Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.5% → 96.5%

Time: 21.9s

Alternatives: 16

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: 76.5% 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: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

3. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

5. Simplified 97.5%

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

6. Final simplification 97.5%

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

Alternative 2: 68.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos M \cdot e^{-{M}^{2}}\\ \mathbf{if}\;M \leq -1.56 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\cos M \cdot t\_0\\ \mathbf{elif}\;M \leq -1.05 \cdot 10^{-296}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;M \leq 5.2 \cdot 10^{-160}:\\ \;\;\;\;\left(\left(0.5 \cdot K\right) \cdot \left(n \cdot \sin M\right)\right) \cdot t\_0\\ \mathbf{elif}\;M \leq 9 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (* (cos M) (exp (- (pow M 2.0))))))
   (if (<= M -1.56e-6)
     t_1
     (if (<= M -1.75e-154)
       (* (cos M) t_0)
       (if (<= M -1.05e-296)
         (* (cos M) (exp (* n (- M (* m 0.5)))))
         (if (<= M 5.2e-160)
           (* (* (* 0.5 K) (* n (sin M))) t_0)
           (if (<= M 9e-57)
             t_0
             (if (<= M 27.0) (* 0.5 (* K (* m (* (sin M) t_0)))) t_1))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double t_1 = cos(M) * exp(-pow(M, 2.0));
	double tmp;
	if (M <= -1.56e-6) {
		tmp = t_1;
	} else if (M <= -1.75e-154) {
		tmp = cos(M) * t_0;
	} else if (M <= -1.05e-296) {
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	} else if (M <= 5.2e-160) {
		tmp = ((0.5 * K) * (n * sin(M))) * t_0;
	} else if (M <= 9e-57) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = 0.5 * (K * (m * (sin(M) * t_0)));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos(m_1) * exp(-(m_1 ** 2.0d0))
    if (m_1 <= (-1.56d-6)) then
        tmp = t_1
    else if (m_1 <= (-1.75d-154)) then
        tmp = cos(m_1) * t_0
    else if (m_1 <= (-1.05d-296)) then
        tmp = cos(m_1) * exp((n * (m_1 - (m * 0.5d0))))
    else if (m_1 <= 5.2d-160) then
        tmp = ((0.5d0 * k) * (n * sin(m_1))) * t_0
    else if (m_1 <= 9d-57) then
        tmp = t_0
    else if (m_1 <= 27.0d0) then
        tmp = 0.5d0 * (k * (m * (sin(m_1) * t_0)))
    else
        tmp = t_1
    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 t_1 = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	double tmp;
	if (M <= -1.56e-6) {
		tmp = t_1;
	} else if (M <= -1.75e-154) {
		tmp = Math.cos(M) * t_0;
	} else if (M <= -1.05e-296) {
		tmp = Math.cos(M) * Math.exp((n * (M - (m * 0.5))));
	} else if (M <= 5.2e-160) {
		tmp = ((0.5 * K) * (n * Math.sin(M))) * t_0;
	} else if (M <= 9e-57) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = 0.5 * (K * (m * (Math.sin(M) * t_0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-l)
	t_1 = math.cos(M) * math.exp(-math.pow(M, 2.0))
	tmp = 0
	if M <= -1.56e-6:
		tmp = t_1
	elif M <= -1.75e-154:
		tmp = math.cos(M) * t_0
	elif M <= -1.05e-296:
		tmp = math.cos(M) * math.exp((n * (M - (m * 0.5))))
	elif M <= 5.2e-160:
		tmp = ((0.5 * K) * (n * math.sin(M))) * t_0
	elif M <= 9e-57:
		tmp = t_0
	elif M <= 27.0:
		tmp = 0.5 * (K * (m * (math.sin(M) * t_0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	t_1 = Float64(cos(M) * exp(Float64(-(M ^ 2.0))))
	tmp = 0.0
	if (M <= -1.56e-6)
		tmp = t_1;
	elseif (M <= -1.75e-154)
		tmp = Float64(cos(M) * t_0);
	elseif (M <= -1.05e-296)
		tmp = Float64(cos(M) * exp(Float64(n * Float64(M - Float64(m * 0.5)))));
	elseif (M <= 5.2e-160)
		tmp = Float64(Float64(Float64(0.5 * K) * Float64(n * sin(M))) * t_0);
	elseif (M <= 9e-57)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = Float64(0.5 * Float64(K * Float64(m * Float64(sin(M) * t_0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-l);
	t_1 = cos(M) * exp(-(M ^ 2.0));
	tmp = 0.0;
	if (M <= -1.56e-6)
		tmp = t_1;
	elseif (M <= -1.75e-154)
		tmp = cos(M) * t_0;
	elseif (M <= -1.05e-296)
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	elseif (M <= 5.2e-160)
		tmp = ((0.5 * K) * (n * sin(M))) * t_0;
	elseif (M <= 9e-57)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = 0.5 * (K * (m * (sin(M) * t_0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -1.56e-6], t$95$1, If[LessEqual[M, -1.75e-154], N[(N[Cos[M], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[M, -1.05e-296], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 5.2e-160], N[(N[(N[(0.5 * K), $MachinePrecision] * N[(n * N[Sin[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[M, 9e-57], t$95$0, If[LessEqual[M, 27.0], N[(0.5 * N[(K * N[(m * N[(N[Sin[M], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if M < -1.5600000000000001e-6 or 27 < M

   1. Initial program 73.1%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around inf 98.5%

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

      1. mul-1-neg 98.5%

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

   8. Simplified 98.5%

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

   if -1.5600000000000001e-6 < M < -1.75e-154

   1. Initial program 62.6%

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

   3. Taylor expanded in K around 0 96.6%

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

      1. cos-neg 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in l around inf 50.0%

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

      1. mul-1-neg 40.9%

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

   8. Simplified 50.0%

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

   if -1.75e-154 < M < -1.05e-296

   1. Initial program 68.1%

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

   3. Taylor expanded in n around 0 46.6%

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

      1. +-commutative 46.6%

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

      2. unpow2 46.6%

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

      3. distribute-rgt-out 46.6%

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

      4. *-commutative 46.6%

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

      5. *-commutative 46.6%

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

   5. Simplified 46.6%

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

   6. Taylor expanded in K around 0 63.7%

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

      1. cos-neg 91.4%

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

   8. Simplified 63.7%

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

   9. Taylor expanded in n around inf 51.8%

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

   if -1.05e-296 < M < 5.20000000000000007e-160

   1. Initial program 81.3%

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

   3. Taylor expanded in K around 0 93.8%

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

      1. cos-neg 93.8%

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

      2. *-commutative 93.8%

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

      3. associate-*l* 93.8%

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

      4. metadata-eval 93.8%

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

      5. distribute-rgt-neg-in 93.8%

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

      6. *-commutative 93.8%

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

      7. associate-*r* 93.8%

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

      8. distribute-lft-neg-in 93.8%

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

      9. sin-neg 93.8%

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

      10. distribute-rgt-neg-out 93.8%

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

      11. remove-double-neg 93.8%

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

   5. Simplified 93.8%

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

   6. Taylor expanded in l around inf 39.0%

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

      1. mul-1-neg 39.0%

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

   8. Simplified 39.0%

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

   9. Taylor expanded in n around inf 63.6%

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

       1. associate-*r* 63.6%

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

   11. Simplified 63.6%

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

   if 5.20000000000000007e-160 < M < 8.99999999999999945e-57

   1. Initial program 90.9%

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

   3. Taylor expanded in K around 0 96.1%

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

      1. cos-neg 96.1%

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

   5. Simplified 96.1%

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

   6. Taylor expanded in l around inf 47.6%

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

      1. mul-1-neg 47.5%

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

   8. Simplified 47.6%

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

   9. Taylor expanded in M around 0 47.6%

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

   if 8.99999999999999945e-57 < M < 27

   1. Initial program 83.3%

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

   3. Taylor expanded in K around 0 83.3%

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

      1. cos-neg 83.3%

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

      2. *-commutative 83.3%

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

      3. associate-*l* 83.3%

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

      4. metadata-eval 83.3%

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

      5. distribute-rgt-neg-in 83.3%

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

      6. *-commutative 83.3%

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

      7. associate-*r* 83.3%

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

      8. distribute-lft-neg-in 83.3%

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

      9. sin-neg 83.3%

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

      10. distribute-rgt-neg-out 83.3%

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

      11. remove-double-neg 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in l around inf 51.1%

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

      1. mul-1-neg 51.1%

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

   8. Simplified 51.1%

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

   9. Taylor expanded in m around inf 51.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(e^{-\ell} \cdot \sin M\right)\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.56 \cdot 10^{-6}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;M \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;M \leq -1.05 \cdot 10^{-296}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;M \leq 5.2 \cdot 10^{-160}:\\ \;\;\;\;\left(\left(0.5 \cdot K\right) \cdot \left(n \cdot \sin M\right)\right) \cdot e^{-\ell}\\ \mathbf{elif}\;M \leq 9 \cdot 10^{-57}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot e^{-\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ t_1 := e^{-\ell}\\ t_2 := \cos M \cdot e^{-{M}^{2}}\\ \mathbf{if}\;M \leq -28.5:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;M \leq 1.5 \cdot 10^{-304}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 3.5 \cdot 10^{-227}:\\ \;\;\;\;\left(\left(0.5 \cdot K\right) \cdot \left(n \cdot \sin M\right)\right) \cdot t\_1\\ \mathbf{elif}\;M \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot t\_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (* -0.25 (pow m 2.0)))))
        (t_1 (exp (- l)))
        (t_2 (* (cos M) (exp (- (pow M 2.0))))))
   (if (<= M -28.5)
     t_2
     (if (<= M 1.5e-304)
       t_0
       (if (<= M 3.5e-227)
         (* (* (* 0.5 K) (* n (sin M))) t_1)
         (if (<= M 2.4e-68)
           t_0
           (if (<= M 27.0) (* 0.5 (* K (* m (* (sin M) t_1)))) t_2)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp((-0.25 * pow(m, 2.0)));
	double t_1 = exp(-l);
	double t_2 = cos(M) * exp(-pow(M, 2.0));
	double tmp;
	if (M <= -28.5) {
		tmp = t_2;
	} else if (M <= 1.5e-304) {
		tmp = t_0;
	} else if (M <= 3.5e-227) {
		tmp = ((0.5 * K) * (n * sin(M))) * t_1;
	} else if (M <= 2.4e-68) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = 0.5 * (K * (m * (sin(M) * t_1)));
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    t_1 = exp(-l)
    t_2 = cos(m_1) * exp(-(m_1 ** 2.0d0))
    if (m_1 <= (-28.5d0)) then
        tmp = t_2
    else if (m_1 <= 1.5d-304) then
        tmp = t_0
    else if (m_1 <= 3.5d-227) then
        tmp = ((0.5d0 * k) * (n * sin(m_1))) * t_1
    else if (m_1 <= 2.4d-68) then
        tmp = t_0
    else if (m_1 <= 27.0d0) then
        tmp = 0.5d0 * (k * (m * (sin(m_1) * t_1)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	double t_1 = Math.exp(-l);
	double t_2 = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	double tmp;
	if (M <= -28.5) {
		tmp = t_2;
	} else if (M <= 1.5e-304) {
		tmp = t_0;
	} else if (M <= 3.5e-227) {
		tmp = ((0.5 * K) * (n * Math.sin(M))) * t_1;
	} else if (M <= 2.4e-68) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = 0.5 * (K * (m * (Math.sin(M) * t_1)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	t_1 = math.exp(-l)
	t_2 = math.cos(M) * math.exp(-math.pow(M, 2.0))
	tmp = 0
	if M <= -28.5:
		tmp = t_2
	elif M <= 1.5e-304:
		tmp = t_0
	elif M <= 3.5e-227:
		tmp = ((0.5 * K) * (n * math.sin(M))) * t_1
	elif M <= 2.4e-68:
		tmp = t_0
	elif M <= 27.0:
		tmp = 0.5 * (K * (m * (math.sin(M) * t_1)))
	else:
		tmp = t_2
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))))
	t_1 = exp(Float64(-l))
	t_2 = Float64(cos(M) * exp(Float64(-(M ^ 2.0))))
	tmp = 0.0
	if (M <= -28.5)
		tmp = t_2;
	elseif (M <= 1.5e-304)
		tmp = t_0;
	elseif (M <= 3.5e-227)
		tmp = Float64(Float64(Float64(0.5 * K) * Float64(n * sin(M))) * t_1);
	elseif (M <= 2.4e-68)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = Float64(0.5 * Float64(K * Float64(m * Float64(sin(M) * t_1))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp((-0.25 * (m ^ 2.0)));
	t_1 = exp(-l);
	t_2 = cos(M) * exp(-(M ^ 2.0));
	tmp = 0.0;
	if (M <= -28.5)
		tmp = t_2;
	elseif (M <= 1.5e-304)
		tmp = t_0;
	elseif (M <= 3.5e-227)
		tmp = ((0.5 * K) * (n * sin(M))) * t_1;
	elseif (M <= 2.4e-68)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = 0.5 * (K * (m * (sin(M) * t_1)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -28.5], t$95$2, If[LessEqual[M, 1.5e-304], t$95$0, If[LessEqual[M, 3.5e-227], N[(N[(N[(0.5 * K), $MachinePrecision] * N[(n * N[Sin[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[M, 2.4e-68], t$95$0, If[LessEqual[M, 27.0], N[(0.5 * N[(K * N[(m * N[(N[Sin[M], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -28.5 or 27 < M

   1. Initial program 73.5%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around inf 99.3%

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

      1. mul-1-neg 99.3%

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

   8. Simplified 99.3%

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

   if -28.5 < M < 1.5000000000000001e-304 or 3.5000000000000001e-227 < M < 2.39999999999999991e-68

   1. Initial program 71.5%

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

   3. Taylor expanded in K around 0 95.2%

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

      1. cos-neg 95.2%

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

   5. Simplified 95.2%

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

   6. Taylor expanded in m around inf 71.8%

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

   if 1.5000000000000001e-304 < M < 3.5000000000000001e-227

   1. Initial program 87.5%

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

   3. Taylor expanded in K around 0 87.5%

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

      1. cos-neg 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*l* 87.5%

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

      4. metadata-eval 87.5%

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

      5. distribute-rgt-neg-in 87.5%

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

      6. *-commutative 87.5%

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

      7. associate-*r* 87.5%

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

      8. distribute-lft-neg-in 87.5%

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

      9. sin-neg 87.5%

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

      10. distribute-rgt-neg-out 87.5%

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

      11. remove-double-neg 87.5%

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

   5. Simplified 87.5%

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

   6. Taylor expanded in l around inf 45.2%

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

      1. mul-1-neg 45.2%

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

   8. Simplified 45.2%

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

   9. Taylor expanded in n around inf 70.4%

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

       1. associate-*r* 70.4%

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

   11. Simplified 70.4%

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

   if 2.39999999999999991e-68 < M < 27

   1. Initial program 87.5%

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

   3. Taylor expanded in K around 0 87.5%

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

      1. cos-neg 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*l* 87.5%

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

      4. metadata-eval 87.5%

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

      5. distribute-rgt-neg-in 87.5%

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

      6. *-commutative 87.5%

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

      7. associate-*r* 87.5%

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

      8. distribute-lft-neg-in 87.5%

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

      9. sin-neg 87.5%

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

      10. distribute-rgt-neg-out 87.5%

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

      11. remove-double-neg 87.5%

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

   5. Simplified 87.5%

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

   6. Taylor expanded in l around inf 57.3%

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

      1. mul-1-neg 57.3%

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

   8. Simplified 57.3%

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

   9. Taylor expanded in m around inf 57.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(e^{-\ell} \cdot \sin M\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -28.5:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;M \leq 1.5 \cdot 10^{-304}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;M \leq 3.5 \cdot 10^{-227}:\\ \;\;\;\;\left(\left(0.5 \cdot K\right) \cdot \left(n \cdot \sin M\right)\right) \cdot e^{-\ell}\\ \mathbf{elif}\;M \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot e^{-\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M - n \cdot 0.5\\ \mathbf{if}\;m \leq -29.5:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - t\_0\right) \cdot t\_0 + \left(\left|n - m\right| - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* n 0.5))))
   (if (<= m -29.5)
     (* (cos M) (exp (* -0.25 (pow m 2.0))))
     (* (cos M) (exp (+ (* (- m t_0) t_0) (- (fabs (- n m)) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (n * 0.5);
	double tmp;
	if (m <= -29.5) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp((((m - t_0) * t_0) + (fabs((n - m)) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_1 - (n * 0.5d0)
    if (m <= (-29.5d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp((((m - t_0) * t_0) + (abs((n - m)) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (n * 0.5);
	double tmp;
	if (m <= -29.5) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp((((m - t_0) * t_0) + (Math.abs((n - m)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M - (n * 0.5)
	tmp = 0
	if m <= -29.5:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp((((m - t_0) * t_0) + (math.fabs((n - m)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M - Float64(n * 0.5))
	tmp = 0.0
	if (m <= -29.5)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(m - t_0) * t_0) + Float64(abs(Float64(n - m)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = M - (n * 0.5);
	tmp = 0.0;
	if (m <= -29.5)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp((((m - t_0) * t_0) + (abs((n - m)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -29.5], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(m - t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -29.5

   1. Initial program 70.0%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around inf 98.7%

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

   if -29.5 < m

   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 58.1%

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

      1. +-commutative 58.1%

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

      2. unpow2 58.1%

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

      3. distribute-rgt-out 64.1%

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

      4. *-commutative 64.1%

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

      5. *-commutative 64.1%

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

   5. Simplified 64.1%

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

   6. Taylor expanded in K around 0 79.6%

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

      1. cos-neg 96.6%

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

   8. Simplified 79.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -29.5:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - \left(M - n \cdot 0.5\right)\right) \cdot \left(M - n \cdot 0.5\right) + \left(\left|n - m\right| - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M - m \cdot 0.5\\ \mathbf{if}\;n \leq 19:\\ \;\;\;\;\cos M \cdot e^{\left(n - t\_0\right) \cdot t\_0 + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* m 0.5))))
   (if (<= n 19.0)
     (* (cos M) (exp (+ (* (- n t_0) t_0) (- (fabs (- n m)) l))))
     (* (cos M) (exp (* -0.25 (pow n 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (m * 0.5);
	double tmp;
	if (n <= 19.0) {
		tmp = cos(M) * exp((((n - t_0) * t_0) + (fabs((n - m)) - l)));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = m_1 - (m * 0.5d0)
    if (n <= 19.0d0) then
        tmp = cos(m_1) * exp((((n - t_0) * t_0) + (abs((n - m)) - l)))
    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 t_0 = M - (m * 0.5);
	double tmp;
	if (n <= 19.0) {
		tmp = Math.cos(M) * Math.exp((((n - t_0) * t_0) + (Math.abs((n - m)) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M - (m * 0.5)
	tmp = 0
	if n <= 19.0:
		tmp = math.cos(M) * math.exp((((n - t_0) * t_0) + (math.fabs((n - m)) - l)))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M - Float64(m * 0.5))
	tmp = 0.0
	if (n <= 19.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(n - t_0) * t_0) + Float64(abs(Float64(n - m)) - l))));
	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)
	t_0 = M - (m * 0.5);
	tmp = 0.0;
	if (n <= 19.0)
		tmp = cos(M) * exp((((n - t_0) * t_0) + (abs((n - m)) - l)));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 19.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(n - t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $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 2 regimes

2. if n < 19

   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 n around 0 64.1%

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

      1. +-commutative 64.1%

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

      2. unpow2 64.1%

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

      3. distribute-rgt-out 66.3%

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

      4. *-commutative 66.3%

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

      5. *-commutative 66.3%

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

   5. Simplified 66.3%

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

   6. Taylor expanded in K around 0 84.5%

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

      1. cos-neg 97.0%

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

   8. Simplified 84.5%

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

   if 19 < n

   1. Initial program 73.3%

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

   3. Taylor expanded in K around 0 98.7%

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

      1. cos-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in n around inf 97.4%

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

4. Final simplification 88.3%

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

Alternative 6: 68.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 8.2 \cdot 10^{-308}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 19:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 8.2e-308)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= n 19.0)
     (* (cos M) (exp (+ (* M (- m M)) (- (fabs (- n m)) l))))
     (* (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 <= 8.2e-308) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (n <= 19.0) {
		tmp = cos(M) * exp(((M * (m - M)) + (fabs((n - m)) - l)));
	} 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 <= 8.2d-308) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (n <= 19.0d0) then
        tmp = cos(m_1) * exp(((m_1 * (m - m_1)) + (abs((n - m)) - l)))
    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 <= 8.2e-308) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (n <= 19.0) {
		tmp = Math.cos(M) * Math.exp(((M * (m - M)) + (Math.abs((n - m)) - l)));
	} 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 <= 8.2e-308:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif n <= 19.0:
		tmp = math.cos(M) * math.exp(((M * (m - M)) + (math.fabs((n - m)) - l)))
	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 <= 8.2e-308)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (n <= 19.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(m - M)) + Float64(abs(Float64(n - m)) - l))));
	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 <= 8.2e-308)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (n <= 19.0)
		tmp = cos(M) * exp(((M * (m - M)) + (abs((n - m)) - l)));
	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, 8.2e-308], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 19.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $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 < 8.19999999999999965e-308

   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 97.1%

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

      1. cos-neg 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in m around inf 63.3%

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

   if 8.19999999999999965e-308 < n < 19

   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. Add Preprocessing

   3. Taylor expanded in m around 0 57.1%

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

      1. +-commutative 57.1%

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

      2. unpow2 57.1%

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

      3. distribute-rgt-out 57.1%

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

      4. *-commutative 57.1%

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

      5. *-commutative 57.1%

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

   5. Simplified 57.1%

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

   6. Taylor expanded in K around 0 68.1%

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

      1. cos-neg 96.7%

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

   8. Simplified 68.1%

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

   9. Taylor expanded in n around 0 68.1%

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

       1. associate--r+ 68.1%

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

       2. associate-*r* 68.1%

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

       3. neg-mul-1 68.1%

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

       4. cancel-sign-sub 68.1%

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

   11. Simplified 68.1%

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

   if 19 < n

   1. Initial program 73.3%

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

   3. Taylor expanded in K around 0 98.7%

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

      1. cos-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in n around inf 97.4%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 8.2 \cdot 10^{-308}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 19:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 19.0)
   (* (cos M) (exp (- (- (fabs (- n m)) l) (* (* m 0.5) (+ n (* m 0.5))))))
   (* (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 <= 19.0) {
		tmp = cos(M) * exp(((fabs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	} 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 <= 19.0d0) then
        tmp = cos(m_1) * exp(((abs((n - m)) - l) - ((m * 0.5d0) * (n + (m * 0.5d0)))))
    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 <= 19.0) {
		tmp = Math.cos(M) * Math.exp(((Math.abs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	} 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 <= 19.0:
		tmp = math.cos(M) * math.exp(((math.fabs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))))
	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 <= 19.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(Float64(m * 0.5) * Float64(n + Float64(m * 0.5))))));
	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 <= 19.0)
		tmp = cos(M) * exp(((abs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	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, 19.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[(m * 0.5), $MachinePrecision] * N[(n + N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 2 regimes

2. if n < 19

   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 n around 0 64.1%

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

      1. +-commutative 64.1%

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

      2. unpow2 64.1%

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

      3. distribute-rgt-out 66.3%

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

      4. *-commutative 66.3%

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

      5. *-commutative 66.3%

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

   5. Simplified 66.3%

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

   6. Taylor expanded in K around 0 84.5%

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

      1. cos-neg 97.0%

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

   8. Simplified 84.5%

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

   9. Taylor expanded in M around 0 67.2%

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

       1. associate--r+ 67.2%

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

       2. associate-*r* 67.2%

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

       3. *-commutative 67.2%

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

       4. *-commutative 67.2%

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

   11. Simplified 67.2%

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

   if 19 < n

   1. Initial program 73.3%

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

   3. Taylor expanded in K around 0 98.7%

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

      1. cos-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in n around inf 97.4%

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

4. Final simplification 76.0%

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

Alternative 8: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -29.5:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -2.55 \cdot 10^{-306}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -29.5)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= m -2.55e-306)
     (* (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 (m <= -29.5) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (m <= -2.55e-306) {
		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 (m <= (-29.5d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= (-2.55d-306)) 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 (m <= -29.5) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= -2.55e-306) {
		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 m <= -29.5:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= -2.55e-306:
		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 (m <= -29.5)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (m <= -2.55e-306)
		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 (m <= -29.5)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (m <= -2.55e-306)
		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[m, -29.5], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -2.55e-306], 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 m < -29.5

   1. Initial program 70.0%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around inf 98.7%

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

   if -29.5 < m < -2.54999999999999986e-306

   1. Initial program 75.5%

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

   3. Taylor expanded in K around 0 92.9%

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

      1. cos-neg 92.9%

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

   5. Simplified 92.9%

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

   6. Taylor expanded in M around inf 64.2%

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

      1. mul-1-neg 64.2%

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

   8. Simplified 64.2%

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

   if -2.54999999999999986e-306 < m

   1. Initial program 76.6%

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

   3. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in n around inf 58.6%

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

Alternative 9: 53.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-242}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 3.65:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot e^{-\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 1.1e-242)
   (* (cos M) (exp (* m (- M (* n 0.5)))))
   (if (<= l 3.65)
     (* (cos M) (exp (* n (- M (* m 0.5)))))
     (* 0.5 (* K (* m (* (sin M) (exp (- l)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1.1e-242) {
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	} else if (l <= 3.65) {
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	} else {
		tmp = 0.5 * (K * (m * (sin(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 <= 1.1d-242) then
        tmp = cos(m_1) * exp((m * (m_1 - (n * 0.5d0))))
    else if (l <= 3.65d0) then
        tmp = cos(m_1) * exp((n * (m_1 - (m * 0.5d0))))
    else
        tmp = 0.5d0 * (k * (m * (sin(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 <= 1.1e-242) {
		tmp = Math.cos(M) * Math.exp((m * (M - (n * 0.5))));
	} else if (l <= 3.65) {
		tmp = Math.cos(M) * Math.exp((n * (M - (m * 0.5))));
	} else {
		tmp = 0.5 * (K * (m * (Math.sin(M) * Math.exp(-l))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 1.1e-242:
		tmp = math.cos(M) * math.exp((m * (M - (n * 0.5))))
	elif l <= 3.65:
		tmp = math.cos(M) * math.exp((n * (M - (m * 0.5))))
	else:
		tmp = 0.5 * (K * (m * (math.sin(M) * math.exp(-l))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 1.1e-242)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	elseif (l <= 3.65)
		tmp = Float64(cos(M) * exp(Float64(n * Float64(M - Float64(m * 0.5)))));
	else
		tmp = Float64(0.5 * Float64(K * Float64(m * Float64(sin(M) * exp(Float64(-l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 1.1e-242)
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	elseif (l <= 3.65)
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	else
		tmp = 0.5 * (K * (m * (sin(M) * exp(-l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 1.1e-242], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.65], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(K * N[(m * N[(N[Sin[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.10000000000000001e-242

   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. Add Preprocessing

   3. Taylor expanded in m around 0 55.2%

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

      1. +-commutative 55.2%

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

      2. unpow2 55.2%

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

      3. distribute-rgt-out 59.5%

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

      4. *-commutative 59.5%

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

      5. *-commutative 59.5%

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

   5. Simplified 59.5%

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

   6. Taylor expanded in K around 0 71.1%

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

      1. cos-neg 95.4%

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

   8. Simplified 71.1%

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

   9. Taylor expanded in m around inf 41.0%

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

   if 1.10000000000000001e-242 < l < 3.64999999999999991

   1. Initial program 69.3%

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

   3. Taylor expanded in n around 0 47.1%

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

      1. +-commutative 47.1%

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

      2. unpow2 47.1%

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

      3. distribute-rgt-out 53.3%

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

      4. *-commutative 53.3%

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

      5. *-commutative 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in K around 0 75.8%

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

      1. cos-neg 99.9%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in n around inf 29.2%

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

   if 3.64999999999999991 < l

   1. Initial program 73.9%

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

   3. Taylor expanded in K around 0 81.2%

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

      1. cos-neg 81.2%

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

      2. *-commutative 81.2%

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

      3. associate-*l* 81.2%

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

      4. metadata-eval 81.2%

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

      5. distribute-rgt-neg-in 81.2%

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

      6. *-commutative 81.2%

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

      7. associate-*r* 81.2%

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

      8. distribute-lft-neg-in 81.2%

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

      9. sin-neg 81.2%

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

      10. distribute-rgt-neg-out 81.2%

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

      11. remove-double-neg 81.2%

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

   5. Simplified 81.2%

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

   6. Taylor expanded in l around inf 79.8%

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

      1. mul-1-neg 79.8%

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

   8. Simplified 79.8%

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

   9. Taylor expanded in m around inf 98.6%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-242}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 3.65:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot e^{-\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-242}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 3.7:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 1.9e-242)
   (* (cos M) (exp (* m (- M (* n 0.5)))))
   (if (<= l 3.7) (* (cos M) (exp (* n (- M (* m 0.5))))) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1.9e-242) {
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	} else if (l <= 3.7) {
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	} else {
		tmp = 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 <= 1.9d-242) then
        tmp = cos(m_1) * exp((m * (m_1 - (n * 0.5d0))))
    else if (l <= 3.7d0) then
        tmp = cos(m_1) * exp((n * (m_1 - (m * 0.5d0))))
    else
        tmp = 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 <= 1.9e-242) {
		tmp = Math.cos(M) * Math.exp((m * (M - (n * 0.5))));
	} else if (l <= 3.7) {
		tmp = Math.cos(M) * Math.exp((n * (M - (m * 0.5))));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 1.9e-242:
		tmp = math.cos(M) * math.exp((m * (M - (n * 0.5))))
	elif l <= 3.7:
		tmp = math.cos(M) * math.exp((n * (M - (m * 0.5))))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 1.9e-242)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	elseif (l <= 3.7)
		tmp = Float64(cos(M) * exp(Float64(n * Float64(M - Float64(m * 0.5)))));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 1.9e-242)
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	elseif (l <= 3.7)
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 1.9e-242], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.7], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.9000000000000001e-242

   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. Add Preprocessing

   3. Taylor expanded in m around 0 55.2%

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

      1. +-commutative 55.2%

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

      2. unpow2 55.2%

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

      3. distribute-rgt-out 59.5%

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

      4. *-commutative 59.5%

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

      5. *-commutative 59.5%

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

   5. Simplified 59.5%

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

   6. Taylor expanded in K around 0 71.1%

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

      1. cos-neg 95.4%

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

   8. Simplified 71.1%

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

   9. Taylor expanded in m around inf 41.0%

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

   if 1.9000000000000001e-242 < l < 3.7000000000000002

   1. Initial program 69.3%

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

   3. Taylor expanded in n around 0 47.1%

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

      1. +-commutative 47.1%

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

      2. unpow2 47.1%

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

      3. distribute-rgt-out 53.3%

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

      4. *-commutative 53.3%

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

      5. *-commutative 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in K around 0 75.8%

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

      1. cos-neg 99.9%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in n around inf 29.2%

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

   if 3.7000000000000002 < l

   1. Initial program 73.9%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   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. mul-1-neg 79.8%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in M around 0 98.6%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-242}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 3.7:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 1.6e-9)
   (* (cos M) (exp (* m (- M (* n 0.5)))))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1.6e-9) {
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	} 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 <= 1.6d-9) then
        tmp = cos(m_1) * exp((m * (m_1 - (n * 0.5d0))))
    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 <= 1.6e-9) {
		tmp = Math.cos(M) * Math.exp((m * (M - (n * 0.5))));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 1.6e-9:
		tmp = math.cos(M) * math.exp((m * (M - (n * 0.5))))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 1.6e-9)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	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 <= 1.6e-9)
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 1.6e-9], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $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 < 1.60000000000000006e-9

   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. Add Preprocessing

   3. Taylor expanded in m around 0 52.3%

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

      1. +-commutative 52.3%

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

      2. unpow2 52.3%

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

      3. distribute-rgt-out 57.8%

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

      4. *-commutative 57.8%

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

      5. *-commutative 57.8%

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

   5. Simplified 57.8%

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

   6. Taylor expanded in K around 0 73.0%

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

      1. cos-neg 96.5%

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

   8. Simplified 73.0%

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

   9. Taylor expanded in m around inf 40.9%

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

   if 1.60000000000000006e-9 < l

   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 K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 96.0%

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

      1. mul-1-neg 77.9%

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

   8. Simplified 96.0%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 36.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

3. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

5. Simplified 97.5%

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

6. Taylor expanded in l around inf 33.5%

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

   1. mul-1-neg 28.0%

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

8. Simplified 33.5%

   \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]
9. Add Preprocessing

Alternative 13: 35.7% 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 74.5%

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

3. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

5. Simplified 97.5%

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

6. Taylor expanded in l around inf 33.5%

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

   1. mul-1-neg 28.0%

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

8. Simplified 33.5%

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

9. Taylor expanded in M around 0 33.5%

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

Alternative 14: 7.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

3. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

5. Simplified 97.5%

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

6. Taylor expanded in l around inf 33.5%

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

   1. mul-1-neg 28.0%

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

8. Simplified 33.5%

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

9. Taylor expanded in l around 0 5.8%

   \[\leadsto \color{blue}{\cos M} \]
10. Add Preprocessing

Alternative 15: 6.4% accurate, 38.6× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot \left(\left(m + n\right) \cdot \left(M \cdot K\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (+ 1.0 (* 0.5 (* (+ m n) (* M K)))))

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 + (0.5 * ((m + n) * (M * K)));
}

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 + (0.5d0 * ((m + n) * (m_1 * k)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 + (0.5 * ((m + n) * (M * K)));
}

Python

def code(K, m, n, M, l):
	return 1.0 + (0.5 * ((m + n) * (M * K)))

Julia

function code(K, m, n, M, l)
	return Float64(1.0 + Float64(0.5 * Float64(Float64(m + n) * Float64(M * K))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 + (0.5 * ((m + n) * (M * K)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 + N[(0.5 * N[(N[(m + n), $MachinePrecision] * N[(M * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.5%

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

   1. *-un-lft-identity 74.5%

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

   2. *-commutative 74.5%

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

4. Applied egg-rr 13.6%

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

5. Taylor expanded in M around inf 4.6%

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

6. Taylor expanded in M around 0 4.9%

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

   1. *-commutative 4.9%

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

   2. *-commutative 4.9%

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

8. Simplified 4.9%

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

9. Taylor expanded in K around 0 5.0%

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

    1. associate-*r* 5.4%

       \[\leadsto 1 + 0.5 \cdot \color{blue}{\left(\left(K \cdot M\right) \cdot \left(m + n\right)\right)} \]

11. Simplified 5.4%

    \[\leadsto \color{blue}{1 + 0.5 \cdot \left(\left(K \cdot M\right) \cdot \left(m + n\right)\right)} \]

12. Final simplification 5.4%

    \[\leadsto 1 + 0.5 \cdot \left(\left(m + n\right) \cdot \left(M \cdot K\right)\right) \]
13. Add Preprocessing

Alternative 16: 6.3% accurate, 38.6× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot \left(K \cdot \left(M \cdot \left(m + n\right)\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (+ 1.0 (* 0.5 (* K (* M (+ m n))))))

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 + (0.5 * (K * (M * (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 = 1.0d0 + (0.5d0 * (k * (m_1 * (m + n))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 + (0.5 * (K * (M * (m + n))));
}

Python

def code(K, m, n, M, l):
	return 1.0 + (0.5 * (K * (M * (m + n))))

Julia

function code(K, m, n, M, l)
	return Float64(1.0 + Float64(0.5 * Float64(K * Float64(M * Float64(m + n)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 + (0.5 * (K * (M * (m + n))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 + N[(0.5 * N[(K * N[(M * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.5%

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

   1. *-un-lft-identity 74.5%

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

   2. *-commutative 74.5%

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

4. Applied egg-rr 13.6%

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

5. Taylor expanded in M around inf 4.6%

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

6. Taylor expanded in M around 0 4.9%

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

   1. *-commutative 4.9%

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

   2. *-commutative 4.9%

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

8. Simplified 4.9%

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

9. Taylor expanded in K around 0 5.0%

   \[\leadsto \color{blue}{1 + 0.5 \cdot \left(K \cdot \left(M \cdot \left(m + n\right)\right)\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024088 
(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)))))))