutative 66.3%

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

      5. *-commutative 66.3%

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

   5. Simplified 66.3%

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

   6. Taylor expanded in n around 0 62.9%

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

      1. associate--r+ 62.9%

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

      2. associate-*r* 62.9%

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

      3. neg-mul-1 62.9%

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

      4. cancel-sign-sub 62.9%

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

   8. Simplified 62.9%

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

      1. associate-+l- 62.9%

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

      2. add-sqr-sqrt 35.6%

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

      3. fabs-sqr 35.6%

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

      4. add-sqr-sqrt 62.9%

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

      5. sub-neg 62.9%

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

      6. add-sqr-sqrt 62.9%

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

      7. sqrt-unprod 66.3%

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

      8. sqr-neg 66.3%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 11.2%

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

   10. Applied egg-rr 11.2%

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

   11. Taylor expanded in M around 0 39.7%

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

       1. +-commutative 39.7%

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

   13. Simplified 39.7%

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

   if -4.1e10 < M < 1.08e96

   1. Initial program 72.7%

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

   3. Taylor expanded in m around 0 52.5%

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

      1. +-commutative 52.5%

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

      2. unpow2 52.5%

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

      3. distribute-rgt-out 52.5%

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

      4. *-commutative 52.5%

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

      5. *-commutative 52.5%

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

   5. Simplified 52.5%

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

   6. Taylor expanded in n around 0 32.4%

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

      1. associate--r+ 32.4%

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

      2. associate-*r* 32.4%

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

      3. neg-mul-1 32.4%

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

      4. cancel-sign-sub 32.4%

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

   8. Simplified 32.4%

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

      1. associate-+l- 32.4%

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

      2. add-sqr-sqrt 16.5%

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

      3. fabs-sqr 16.5%

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

      4. add-sqr-sqrt 51.9%

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

      5. sub-neg 51.9%

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

      6. add-sqr-sqrt 22.3%

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

      7. sqrt-unprod 49.4%

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

      8. sqr-neg 49.4%

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

      9. sqrt-unprod 27.1%

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

      10. add-sqr-sqrt 49.4%

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

   10. Applied egg-rr 49.4%

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

   11. Taylor expanded in K around 0 61.2%

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

       1. cos-neg 61.2%

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

   13. Simplified 61.2%

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

   if 1.08e96 < M

   1. Initial program 78.0%

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

   3. Taylor expanded in m around 0 70.7%

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

      1. +-commutative 70.7%

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

      2. unpow2 70.7%

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

      3. distribute-rgt-out 73.2%

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

      4. *-commutative 73.2%

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

      5. *-commutative 73.2%

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

   5. Simplified 73.2%

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

   6. Taylor expanded in m around inf 32.6%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -41000000000:\\ \;\;\;\;\cos \left(0.5 \cdot \left(\left(m + n\right) \cdot K\right)\right) \cdot e^{m - \left(n + \ell\right)}\\ \mathbf{elif}\;M \leq 1.08 \cdot 10^{+96}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -780000000000 \lor \neg \left(M \leq 2.6 \cdot 10^{+95}\right):\\ \;\;\;\;\cos \left(0.5 \cdot \left(\left(m + n\right) \cdot K\right)\right) \cdot e^{m - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -780000000000.0) (not (<= M 2.6e+95)))
   (* (cos (* 0.5 (* (+ m n) K))) (exp (- m (+ n l))))
   (* (cos M) (exp (+ (- m n) (- (* M (+ M m)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -780000000000.0) || !(M <= 2.6e+95)) {
		tmp = cos((0.5 * ((m + n) * K))) * exp((m - (n + l)));
	} else {
		tmp = cos(M) * exp(((m - n) + ((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) :: tmp
    if ((m_1 <= (-780000000000.0d0)) .or. (.not. (m_1 <= 2.6d+95))) then
        tmp = cos((0.5d0 * ((m + n) * k))) * exp((m - (n + l)))
    else
        tmp = cos(m_1) * exp(((m - n) + ((m_1 * (m_1 + m)) - 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 ((M <= -780000000000.0) || !(M <= 2.6e+95)) {
		tmp = Math.cos((0.5 * ((m + n) * K))) * Math.exp((m - (n + l)));
	} else {
		tmp = Math.cos(M) * Math.exp(((m - n) + ((M * (M + m)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -780000000000.0) or not (M <= 2.6e+95):
		tmp = math.cos((0.5 * ((m + n) * K))) * math.exp((m - (n + l)))
	else:
		tmp = math.cos(M) * math.exp(((m - n) + ((M * (M + m)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -780000000000.0) || !(M <= 2.6e+95))
		tmp = Float64(cos(Float64(0.5 * Float64(Float64(m + n) * K))) * exp(Float64(m - Float64(n + l))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(m - n) + Float64(Float64(M * Float64(M + m)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -780000000000.0) || ~((M <= 2.6e+95)))
		tmp = cos((0.5 * ((m + n) * K))) * exp((m - (n + l)));
	else
		tmp = cos(M) * exp(((m - n) + ((M * (M + m)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -780000000000.0], N[Not[LessEqual[M, 2.6e+95]], $MachinePrecision]], N[(N[Cos[N[(0.5 * N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(m - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(M * N[(M + m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -7.8e11 or 2.5999999999999999e95 < M

   1. Initial program 77.0%

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

   3. Taylor expanded in m around 0 66.1%

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

      1. +-commutative 66.1%

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

      2. unpow2 66.1%

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

      3. distribute-rgt-out 69.1%

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

      4. *-commutative 69.1%

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

      5. *-commutative 69.1%

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

   5. Simplified 69.1%

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

   6. Taylor expanded in n around 0 66.2%

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

      1. associate--r+ 66.2%

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

      2. associate-*r* 66.2%

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

      3. neg-mul-1 66.2%

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

      4. cancel-sign-sub 66.2%

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

   8. Simplified 66.2%

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

      1. associate-+l- 66.2%

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

      2. add-sqr-sqrt 35.1%

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

      3. fabs-sqr 35.1%

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

      4. add-sqr-sqrt 66.2%

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

      5. sub-neg 66.2%

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

      6. add-sqr-sqrt 37.1%

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

      7. sqrt-unprod 39.6%

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

      8. sqr-neg 39.6%

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

      9. sqrt-unprod 1.5%

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

      10. add-sqr-sqrt 8.1%

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

   10. Applied egg-rr 8.1%

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

   11. Taylor expanded in M around 0 36.8%

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

       1. +-commutative 36.8%

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

   13. Simplified 36.8%

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

   if -7.8e11 < M < 2.5999999999999999e95

   1. Initial program 72.7%

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

   3. Taylor expanded in m around 0 52.5%

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

      1. +-commutative 52.5%

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

      2. unpow2 52.5%

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

      3. distribute-rgt-out 52.5%

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

      4. *-commutative 52.5%

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

      5. *-commutative 52.5%

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

   5. Simplified 52.5%

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

   6. Taylor expanded in n around 0 32.4%

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

      1. associate--r+ 32.4%

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

      2. associate-*r* 32.4%

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

      3. neg-mul-1 32.4%

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

      4. cancel-sign-sub 32.4%

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

   8. Simplified 32.4%

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

      1. associate-+l- 32.4%

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

      2. add-sqr-sqrt 16.5%

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

      3. fabs-sqr 16.5%

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

      4. add-sqr-sqrt 51.9%

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

      5. sub-neg 51.9%

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

      6. add-sqr-sqrt 22.3%

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

      7. sqrt-unprod 49.4%

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

      8. sqr-neg 49.4%

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

      9. sqrt-unprod 27.1%

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

      10. add-sqr-sqrt 49.4%

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

   10. Applied egg-rr 49.4%

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

   11. Taylor expanded in K around 0 61.2%

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

       1. cos-neg 61.2%

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

   13. Simplified 61.2%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -780000000000 \lor \neg \left(M \leq 2.6 \cdot 10^{+95}\right):\\ \;\;\;\;\cos \left(0.5 \cdot \left(\left(m + n\right) \cdot K\right)\right) \cdot e^{m - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -2.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-150}:\\ \;\;\;\;t\_0 \cdot \left(-0.125 \cdot {\left(\left(m + n\right) \cdot K\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))))
   (if (<= l -2.5)
     t_0
     (if (<= l 1.95e-150)
       (* t_0 (* -0.125 (pow (* (+ m n) K) 2.0)))
       (* (cos M) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double tmp;
	if (l <= -2.5) {
		tmp = t_0;
	} else if (l <= 1.95e-150) {
		tmp = t_0 * (-0.125 * pow(((m + n) * K), 2.0));
	} else {
		tmp = cos(M) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-l)
    if (l <= (-2.5d0)) then
        tmp = t_0
    else if (l <= 1.95d-150) then
        tmp = t_0 * ((-0.125d0) * (((m + n) * k) ** 2.0d0))
    else
        tmp = cos(m_1) * t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(-l);
	double tmp;
	if (l <= -2.5) {
		tmp = t_0;
	} else if (l <= 1.95e-150) {
		tmp = t_0 * (-0.125 * Math.pow(((m + n) * K), 2.0));
	} else {
		tmp = Math.cos(M) * t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-l)
	tmp = 0
	if l <= -2.5:
		tmp = t_0
	elif l <= 1.95e-150:
		tmp = t_0 * (-0.125 * math.pow(((m + n) * K), 2.0))
	else:
		tmp = math.cos(M) * t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	tmp = 0.0
	if (l <= -2.5)
		tmp = t_0;
	elseif (l <= 1.95e-150)
		tmp = Float64(t_0 * Float64(-0.125 * (Float64(Float64(m + n) * K) ^ 2.0)));
	else
		tmp = Float64(cos(M) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-l);
	tmp = 0.0;
	if (l <= -2.5)
		tmp = t_0;
	elseif (l <= 1.95e-150)
		tmp = t_0 * (-0.125 * (((m + n) * K) ^ 2.0));
	else
		tmp = cos(M) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[l, -2.5], t$95$0, If[LessEqual[l, 1.95e-150], N[(t$95$0 * N[(-0.125 * N[Power[N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.5

   1. Initial program 75.0%

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

   3. Taylor expanded in l around inf 18.8%

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

      1. mul-1-neg 18.8%

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

   5. Simplified 18.8%

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

   6. Taylor expanded in M around 0 18.8%

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

      1. *-commutative 18.8%

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

      2. associate-*r* 18.8%

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

      3. remove-double-neg 18.8%

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

      4. mul-1-neg 18.8%

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

      5. sub-neg 18.8%

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

      6. associate-*r* 18.8%

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

      7. *-commutative 18.8%

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

      8. sub-neg 18.8%

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

      9. mul-1-neg 18.8%

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

      10. remove-double-neg 18.8%

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

      11. +-commutative 18.8%

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

   8. Simplified 18.8%

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

   9. Taylor expanded in K around 0 20.6%

      \[\leadsto e^{-\ell} \cdot \color{blue}{\left(1 + -0.125 \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 20.6%

          \[\leadsto e^{-\ell} \cdot \left(1 + -0.125 \cdot \color{blue}{\left({\left(m + n\right)}^{2} \cdot {K}^{2}\right)}\right) \]

   11. Simplified 20.6%

       \[\leadsto e^{-\ell} \cdot \color{blue}{\left(1 + -0.125 \cdot \left({\left(m + n\right)}^{2} \cdot {K}^{2}\right)\right)} \]

   12. Taylor expanded in K around 0 20.8%

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

   if -2.5 < l < 1.9500000000000001e-150

   1. Initial program 73.6%

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

   3. Taylor expanded in l around inf 9.7%

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

      1. mul-1-neg 9.7%

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

   5. Simplified 9.7%

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

   6. Taylor expanded in M around 0 9.7%

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

      1. *-commutative 9.7%

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

      2. associate-*r* 9.7%

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

      3. remove-double-neg 9.7%

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

      4. mul-1-neg 9.7%

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

      5. sub-neg 9.7%

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

      6. associate-*r* 9.7%

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

      7. *-commutative 9.7%

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

      8. sub-neg 9.7%

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

      9. mul-1-neg 9.7%

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

      10. remove-double-neg 9.7%

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

      11. +-commutative 9.7%

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

   8. Simplified 9.7%

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

   9. Taylor expanded in K around 0 8.0%

      \[\leadsto e^{-\ell} \cdot \color{blue}{\left(1 + -0.125 \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 8.0%

          \[\leadsto e^{-\ell} \cdot \left(1 + -0.125 \cdot \color{blue}{\left({\left(m + n\right)}^{2} \cdot {K}^{2}\right)}\right) \]

   11. Simplified 8.0%

       \[\leadsto e^{-\ell} \cdot \color{blue}{\left(1 + -0.125 \cdot \left({\left(m + n\right)}^{2} \cdot {K}^{2}\right)\right)} \]

   12. Taylor expanded in K around inf 21.5%

       \[\leadsto e^{-\ell} \cdot \color{blue}{\left(-0.125 \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. unpow2 21.5%

          \[\leadsto e^{-\ell} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot {\left(m + n\right)}^{2}\right)\right) \]

       2. unpow2 21.5%

          \[\leadsto e^{-\ell} \cdot \left(-0.125 \cdot \left(\left(K \cdot K\right) \cdot \color{blue}{\left(\left(m + n\right) \cdot \left(m + n\right)\right)}\right)\right) \]

       3. swap-sqr 19.2%

          \[\leadsto e^{-\ell} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(K \cdot \left(m + n\right)\right) \cdot \left(K \cdot \left(m + n\right)\right)\right)}\right) \]

       4. unpow2 19.2%

          \[\leadsto e^{-\ell} \cdot \left(-0.125 \cdot \color{blue}{{\left(K \cdot \left(m + n\right)\right)}^{2}}\right) \]

   14. Simplified 19.2%

       \[\leadsto e^{-\ell} \cdot \color{blue}{\left(-0.125 \cdot {\left(K \cdot \left(m + n\right)\right)}^{2}\right)} \]

   if 1.9500000000000001e-150 < l

   1. Initial program 74.8%

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

   3. Taylor expanded in K around 0 97.6%

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

      1. cos-neg 97.6%

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

      2. sub-neg 97.6%

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

      3. sub-neg 97.6%

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

      4. associate--r+ 97.6%

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

      5. *-commutative 97.6%

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

      6. associate--r+ 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in l around inf 68.3%

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

      1. neg-mul-1 68.3%

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

   8. Simplified 68.3%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-150}:\\ \;\;\;\;e^{-\ell} \cdot \left(-0.125 \cdot {\left(\left(m + n\right) \cdot K\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.62:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 1.62)
   (* (cos M) (exp (+ (- m n) (- (* M (+ M m)) l))))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1.62) {
		tmp = cos(M) * exp(((m - n) + ((M * (M + m)) - l)));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 1.62d0) then
        tmp = cos(m_1) * exp(((m - n) + ((m_1 * (m_1 + m)) - l)))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1.62) {
		tmp = Math.cos(M) * Math.exp(((m - n) + ((M * (M + m)) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 1.62:
		tmp = math.cos(M) * math.exp(((m - n) + ((M * (M + m)) - l)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 1.62)
		tmp = Float64(cos(M) * exp(Float64(Float64(m - n) + Float64(Float64(M * Float64(M + m)) - l))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 1.62)
		tmp = cos(M) * exp(((m - n) + ((M * (M + m)) - l)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 1.62], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(M * N[(M + m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.6200000000000001

   1. Initial program 74.4%

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

   3. Taylor expanded in m around 0 55.9%

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

      1. +-commutative 55.9%

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

      2. unpow2 55.9%

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

      3. distribute-rgt-out 57.0%

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

      4. *-commutative 57.0%

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

      5. *-commutative 57.0%

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

   5. Simplified 57.0%

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

   6. Taylor expanded in n around 0 40.6%

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

      1. associate--r+ 40.6%

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

      2. associate-*r* 40.6%

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

      3. neg-mul-1 40.6%

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

      4. cancel-sign-sub 40.6%

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

   8. Simplified 40.6%

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

      1. associate-+l- 40.6%

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

      2. add-sqr-sqrt 20.3%

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

      3. fabs-sqr 20.3%

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

      4. add-sqr-sqrt 55.9%

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

      5. sub-neg 55.9%

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

      6. add-sqr-sqrt 25.7%

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

      7. sqrt-unprod 42.7%

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

      8. sqr-neg 42.7%

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

      9. sqrt-unprod 16.5%

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

      10. add-sqr-sqrt 29.5%

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

   10. Applied egg-rr 29.5%

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

   11. Taylor expanded in K around 0 37.9%

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

       1. cos-neg 37.9%

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

   13. Simplified 37.9%

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

   if 1.6200000000000001 < l

   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 K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--r+ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate--r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 98.6%

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

      1. neg-mul-1 98.6%

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

   8. Simplified 98.6%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.62:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

3. Taylor expanded in l around inf 28.8%

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

   1. mul-1-neg 28.8%

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

5. Simplified 28.8%

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

6. Taylor expanded in M around 0 28.8%

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

   1. *-commutative 28.8%

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

   2. associate-*r* 28.8%

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

   3. remove-double-neg 28.8%

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

   4. mul-1-neg 28.8%

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

   5. sub-neg 28.8%

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

   6. associate-*r* 28.8%

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

   7. *-commutative 28.8%

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

   8. sub-neg 28.8%

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

   9. mul-1-neg 28.8%

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

   10. remove-double-neg 28.8%

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

   11. +-commutative 28.8%

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

8. Simplified 28.8%

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

9. Taylor expanded in K around 0 18.2%

   \[\leadsto e^{-\ell} \cdot \color{blue}{\left(1 + -0.125 \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 18.2%

       \[\leadsto e^{-\ell} \cdot \left(1 + -0.125 \cdot \color{blue}{\left({\left(m + n\right)}^{2} \cdot {K}^{2}\right)}\right) \]

11. Simplified 18.2%

    \[\leadsto e^{-\ell} \cdot \color{blue}{\left(1 + -0.125 \cdot \left({\left(m + n\right)}^{2} \cdot {K}^{2}\right)\right)} \]

12. Taylor expanded in K around 0 36.7%

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

Reproduce

herbie shell --seed 2024109 
(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)))))))