Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.9% → 96.5%

Time: 21.2s

Alternatives: 10

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: 76.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in K around 0 95.6%

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

   1. cos-neg 95.6%

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

5. Simplified 95.6%

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

6. Final simplification 95.6%

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

Alternative 2: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{M \cdot \left(m - M\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{if}\;m \leq -18000:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -2.3 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -6 \cdot 10^{-253}:\\ \;\;\;\;\cos \left(0.5 \cdot \left(m \cdot K\right)\right) \cdot e^{\left(m - n\right) - \ell}\\ \mathbf{elif}\;m \leq 1.26 \cdot 10^{-247}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (+ (* M (- m M)) (- (fabs (- m n)) l))))))
   (if (<= m -18000.0)
     (* (cos M) (exp (* (pow m 2.0) -0.25)))
     (if (<= m -2.3e-189)
       t_0
       (if (<= m -6e-253)
         (* (cos (* 0.5 (* m K))) (exp (- (- m n) l)))
         (if (<= m 1.26e-247) t_0 (* (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 = cos(M) * exp(((M * (m - M)) + (fabs((m - n)) - l)));
	double tmp;
	if (m <= -18000.0) {
		tmp = cos(M) * exp((pow(m, 2.0) * -0.25));
	} else if (m <= -2.3e-189) {
		tmp = t_0;
	} else if (m <= -6e-253) {
		tmp = cos((0.5 * (m * K))) * exp(((m - n) - l));
	} else if (m <= 1.26e-247) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = cos(m_1) * exp(((m_1 * (m - m_1)) + (abs((m - n)) - l)))
    if (m <= (-18000.0d0)) then
        tmp = cos(m_1) * exp(((m ** 2.0d0) * (-0.25d0)))
    else if (m <= (-2.3d-189)) then
        tmp = t_0
    else if (m <= (-6d-253)) then
        tmp = cos((0.5d0 * (m * k))) * exp(((m - n) - l))
    else if (m <= 1.26d-247) then
        tmp = t_0
    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 = Math.cos(M) * Math.exp(((M * (m - M)) + (Math.abs((m - n)) - l)));
	double tmp;
	if (m <= -18000.0) {
		tmp = Math.cos(M) * Math.exp((Math.pow(m, 2.0) * -0.25));
	} else if (m <= -2.3e-189) {
		tmp = t_0;
	} else if (m <= -6e-253) {
		tmp = Math.cos((0.5 * (m * K))) * Math.exp(((m - n) - l));
	} else if (m <= 1.26e-247) {
		tmp = t_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):
	t_0 = math.cos(M) * math.exp(((M * (m - M)) + (math.fabs((m - n)) - l)))
	tmp = 0
	if m <= -18000.0:
		tmp = math.cos(M) * math.exp((math.pow(m, 2.0) * -0.25))
	elif m <= -2.3e-189:
		tmp = t_0
	elif m <= -6e-253:
		tmp = math.cos((0.5 * (m * K))) * math.exp(((m - n) - l))
	elif m <= 1.26e-247:
		tmp = t_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)
	t_0 = Float64(cos(M) * exp(Float64(Float64(M * Float64(m - M)) + Float64(abs(Float64(m - n)) - l))))
	tmp = 0.0
	if (m <= -18000.0)
		tmp = Float64(cos(M) * exp(Float64((m ^ 2.0) * -0.25)));
	elseif (m <= -2.3e-189)
		tmp = t_0;
	elseif (m <= -6e-253)
		tmp = Float64(cos(Float64(0.5 * Float64(m * K))) * exp(Float64(Float64(m - n) - l)));
	elseif (m <= 1.26e-247)
		tmp = t_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)
	t_0 = cos(M) * exp(((M * (m - M)) + (abs((m - n)) - l)));
	tmp = 0.0;
	if (m <= -18000.0)
		tmp = cos(M) * exp(((m ^ 2.0) * -0.25));
	elseif (m <= -2.3e-189)
		tmp = t_0;
	elseif (m <= -6e-253)
		tmp = cos((0.5 * (m * K))) * exp(((m - n) - l));
	elseif (m <= 1.26e-247)
		tmp = t_0;
	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[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -18000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -2.3e-189], t$95$0, If[LessEqual[m, -6e-253], N[(N[Cos[N[(0.5 * N[(m * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.26e-247], t$95$0, N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -18000

   1. Initial program 73.2%

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

   3. Taylor expanded in K around 0 98.2%

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

      1. cos-neg 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in m around inf 96.5%

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

      1. *-commutative 96.5%

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

   8. Simplified 96.5%

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

   if -18000 < m < -2.2999999999999998e-189 or -6.0000000000000004e-253 < m < 1.25999999999999999e-247

   1. Initial program 87.4%

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

   3. Taylor expanded in K around 0 94.8%

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

      1. cos-neg 94.8%

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

   5. Simplified 94.8%

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

   6. Taylor expanded in m around 0 94.8%

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

      1. +-commutative 94.8%

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

      2. unpow2 94.8%

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

      3. distribute-rgt-out 94.8%

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

      4. *-commutative 94.8%

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

      5. *-commutative 94.8%

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

   8. Simplified 94.8%

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

   9. Taylor expanded in n around 0 78.4%

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

       1. associate--r+ 78.4%

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

       2. associate-*r* 78.4%

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

       3. neg-mul-1 78.4%

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

       4. cancel-sign-sub 78.4%

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

       5. fabs-sub 78.4%

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

   11. Simplified 78.4%

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

   if -2.2999999999999998e-189 < m < -6.0000000000000004e-253

   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. Step-by-step derivation

      1. associate--r- 90.9%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 19.3%

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

      4. sqr-neg 19.3%

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

      5. sqrt-unprod 19.3%

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

      6. add-sqr-sqrt 19.3%

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

      7. div-inv 19.3%

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

      8. metadata-eval 19.3%

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

      9. add-sqr-sqrt 0.3%

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

      10. fabs-sqr 0.3%

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

      11. add-sqr-sqrt 19.3%

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

   4. Applied egg-rr 19.3%

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

   5. Taylor expanded in m around inf 19.5%

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

      1. *-commutative 19.5%

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

   7. Simplified 19.5%

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

   8. Taylor expanded in l around inf 91.1%

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

      1. neg-mul-1 91.1%

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

   10. Simplified 91.1%

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

   if 1.25999999999999999e-247 < 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 94.4%

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

      1. cos-neg 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in n around inf 52.9%

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

      1. *-commutative 52.9%

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

   8. Simplified 52.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -18000:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -2.3 \cdot 10^{-189}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{elif}\;m \leq -6 \cdot 10^{-253}:\\ \;\;\;\;\cos \left(0.5 \cdot \left(m \cdot K\right)\right) \cdot e^{\left(m - n\right) - \ell}\\ \mathbf{elif}\;m \leq 1.26 \cdot 10^{-247}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in K around 0 95.6%

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

   1. cos-neg 95.6%

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

5. Simplified 95.6%

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

   1. sub-neg 95.6%

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

   2. distribute-neg-out 95.6%

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

   3. div-inv 95.6%

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

   4. metadata-eval 95.6%

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

   5. add-sqr-sqrt 51.4%

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

   6. fabs-sqr 51.4%

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

   7. add-sqr-sqrt 95.6%

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

7. Applied egg-rr 95.6%

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

   1. +-commutative 95.6%

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

   2. distribute-neg-in 95.6%

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

   3. sub-neg 95.6%

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

   4. neg-sub0 95.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{\left(0 - \left(\ell - \left(m - n\right)\right)\right)} - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}} \]

   5. associate--r- 95.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left(0 - \ell\right) + \left(m - n\right)\right)} - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}} \]

   6. neg-sub0 95.6%

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

9. Simplified 95.6%

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

10. Final simplification 95.6%

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

Alternative 4: 74.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -16 \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \left(m \cdot K\right)\right) \cdot e^{\left(m - n\right) - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -16.0) (not (<= M 27.0)))
   (* (cos M) (exp (- (pow M 2.0))))
   (* (cos (* 0.5 (* m K))) (exp (- (- m n) l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -16.0) || !(M <= 27.0)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos((0.5 * (m * K))) * exp(((m - n) - l));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-16.0d0)) .or. (.not. (m_1 <= 27.0d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos((0.5d0 * (m * k))) * exp(((m - n) - l))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -16.0) || !(M <= 27.0)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos((0.5 * (m * K))) * Math.exp(((m - n) - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -16.0) or not (M <= 27.0):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos((0.5 * (m * K))) * math.exp(((m - n) - l))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -16.0) || !(M <= 27.0))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(Float64(0.5 * Float64(m * K))) * exp(Float64(Float64(m - n) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -16.0) || ~((M <= 27.0)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos((0.5 * (m * K))) * exp(((m - n) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -16.0], N[Not[LessEqual[M, 27.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(0.5 * N[(m * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -16 or 27 < M

   1. Initial program 78.2%

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

   3. Taylor expanded in 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 98.4%

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

      1. mul-1-neg 98.4%

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

   8. Simplified 98.4%

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

   if -16 < M < 27

   1. Initial program 77.7%

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

      1. associate--r- 77.7%

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

      2. add-sqr-sqrt 1.5%

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

      3. sqrt-unprod 28.9%

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

      4. sqr-neg 28.9%

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

      5. sqrt-unprod 28.9%

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

      6. add-sqr-sqrt 28.9%

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

      7. div-inv 28.9%

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

      8. metadata-eval 28.9%

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

      9. add-sqr-sqrt 8.1%

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

      10. fabs-sqr 8.1%

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

      11. add-sqr-sqrt 28.9%

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

   4. Applied egg-rr 28.9%

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

   5. Taylor expanded in m around inf 28.1%

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

      1. *-commutative 28.1%

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

   7. Simplified 28.1%

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

   8. Taylor expanded in l around inf 56.1%

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

      1. neg-mul-1 56.1%

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

   10. Simplified 56.1%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -16 \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \left(m \cdot K\right)\right) \cdot e^{\left(m - n\right) - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \cdot 10^{+102}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 200:\\ \;\;\;\;\cos \left(0.5 \cdot \left(m \cdot K\right)\right) \cdot e^{\left(m - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -4.1e+102)
   (* (cos M) (exp (* m (- M (* n 0.5)))))
   (if (<= l 200.0)
     (* (cos (* 0.5 (* m K))) (exp (- (- m n) l)))
     (* (cos M) (exp (- l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -4.1e+102) {
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	} else if (l <= 200.0) {
		tmp = cos((0.5 * (m * K))) * exp(((m - n) - l));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-4.1d+102)) then
        tmp = cos(m_1) * exp((m * (m_1 - (n * 0.5d0))))
    else if (l <= 200.0d0) then
        tmp = cos((0.5d0 * (m * k))) * exp(((m - n) - l))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -4.1e+102) {
		tmp = Math.cos(M) * Math.exp((m * (M - (n * 0.5))));
	} else if (l <= 200.0) {
		tmp = Math.cos((0.5 * (m * K))) * Math.exp(((m - n) - l));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -4.1e+102:
		tmp = math.cos(M) * math.exp((m * (M - (n * 0.5))))
	elif l <= 200.0:
		tmp = math.cos((0.5 * (m * K))) * math.exp(((m - n) - l))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -4.1e+102)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	elseif (l <= 200.0)
		tmp = Float64(cos(Float64(0.5 * Float64(m * K))) * exp(Float64(Float64(m - n) - l)));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -4.1e+102)
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	elseif (l <= 200.0)
		tmp = cos((0.5 * (m * K))) * exp(((m - n) - l));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -4.1e+102], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 200.0], N[(N[Cos[N[(0.5 * N[(m * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.1e102

   1. Initial program 63.4%

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

   3. Taylor expanded in K around 0 92.7%

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

      1. cos-neg 92.7%

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

   5. Simplified 92.7%

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

   6. Taylor expanded in m around 0 73.3%

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

      1. +-commutative 73.3%

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

      2. unpow2 73.3%

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

      3. distribute-rgt-out 78.3%

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

      4. *-commutative 78.3%

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

      5. *-commutative 78.3%

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

   8. Simplified 78.3%

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

   9. Taylor expanded in m around inf 45.1%

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

   if -4.1e102 < l < 200

   1. Initial program 78.6%

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

      1. associate--r- 78.6%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 12.9%

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

      4. sqr-neg 12.9%

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

      5. sqrt-unprod 12.9%

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

      6. add-sqr-sqrt 12.9%

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

      7. div-inv 12.9%

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

      8. metadata-eval 12.9%

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

      9. add-sqr-sqrt 3.1%

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

      10. fabs-sqr 3.1%

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

      11. add-sqr-sqrt 12.9%

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

   4. Applied egg-rr 12.9%

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

   5. Taylor expanded in m around inf 12.2%

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

      1. *-commutative 12.2%

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

   7. Simplified 12.2%

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

   8. Taylor expanded in l around inf 37.8%

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

      1. neg-mul-1 37.8%

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

   10. Simplified 37.8%

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

   if 200 < l

   1. Initial program 85.3%

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

   3. Taylor expanded in l around inf 85.3%

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

      1. mul-1-neg 85.3%

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

   5. Simplified 85.3%

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

   6. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \cdot 10^{+102}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 200:\\ \;\;\;\;\cos \left(0.5 \cdot \left(m \cdot K\right)\right) \cdot e^{\left(m - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 150:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 150.0)
   (* (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 <= 150.0) {
		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 <= 150.0d0) 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 <= 150.0) {
		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 <= 150.0:
		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 <= 150.0)
		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 <= 150.0)
		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, 150.0], 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 < 150

   1. Initial program 75.1%

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

   3. Taylor expanded in K around 0 94.0%

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

      1. cos-neg 94.0%

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

   5. Simplified 94.0%

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

   6. Taylor expanded in m around 0 72.3%

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

      1. +-commutative 72.3%

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

      2. unpow2 72.3%

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

      3. distribute-rgt-out 77.1%

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

      4. *-commutative 77.1%

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

      5. *-commutative 77.1%

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

   8. Simplified 77.1%

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

   9. Taylor expanded in m around inf 42.7%

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

   if 150 < l

   1. Initial program 85.5%

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

   3. Taylor expanded in l around inf 84.1%

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

      1. mul-1-neg 84.1%

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

   5. Simplified 84.1%

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

   6. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

   8. Simplified 98.6%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 150:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.9% 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 77.9%

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

3. Taylor expanded in l around inf 33.0%

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

   1. mul-1-neg 33.0%

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

5. Simplified 33.0%

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

6. Taylor expanded in K around 0 37.8%

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

   1. cos-neg 37.8%

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

8. Simplified 37.8%

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

Alternative 8: 6.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(m \cdot \left(0.5 \cdot K\right) - M\right) \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(((m * (0.5 * K)) - 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 * (0.5d0 * k)) - m_1))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in l around inf 33.0%

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

   1. mul-1-neg 33.0%

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

5. Simplified 33.0%

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

6. Taylor expanded in l around 0 7.9%

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

7. Taylor expanded in n around 0 8.0%

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

   1. associate-*r* 8.0%

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

   2. *-commutative 8.0%

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

9. Simplified 8.0%

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

10. Final simplification 8.0%

    \[\leadsto \cos \left(m \cdot \left(0.5 \cdot K\right) - M\right) \]
11. Add Preprocessing

Alternative 9: 6.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(m \cdot \left(0.5 \cdot K\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in l around inf 33.0%

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

   1. mul-1-neg 33.0%

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

5. Simplified 33.0%

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

6. Taylor expanded in l around 0 7.9%

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

7. Taylor expanded in m around inf 8.0%

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

   1. associate-*r* 8.0%

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

   2. *-commutative 8.0%

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

9. Simplified 8.0%

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

10. Final simplification 8.0%

    \[\leadsto \cos \left(m \cdot \left(0.5 \cdot K\right)\right) \]
11. Add Preprocessing

Alternative 10: 7.3% 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 77.9%

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

3. Taylor expanded in l around inf 33.0%

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

   1. mul-1-neg 33.0%

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

5. Simplified 33.0%

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

6. Taylor expanded in l around 0 7.9%

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

7. Taylor expanded in K around 0 8.1%

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

   1. cos-neg 8.1%

      \[\leadsto \color{blue}{\cos M} \]

9. Simplified 8.1%

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

Reproduce

herbie shell --seed 2024092 
(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)))))))