Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 77.0% → 96.9%

Time: 19.3s

Alternatives: 9

Speedup: 1.4×

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: 77.0% 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

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

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 74.9%

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

2. Taylor expanded in K around 0 96.9%

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

   1. cos-neg 96.9%

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

4. Simplified 96.9%

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

5. Final simplification 96.9%

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

Alternative 2: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1260000 \lor \neg \left(m \leq 53\right):\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -1260000.0) (not (<= m 53.0)))
   (* (cos M) (exp (* (pow m 2.0) -0.25)))
   (* (cos M) (exp (- (pow M 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -1260000.0) || !(m <= 53.0)) {
		tmp = cos(M) * exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = cos(M) * exp(-pow(M, 2.0));
	}
	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) :: tmp
    if ((m <= (-1260000.0d0)) .or. (.not. (m <= 53.0d0))) then
        tmp = cos(m_1) * exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    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 <= -1260000.0) || !(m <= 53.0)) {
		tmp = Math.cos(M) * Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -1260000.0) or not (m <= 53.0):
		tmp = math.cos(M) * math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -1260000.0) || !(m <= 53.0))
		tmp = Float64(cos(M) * exp(Float64((m ^ 2.0) * -0.25)));
	else
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -1260000.0) || ~((m <= 53.0)))
		tmp = cos(M) * exp(((m ^ 2.0) * -0.25));
	else
		tmp = cos(M) * exp(-(M ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -1260000.0], N[Not[LessEqual[m, 53.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.26e6 or 53 < m

   1. Initial program 71.4%

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

   2. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in m around inf 99.2%

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

      1. *-commutative 70.6%

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

   7. Simplified 99.2%

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

   if -1.26e6 < m < 53

   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. Taylor expanded in K around 0 94.6%

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

      1. cos-neg 94.6%

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

   4. Simplified 94.6%

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

   5. Taylor expanded in M around inf 63.2%

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

      1. mul-1-neg 63.2%

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

   7. Simplified 63.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1260000 \lor \neg \left(m \leq 53\right):\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \]

Alternative 3: 65.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 4.2 \cdot 10^{-177}:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 4.2e-177)
   (* (cos M) (exp (* (pow m 2.0) -0.25)))
   (if (<= n 54.0)
     (* (cos M) (exp (- (pow M 2.0))))
     (* (cos M) (exp (* -0.25 (pow n 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 4.2e-177) {
		tmp = cos(M) * exp((pow(m, 2.0) * -0.25));
	} else if (n <= 54.0) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp((-0.25 * pow(n, 2.0)));
	}
	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) :: tmp
    if (n <= 4.2d-177) then
        tmp = cos(m_1) * exp(((m ** 2.0d0) * (-0.25d0)))
    else if (n <= 54.0d0) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 4.2e-177) {
		tmp = Math.cos(M) * Math.exp((Math.pow(m, 2.0) * -0.25));
	} else if (n <= 54.0) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 4.2e-177:
		tmp = math.cos(M) * math.exp((math.pow(m, 2.0) * -0.25))
	elif n <= 54.0:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 4.2e-177)
		tmp = Float64(cos(M) * exp(Float64((m ^ 2.0) * -0.25)));
	elseif (n <= 54.0)
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (n ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 4.2e-177)
		tmp = cos(M) * exp(((m ^ 2.0) * -0.25));
	elseif (n <= 54.0)
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 4.2e-177], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 4.20000000000000002e-177

   1. Initial program 76.1%

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

   2. Taylor expanded in K around 0 95.3%

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

      1. cos-neg 95.3%

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

   4. Simplified 95.3%

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

   5. Taylor expanded in m around inf 55.4%

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

      1. *-commutative 43.9%

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

   7. Simplified 55.4%

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

   if 4.20000000000000002e-177 < n < 54

   1. Initial program 85.3%

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

   2. Taylor expanded in K around 0 97.0%

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

      1. cos-neg 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in M around inf 59.7%

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

      1. mul-1-neg 59.7%

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

   7. Simplified 59.7%

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

   if 54 < n

   1. Initial program 69.2%

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

   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 n around inf 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.2 \cdot 10^{-177}:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]

Alternative 4: 66.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 720.0) (* (cos M) (exp (- (pow M 2.0)))) (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 720.0d0) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 720.0:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 720.0)
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 720.0)
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 720.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 720

   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. Taylor expanded in K around 0 95.9%

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

      1. cos-neg 95.9%

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

   4. Simplified 95.9%

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

   5. Taylor expanded in M around inf 53.9%

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

      1. mul-1-neg 53.9%

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

   7. Simplified 53.9%

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

   if 720 < l

   1. Initial program 76.7%

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

   2. Taylor expanded in l around inf 76.7%

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

      1. mul-1-neg 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 5: 35.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{-21}:\\ \;\;\;\;-0.5 \cdot \left(K \cdot \left(m \cdot \sin \left(0.5 \cdot \left(n \cdot K\right) - M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 7.6e-21)
   (* -0.5 (* K (* m (sin (- (* 0.5 (* n K)) M)))))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 7.6e-21) {
		tmp = -0.5 * (K * (m * sin(((0.5 * (n * K)) - M))));
	} 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 <= 7.6d-21) then
        tmp = (-0.5d0) * (k * (m * sin(((0.5d0 * (n * k)) - m_1))))
    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 <= 7.6e-21) {
		tmp = -0.5 * (K * (m * Math.sin(((0.5 * (n * K)) - M))));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 7.6e-21:
		tmp = -0.5 * (K * (m * math.sin(((0.5 * (n * K)) - M))))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 7.6e-21)
		tmp = Float64(-0.5 * Float64(K * Float64(m * sin(Float64(Float64(0.5 * Float64(n * K)) - M)))));
	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 <= 7.6e-21)
		tmp = -0.5 * (K * (m * sin(((0.5 * (n * K)) - M))));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 7.6e-21], N[(-0.5 * N[(K * N[(m * N[Sin[N[(N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.5999999999999995e-21

   1. Initial program 75.3%

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

   2. Taylor expanded in m around inf 40.0%

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

      1. *-commutative 40.0%

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

   4. Simplified 40.0%

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

   5. Taylor expanded in m around 0 7.2%

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

      1. *-commutative 7.2%

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

      2. associate-*r* 7.2%

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

      3. *-commutative 7.2%

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

      4. *-commutative 7.2%

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

   7. Simplified 7.2%

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

   8. Taylor expanded in K around inf 17.1%

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

   if 7.5999999999999995e-21 < l

   1. Initial program 73.8%

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

   2. Taylor expanded in l around inf 70.9%

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

      1. mul-1-neg 70.9%

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

   4. Simplified 70.9%

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

   5. Taylor expanded in K around 0 92.5%

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

      1. cos-neg 92.5%

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

   7. Simplified 92.5%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{-21}:\\ \;\;\;\;-0.5 \cdot \left(K \cdot \left(m \cdot \sin \left(0.5 \cdot \left(n \cdot K\right) - M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 6: 19.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.9 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(n \cdot \sin M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(K \cdot \left(m \cdot \sin \left(0.5 \cdot \left(n \cdot K\right) - M\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -2.9e-19)
   (* 0.5 (* K (* n (sin M))))
   (* -0.5 (* K (* m (sin (- (* 0.5 (* n K)) M)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.9e-19) {
		tmp = 0.5 * (K * (n * sin(M)));
	} else {
		tmp = -0.5 * (K * (m * sin(((0.5 * (n * K)) - M))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-2.9d-19)) then
        tmp = 0.5d0 * (k * (n * sin(m_1)))
    else
        tmp = (-0.5d0) * (k * (m * sin(((0.5d0 * (n * k)) - m_1))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.9e-19) {
		tmp = 0.5 * (K * (n * Math.sin(M)));
	} else {
		tmp = -0.5 * (K * (m * Math.sin(((0.5 * (n * K)) - M))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -2.9e-19:
		tmp = 0.5 * (K * (n * math.sin(M)))
	else:
		tmp = -0.5 * (K * (m * math.sin(((0.5 * (n * K)) - M))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -2.9e-19)
		tmp = Float64(0.5 * Float64(K * Float64(n * sin(M))));
	else
		tmp = Float64(-0.5 * Float64(K * Float64(m * sin(Float64(Float64(0.5 * Float64(n * K)) - M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -2.9e-19)
		tmp = 0.5 * (K * (n * sin(M)));
	else
		tmp = -0.5 * (K * (m * sin(((0.5 * (n * K)) - M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -2.9e-19], N[(0.5 * N[(K * N[(n * N[Sin[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(K * N[(m * N[Sin[N[(N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.9e-19

   1. Initial program 71.6%

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

   2. Taylor expanded in m around inf 62.9%

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

      1. *-commutative 62.9%

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

   4. Simplified 62.9%

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

   5. Taylor expanded in m around 0 2.7%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot n\right) - M\right)} \]
   6. Step-by-step derivation

      1. *-commutative 2.7%

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

   7. Simplified 2.7%

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

   8. Taylor expanded in n around 0 2.7%

      \[\leadsto \color{blue}{\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(n \cdot \sin \left(-M\right)\right)\right)} \]
   9. Step-by-step derivation

      1. cos-neg 2.7%

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

      2. *-commutative 2.7%

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

      3. sin-neg 2.7%

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

   10. Simplified 2.7%

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

   11. Taylor expanded in K around inf 18.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(n \cdot \sin M\right)\right)} \]

   if -2.9e-19 < m

   1. Initial program 76.1%

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

   2. Taylor expanded in m around inf 32.3%

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

      1. *-commutative 32.3%

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

   4. Simplified 32.3%

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

   5. Taylor expanded in m around 0 7.3%

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

      1. *-commutative 7.3%

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

      2. associate-*r* 7.3%

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

      3. *-commutative 7.3%

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

      4. *-commutative 7.3%

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

   7. Simplified 7.3%

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

   8. Taylor expanded in K around inf 20.6%

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

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.9 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(n \cdot \sin M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(K \cdot \left(m \cdot \sin \left(0.5 \cdot \left(n \cdot K\right) - M\right)\right)\right)\\ \end{array} \]

Alternative 7: 20.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(K \cdot \left(n \cdot \sin M\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (* 0.5 (* K (* n (sin M)))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(K, m, n, M, l)
	return Float64(0.5 * Float64(K * Float64(n * sin(M))))
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[(0.5 * N[(K * N[(n * N[Sin[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.9%

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

2. Taylor expanded in m around inf 40.3%

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

   1. *-commutative 40.3%

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

4. Simplified 40.3%

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

5. Taylor expanded in m around 0 6.4%

   \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot n\right) - M\right)} \]
6. Step-by-step derivation

   1. *-commutative 6.4%

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

7. Simplified 6.4%

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

8. Taylor expanded in n around 0 6.0%

   \[\leadsto \color{blue}{\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(n \cdot \sin \left(-M\right)\right)\right)} \]
9. Step-by-step derivation

   1. cos-neg 6.0%

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

   2. *-commutative 6.0%

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

   3. sin-neg 6.0%

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

10. Simplified 6.0%

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

11. Taylor expanded in K around inf 19.2%

    \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(n \cdot \sin M\right)\right)} \]

12. Final simplification 19.2%

    \[\leadsto 0.5 \cdot \left(K \cdot \left(n \cdot \sin M\right)\right) \]

Alternative 8: 6.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \cos \left(-M\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

2. Taylor expanded in m around inf 40.3%

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

   1. *-commutative 40.3%

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

4. Simplified 40.3%

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

5. Taylor expanded in m around 0 6.4%

   \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot n\right) - M\right)} \]
6. Step-by-step derivation

   1. *-commutative 6.4%

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

7. Simplified 6.4%

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

8. Taylor expanded in n around 0 6.6%

   \[\leadsto \cos \color{blue}{\left(-1 \cdot M\right)} \]
9. Step-by-step derivation

   1. neg-mul-1 6.6%

      \[\leadsto \cos \color{blue}{\left(-M\right)} \]

10. Simplified 6.6%

    \[\leadsto \cos \color{blue}{\left(-M\right)} \]

11. Final simplification 6.6%

    \[\leadsto \cos \left(-M\right) \]

Alternative 9: 6.7% accurate, 425.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 1.0)

C

double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

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 = 1.0d0
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

Python

def code(K, m, n, M, l):
	return 1.0

Julia

function code(K, m, n, M, l)
	return 1.0
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0;
end

Wolfram

code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 74.9%

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

2. Taylor expanded in m around inf 40.3%

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

   1. *-commutative 40.3%

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

4. Simplified 40.3%

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

5. Taylor expanded in m around 0 6.4%

   \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot n\right) - M\right)} \]
6. Step-by-step derivation

   1. *-commutative 6.4%

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

7. Simplified 6.4%

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

8. Taylor expanded in n around 0 6.0%

   \[\leadsto \color{blue}{\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(n \cdot \sin \left(-M\right)\right)\right)} \]
9. Step-by-step derivation

   1. cos-neg 6.0%

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

   2. *-commutative 6.0%

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

   3. sin-neg 6.0%

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

10. Simplified 6.0%

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

11. Taylor expanded in M around 0 6.6%

    \[\leadsto \color{blue}{1} \]

12. Final simplification 6.6%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023310 
(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)))))))