Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.6% → 96.9%

Time: 16.8s

Alternatives: 13

Speedup: 3.7×

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: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in K around 0 96.4%

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

   1. cos-neg 96.4%

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

5. Simplified 96.4%

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

6. Final simplification 96.4%

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

Alternative 2: 87.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 72:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 72.0)
   (*
    (cos M)
    (exp (+ (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) (- (fabs (- m n)) 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 <= 72.0) {
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (fabs((m - n)) - 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 <= 72.0d0) then
        tmp = cos(m_1) * exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) + (abs((m - n)) - 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 <= 72.0) {
		tmp = Math.cos(M) * Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (Math.abs((m - n)) - 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 <= 72.0:
		tmp = math.cos(M) * math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (math.fabs((m - n)) - 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 <= 72.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) + Float64(abs(Float64(m - n)) - 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 <= 72.0)
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (abs((m - n)) - 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, 72.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(m - n), $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 < 72

   1. Initial program 78.8%

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

   3. Taylor expanded in K around 0 96.3%

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

      1. cos-neg 96.3%

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

   5. Simplified 96.3%

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

   6. Taylor expanded in n around 0 82.0%

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

      1. +-commutative 82.0%

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

      2. unpow2 82.0%

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

      3. distribute-rgt-out 84.6%

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

      4. *-commutative 84.6%

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

      5. *-commutative 84.6%

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

   8. Simplified 84.6%

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

   if 72 < n

   1. Initial program 68.3%

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

   3. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in n around inf 95.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 72:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3.8 \cdot 10^{+50}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(M - m \cdot 0.5\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 3.8e+50)
   (* (cos M) (exp (- (* (- (* m 0.5) M) (- M (* m 0.5))) 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 <= 3.8e+50) {
		tmp = cos(M) * exp(((((m * 0.5) - M) * (M - (m * 0.5))) - 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 <= 3.8d+50) then
        tmp = cos(m_1) * exp(((((m * 0.5d0) - m_1) * (m_1 - (m * 0.5d0))) - 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 <= 3.8e+50) {
		tmp = Math.cos(M) * Math.exp(((((m * 0.5) - M) * (M - (m * 0.5))) - 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 <= 3.8e+50:
		tmp = math.cos(M) * math.exp(((((m * 0.5) - M) * (M - (m * 0.5))) - 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 <= 3.8e+50)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(M - Float64(m * 0.5))) - 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 <= 3.8e+50)
		tmp = cos(M) * exp(((((m * 0.5) - M) * (M - (m * 0.5))) - 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, 3.8e+50], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $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 < 3.79999999999999987e50

   1. Initial program 78.5%

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

   3. Taylor expanded in K around 0 96.0%

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

      1. cos-neg 96.0%

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

   5. Simplified 96.0%

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

   6. Taylor expanded in n around 0 81.0%

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

      1. +-commutative 81.0%

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

      2. unpow2 81.0%

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

      3. distribute-rgt-out 83.9%

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

      4. *-commutative 83.9%

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

      5. *-commutative 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in l around inf 85.7%

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

   10. Taylor expanded in n around 0 87.7%

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

   if 3.79999999999999987e50 < n

   1. Initial program 67.3%

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

   3. Taylor expanded in K around 0 98.1%

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

      1. cos-neg 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in n around inf 98.1%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.8 \cdot 10^{+50}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(M - m \cdot 0.5\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M - m \cdot 0.5\\ t_1 := m \cdot 0.5 - M\\ \mathbf{if}\;M \leq -1.55 \cdot 10^{-222}:\\ \;\;\;\;e^{t\_1 \cdot \left(t\_0 - n\right) - \ell}\\ \mathbf{elif}\;M \leq 0.00095:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5\right) \cdot t\_0 - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot t\_1 - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* m 0.5))) (t_1 (- (* m 0.5) M)))
   (if (<= M -1.55e-222)
     (exp (- (* t_1 (- t_0 n)) l))
     (if (<= M 0.00095)
       (* (cos M) (exp (- (* (* m 0.5) t_0) l)))
       (* (cos M) (exp (- (* M t_1) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (m * 0.5);
	double t_1 = (m * 0.5) - M;
	double tmp;
	if (M <= -1.55e-222) {
		tmp = exp(((t_1 * (t_0 - n)) - l));
	} else if (M <= 0.00095) {
		tmp = cos(M) * exp((((m * 0.5) * t_0) - l));
	} else {
		tmp = cos(M) * exp(((M * t_1) - 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) :: t_1
    real(8) :: tmp
    t_0 = m_1 - (m * 0.5d0)
    t_1 = (m * 0.5d0) - m_1
    if (m_1 <= (-1.55d-222)) then
        tmp = exp(((t_1 * (t_0 - n)) - l))
    else if (m_1 <= 0.00095d0) then
        tmp = cos(m_1) * exp((((m * 0.5d0) * t_0) - l))
    else
        tmp = cos(m_1) * exp(((m_1 * t_1) - l))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (m * 0.5);
	double t_1 = (m * 0.5) - M;
	double tmp;
	if (M <= -1.55e-222) {
		tmp = Math.exp(((t_1 * (t_0 - n)) - l));
	} else if (M <= 0.00095) {
		tmp = Math.cos(M) * Math.exp((((m * 0.5) * t_0) - l));
	} else {
		tmp = Math.cos(M) * Math.exp(((M * t_1) - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M - (m * 0.5)
	t_1 = (m * 0.5) - M
	tmp = 0
	if M <= -1.55e-222:
		tmp = math.exp(((t_1 * (t_0 - n)) - l))
	elif M <= 0.00095:
		tmp = math.cos(M) * math.exp((((m * 0.5) * t_0) - l))
	else:
		tmp = math.cos(M) * math.exp(((M * t_1) - l))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M - Float64(m * 0.5))
	t_1 = Float64(Float64(m * 0.5) - M)
	tmp = 0.0
	if (M <= -1.55e-222)
		tmp = exp(Float64(Float64(t_1 * Float64(t_0 - n)) - l));
	elseif (M <= 0.00095)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(m * 0.5) * t_0) - l)));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * t_1) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = M - (m * 0.5);
	t_1 = (m * 0.5) - M;
	tmp = 0.0;
	if (M <= -1.55e-222)
		tmp = exp(((t_1 * (t_0 - n)) - l));
	elseif (M <= 0.00095)
		tmp = cos(M) * exp((((m * 0.5) * t_0) - l));
	else
		tmp = cos(M) * exp(((M * t_1) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]}, If[LessEqual[M, -1.55e-222], N[Exp[N[(N[(t$95$1 * N[(t$95$0 - n), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision], If[LessEqual[M, 0.00095], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(m * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * t$95$1), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -1.5499999999999999e-222

   1. Initial program 76.8%

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

   3. Taylor expanded in K around 0 96.7%

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

      1. cos-neg 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in n around 0 73.3%

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

      1. +-commutative 73.3%

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

      2. unpow2 73.3%

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

      3. distribute-rgt-out 77.0%

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

      4. *-commutative 77.0%

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

      5. *-commutative 77.0%

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

   8. Simplified 77.0%

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

   9. Taylor expanded in l around inf 82.1%

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

   10. Taylor expanded in M around 0 82.0%

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

   if -1.5499999999999999e-222 < M < 9.49999999999999998e-4

   1. Initial program 70.8%

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

   3. Taylor expanded in K around 0 92.5%

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

      1. cos-neg 92.5%

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

   5. Simplified 92.5%

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

   6. Taylor expanded in n around 0 62.7%

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

      1. +-commutative 62.7%

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

      2. unpow2 62.7%

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

      3. distribute-rgt-out 65.5%

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

      4. *-commutative 65.5%

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

      5. *-commutative 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in l around inf 71.0%

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

   10. Taylor expanded in m around inf 72.4%

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

   if 9.49999999999999998e-4 < M

   1. Initial program 80.8%

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

   3. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 n around 0 80.9%

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

      1. +-commutative 80.9%

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

      2. unpow2 80.9%

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

      3. distribute-rgt-out 96.0%

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

      4. *-commutative 96.0%

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

      5. *-commutative 96.0%

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

   8. Simplified 96.0%

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

   9. Taylor expanded in l around inf 96.0%

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

   10. Taylor expanded in M around inf 85.2%

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

       1. neg-mul-1 85.2%

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

   12. Simplified 85.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.55 \cdot 10^{-222}:\\ \;\;\;\;e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{elif}\;M \leq 0.00095:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5\right) \cdot \left(M - m \cdot 0.5\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m \cdot 0.5 - M\right) - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M - m \cdot 0.5\\ t_1 := m \cdot 0.5 - M\\ \mathbf{if}\;m \leq -2.6 \cdot 10^{-197}:\\ \;\;\;\;\cos M \cdot e^{t\_1 \cdot t\_0 - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_1 \cdot \left(t\_0 - n\right) - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* m 0.5))) (t_1 (- (* m 0.5) M)))
   (if (<= m -2.6e-197)
     (* (cos M) (exp (- (* t_1 t_0) l)))
     (exp (- (* t_1 (- t_0 n)) l)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (m * 0.5);
	double t_1 = (m * 0.5) - M;
	double tmp;
	if (m <= -2.6e-197) {
		tmp = cos(M) * exp(((t_1 * t_0) - l));
	} else {
		tmp = exp(((t_1 * (t_0 - 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = m_1 - (m * 0.5d0)
    t_1 = (m * 0.5d0) - m_1
    if (m <= (-2.6d-197)) then
        tmp = cos(m_1) * exp(((t_1 * t_0) - l))
    else
        tmp = exp(((t_1 * (t_0 - n)) - 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 - (m * 0.5);
	double t_1 = (m * 0.5) - M;
	double tmp;
	if (m <= -2.6e-197) {
		tmp = Math.cos(M) * Math.exp(((t_1 * t_0) - l));
	} else {
		tmp = Math.exp(((t_1 * (t_0 - n)) - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M - (m * 0.5)
	t_1 = (m * 0.5) - M
	tmp = 0
	if m <= -2.6e-197:
		tmp = math.cos(M) * math.exp(((t_1 * t_0) - l))
	else:
		tmp = math.exp(((t_1 * (t_0 - n)) - l))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M - Float64(m * 0.5))
	t_1 = Float64(Float64(m * 0.5) - M)
	tmp = 0.0
	if (m <= -2.6e-197)
		tmp = Float64(cos(M) * exp(Float64(Float64(t_1 * t_0) - l)));
	else
		tmp = exp(Float64(Float64(t_1 * Float64(t_0 - n)) - l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = M - (m * 0.5);
	t_1 = (m * 0.5) - M;
	tmp = 0.0;
	if (m <= -2.6e-197)
		tmp = cos(M) * exp(((t_1 * t_0) - l));
	else
		tmp = exp(((t_1 * (t_0 - n)) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]}, If[LessEqual[m, -2.6e-197], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(t$95$1 * t$95$0), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(t$95$1 * N[(t$95$0 - n), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.6000000000000001e-197

   1. Initial program 77.6%

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

   3. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in n around 0 72.5%

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

      1. +-commutative 72.5%

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

      2. unpow2 72.5%

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

      3. distribute-rgt-out 79.6%

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

      4. *-commutative 79.6%

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

      5. *-commutative 79.6%

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

   8. Simplified 79.6%

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

   9. Taylor expanded in l around inf 81.0%

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

   10. Taylor expanded in n around 0 84.5%

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

   if -2.6000000000000001e-197 < m

   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 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 n around 0 72.4%

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

      1. +-commutative 72.4%

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

      2. unpow2 72.4%

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

      3. distribute-rgt-out 78.8%

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

      4. *-commutative 78.8%

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

      5. *-commutative 78.8%

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

   8. Simplified 78.8%

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

   9. Taylor expanded in l around inf 84.4%

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

   10. Taylor expanded in M around 0 84.4%

       \[\leadsto \color{blue}{1} \cdot e^{\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)\right) - \ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.6 \cdot 10^{-197}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(M - m \cdot 0.5\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot 0.5 - M\\ \mathbf{if}\;M \leq 1.1 \cdot 10^{-49}:\\ \;\;\;\;e^{t\_0 \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot t\_0 - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* m 0.5) M)))
   (if (<= M 1.1e-49)
     (exp (- (* t_0 (- (- M (* m 0.5)) n)) l))
     (* (cos M) (exp (- (* M t_0) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (m * 0.5) - M;
	double tmp;
	if (M <= 1.1e-49) {
		tmp = exp(((t_0 * ((M - (m * 0.5)) - n)) - l));
	} else {
		tmp = cos(M) * exp(((M * t_0) - l));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m * 0.5d0) - m_1
    if (m_1 <= 1.1d-49) then
        tmp = exp(((t_0 * ((m_1 - (m * 0.5d0)) - n)) - l))
    else
        tmp = cos(m_1) * exp(((m_1 * t_0) - l))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (m * 0.5) - M;
	double tmp;
	if (M <= 1.1e-49) {
		tmp = Math.exp(((t_0 * ((M - (m * 0.5)) - n)) - l));
	} else {
		tmp = Math.cos(M) * Math.exp(((M * t_0) - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (m * 0.5) - M
	tmp = 0
	if M <= 1.1e-49:
		tmp = math.exp(((t_0 * ((M - (m * 0.5)) - n)) - l))
	else:
		tmp = math.cos(M) * math.exp(((M * t_0) - l))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(m * 0.5) - M)
	tmp = 0.0
	if (M <= 1.1e-49)
		tmp = exp(Float64(Float64(t_0 * Float64(Float64(M - Float64(m * 0.5)) - n)) - l));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * t_0) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (m * 0.5) - M;
	tmp = 0.0;
	if (M <= 1.1e-49)
		tmp = exp(((t_0 * ((M - (m * 0.5)) - n)) - l));
	else
		tmp = cos(M) * exp(((M * t_0) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]}, If[LessEqual[M, 1.1e-49], N[Exp[N[(N[(t$95$0 * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * t$95$0), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.09999999999999995e-49

   1. Initial program 76.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 n around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. unpow2 69.2%

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

      3. distribute-rgt-out 72.2%

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

      4. *-commutative 72.2%

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

      5. *-commutative 72.2%

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

   8. Simplified 72.2%

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

   9. Taylor expanded in l around inf 76.7%

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

   10. Taylor expanded in M around 0 76.6%

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

   if 1.09999999999999995e-49 < M

   1. Initial program 76.5%

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

   3. Taylor expanded in K around 0 98.8%

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

      1. cos-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in n around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. unpow2 78.9%

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

      3. distribute-rgt-out 93.1%

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

      4. *-commutative 93.1%

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

      5. *-commutative 93.1%

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

   8. Simplified 93.1%

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

   9. Taylor expanded in l around inf 95.4%

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

   10. Taylor expanded in M around inf 76.9%

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

       1. neg-mul-1 76.9%

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

   12. Simplified 76.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{-49}:\\ \;\;\;\;e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m \cdot 0.5 - M\right) - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 740:\\ \;\;\;\;e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 740.0)
   (exp (- (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) l))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 740.0) {
		tmp = exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - 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 <= 740.0d0) then
        tmp = exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - 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 <= 740.0) {
		tmp = Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 740.0:
		tmp = math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - 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 <= 740.0)
		tmp = exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - 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 <= 740.0)
		tmp = exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 740.0], N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 740

   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 K around 0 95.3%

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

      1. cos-neg 95.3%

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

   5. Simplified 95.3%

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

   6. Taylor expanded in n around 0 68.4%

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

      1. +-commutative 68.4%

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

      2. unpow2 68.4%

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

      3. distribute-rgt-out 75.1%

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

      4. *-commutative 75.1%

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

      5. *-commutative 75.1%

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

   8. Simplified 75.1%

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

   9. Taylor expanded in l around inf 79.0%

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

   10. Taylor expanded in M around 0 78.0%

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

   if 740 < l

   1. Initial program 80.3%

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

   3. Taylor expanded in l around inf 80.3%

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

      1. mul-1-neg 80.3%

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

   5. Simplified 80.3%

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

   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 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 740:\\ \;\;\;\;e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.45 \cdot 10^{-19}:\\ \;\;\;\;e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-{M}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M 1.45e-19)
   (exp (- (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) l))
   (exp (- (pow M 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= 1.45e-19) {
		tmp = exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	} else {
		tmp = exp(-pow(M, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m_1 <= 1.45d-19) then
        tmp = exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) - l))
    else
        tmp = exp(-(m_1 ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= 1.45e-19) {
		tmp = Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	} else {
		tmp = Math.exp(-Math.pow(M, 2.0));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= 1.45e-19:
		tmp = math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l))
	else:
		tmp = math.exp(-math.pow(M, 2.0))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= 1.45e-19)
		tmp = exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) - l));
	else
		tmp = exp(Float64(-(M ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= 1.45e-19)
		tmp = exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	else
		tmp = exp(-(M ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, 1.45e-19], N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.45e-19

   1. Initial program 74.7%

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

   3. Taylor expanded in K around 0 95.0%

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

      1. cos-neg 95.0%

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

   5. Simplified 95.0%

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

   6. Taylor expanded in n around 0 68.8%

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

      1. +-commutative 68.8%

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

      2. unpow2 68.8%

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

      3. distribute-rgt-out 72.1%

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

      4. *-commutative 72.1%

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

      5. *-commutative 72.1%

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

   8. Simplified 72.1%

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

   9. Taylor expanded in l around inf 77.4%

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

   10. Taylor expanded in M around 0 77.4%

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

   if 1.45e-19 < M

   1. Initial program 80.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 90.9%

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

      1. mul-1-neg 90.9%

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

   8. Simplified 90.9%

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

   9. Taylor expanded in M around 0 90.9%

      \[\leadsto \color{blue}{1} \cdot e^{-{M}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.45 \cdot 10^{-19}:\\ \;\;\;\;e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in K around 0 96.4%

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

   1. cos-neg 96.4%

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

5. Simplified 96.4%

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

6. Taylor expanded in n around 0 72.5%

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

   1. +-commutative 72.5%

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

   2. unpow2 72.5%

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

   3. distribute-rgt-out 79.1%

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

   4. *-commutative 79.1%

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

   5. *-commutative 79.1%

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

8. Simplified 79.1%

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

9. Taylor expanded in l around inf 82.9%

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

10. Taylor expanded in M around 0 82.1%

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

11. Final simplification 82.1%

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

Alternative 10: 34.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -5 \cdot 10^{-222}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.5 \cdot \left(m \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -5e-222) (exp (* n (- M (* m 0.5)))) (exp (* -0.5 (* m n)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -5e-222) {
		tmp = exp((n * (M - (m * 0.5))));
	} else {
		tmp = exp((-0.5 * (m * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m_1 <= (-5d-222)) then
        tmp = exp((n * (m_1 - (m * 0.5d0))))
    else
        tmp = exp(((-0.5d0) * (m * n)))
    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 <= -5e-222) {
		tmp = Math.exp((n * (M - (m * 0.5))));
	} else {
		tmp = Math.exp((-0.5 * (m * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -5e-222:
		tmp = math.exp((n * (M - (m * 0.5))))
	else:
		tmp = math.exp((-0.5 * (m * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -5e-222)
		tmp = exp(Float64(n * Float64(M - Float64(m * 0.5))));
	else
		tmp = exp(Float64(-0.5 * Float64(m * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -5e-222)
		tmp = exp((n * (M - (m * 0.5))));
	else
		tmp = exp((-0.5 * (m * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -5e-222], N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.5 * N[(m * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -5.00000000000000008e-222

   1. Initial program 76.8%

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

   3. Taylor expanded in K around 0 96.7%

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

      1. cos-neg 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in n around 0 73.3%

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

      1. +-commutative 73.3%

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

      2. unpow2 73.3%

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

      3. distribute-rgt-out 77.0%

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

      4. *-commutative 77.0%

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

      5. *-commutative 77.0%

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

   8. Simplified 77.0%

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

   9. Taylor expanded in n around inf 45.6%

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

   10. Taylor expanded in M around 0 45.6%

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

   if -5.00000000000000008e-222 < M

   1. Initial program 75.8%

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

   3. Taylor expanded in K around 0 96.2%

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

      1. cos-neg 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in n around 0 71.8%

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

      1. +-commutative 71.8%

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

      2. unpow2 71.8%

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

      3. distribute-rgt-out 80.7%

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

      4. *-commutative 80.7%

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

      5. *-commutative 80.7%

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

   8. Simplified 80.7%

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

   9. Taylor expanded in n around inf 43.3%

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

   10. Taylor expanded in M around 0 34.1%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -5 \cdot 10^{-222}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.5 \cdot \left(m \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in K around 0 96.4%

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

   1. cos-neg 96.4%

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

5. Simplified 96.4%

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

6. Taylor expanded in n around 0 72.5%

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

   1. +-commutative 72.5%

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

   2. unpow2 72.5%

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

   3. distribute-rgt-out 79.1%

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

   4. *-commutative 79.1%

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

   5. *-commutative 79.1%

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

8. Simplified 79.1%

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

9. Taylor expanded in n around inf 44.3%

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

10. Taylor expanded in M around 0 34.6%

    \[\leadsto \color{blue}{e^{-0.5 \cdot \left(m \cdot n\right)}} \]
11. Add Preprocessing

Alternative 12: 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 76.2%

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

3. Taylor expanded in l around inf 28.1%

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

   1. mul-1-neg 28.1%

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

5. Simplified 28.1%

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

6. Taylor expanded in l around 0 6.8%

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

   1. *-commutative 6.8%

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

   2. *-commutative 6.8%

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

   3. associate-*r* 6.8%

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

   4. *-lft-identity 6.8%

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

   5. *-lft-identity 6.8%

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

   6. *-commutative 6.8%

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

8. Simplified 6.8%

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

9. Taylor expanded in K around 0 7.3%

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

    1. cos-neg 7.3%

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

11. Simplified 7.3%

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

Alternative 13: 7.1% accurate, 425.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in K around 0 96.4%

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

   1. cos-neg 96.4%

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

5. Simplified 96.4%

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

6. Taylor expanded in M around inf 53.1%

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

   1. mul-1-neg 53.1%

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

8. Simplified 53.1%

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

9. Taylor expanded in M around 0 7.3%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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)))))))