Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 96.7%

Time: 14.3s

Alternatives: 9

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: 75.9% 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: 96.7% 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 80.2%

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

2. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

4. Simplified 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 94.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.4 \cdot 10^{+31} \lor \neg \left(M \leq 2.2 \cdot 10^{-11}\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1.4e+31) (not (<= M 2.2e-11)))
   (* (cos M) (exp (* M (- M))))
   (exp (- (fabs (- m n)) (+ l (* 0.25 (pow (+ m n) 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.4e+31) || !(M <= 2.2e-11)) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = exp((fabs((m - n)) - (l + (0.25 * pow((m + n), 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-1.4d+31)) .or. (.not. (m_1 <= 2.2d-11))) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = exp((abs((m - n)) - (l + (0.25d0 * ((m + n) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.4e+31) || !(M <= 2.2e-11)) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.exp((Math.abs((m - n)) - (l + (0.25 * Math.pow((m + n), 2.0)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1.4e+31) or not (M <= 2.2e-11):
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.exp((math.fabs((m - n)) - (l + (0.25 * math.pow((m + n), 2.0)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1.4e+31) || !(M <= 2.2e-11))
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(0.25 * (Float64(m + n) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1.4e+31) || ~((M <= 2.2e-11)))
		tmp = cos(M) * exp((M * -M));
	else
		tmp = exp((abs((m - n)) - (l + (0.25 * ((m + n) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1.4e+31], N[Not[LessEqual[M, 2.2e-11]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1.40000000000000008e31 or 2.2000000000000002e-11 < M

   1. Initial program 82.4%

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

   2. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in M around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. unpow2 97.7%

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

      3. distribute-rgt-neg-in 97.7%

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

   7. Simplified 97.7%

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

   if -1.40000000000000008e31 < M < 2.2000000000000002e-11

   1. Initial program 78.3%

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

   2. Taylor expanded in K around 0 97.7%

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

      1. cos-neg 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in M around 0 97.7%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.4 \cdot 10^{+31} \lor \neg \left(M \leq 2.2 \cdot 10^{-11}\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \]

Alternative 3: 65.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-M \cdot M}\\ \mathbf{if}\;n \leq 1.65 \cdot 10^{-280}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.02 \cdot 10^{-109}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- (* M M))))))
   (if (<= n 1.65e-280)
     (exp (* m (* m -0.25)))
     (if (<= n 3.2e-154)
       t_0
       (if (<= n 1.02e-109)
         (* (cos M) (exp (- l)))
         (if (<= n 54.0) t_0 (exp (* (* n n) -0.25))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp(-(M * M));
	double tmp;
	if (n <= 1.65e-280) {
		tmp = exp((m * (m * -0.25)));
	} else if (n <= 3.2e-154) {
		tmp = t_0;
	} else if (n <= 1.02e-109) {
		tmp = cos(M) * exp(-l);
	} else if (n <= 54.0) {
		tmp = t_0;
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	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_1))
    if (n <= 1.65d-280) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (n <= 3.2d-154) then
        tmp = t_0
    else if (n <= 1.02d-109) then
        tmp = cos(m_1) * exp(-l)
    else if (n <= 54.0d0) then
        tmp = t_0
    else
        tmp = exp(((n * n) * (-0.25d0)))
    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));
	double tmp;
	if (n <= 1.65e-280) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (n <= 3.2e-154) {
		tmp = t_0;
	} else if (n <= 1.02e-109) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else if (n <= 54.0) {
		tmp = t_0;
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp(-(M * M))
	tmp = 0
	if n <= 1.65e-280:
		tmp = math.exp((m * (m * -0.25)))
	elif n <= 3.2e-154:
		tmp = t_0
	elif n <= 1.02e-109:
		tmp = math.cos(M) * math.exp(-l)
	elif n <= 54.0:
		tmp = t_0
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-Float64(M * M))))
	tmp = 0.0
	if (n <= 1.65e-280)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (n <= 3.2e-154)
		tmp = t_0;
	elseif (n <= 1.02e-109)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	elseif (n <= 54.0)
		tmp = t_0;
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp(-(M * M));
	tmp = 0.0;
	if (n <= 1.65e-280)
		tmp = exp((m * (m * -0.25)));
	elseif (n <= 3.2e-154)
		tmp = t_0;
	elseif (n <= 1.02e-109)
		tmp = cos(M) * exp(-l);
	elseif (n <= 54.0)
		tmp = t_0;
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[(M * M), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 1.65e-280], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.2e-154], t$95$0, If[LessEqual[n, 1.02e-109], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], t$95$0, N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < 1.64999999999999995e-280

   1. Initial program 82.3%

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

   2. Taylor expanded in K around 0 97.2%

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

      1. cos-neg 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in M around 0 90.8%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

   6. Taylor expanded in m around inf 60.7%

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

      1. *-commutative 60.7%

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

      2. unpow2 60.7%

         \[\leadsto e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

      3. associate-*l* 60.7%

         \[\leadsto e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   8. Simplified 60.7%

      \[\leadsto e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   if 1.64999999999999995e-280 < n < 3.20000000000000005e-154 or 1.02e-109 < n < 54

   1. Initial program 88.3%

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

   2. Taylor expanded in K around 0 97.0%

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

      1. cos-neg 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in M around inf 68.5%

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

      1. mul-1-neg 68.5%

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

      2. unpow2 68.5%

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

      3. distribute-rgt-neg-in 68.5%

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

   7. Simplified 68.5%

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

   if 3.20000000000000005e-154 < n < 1.02e-109

   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. Taylor expanded in l around inf 70.3%

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

      1. mul-1-neg 70.3%

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

   4. Simplified 70.3%

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

   5. Taylor expanded in K around 0 70.8%

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

      1. cos-neg 70.8%

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

   7. Simplified 70.8%

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

   if 54 < n

   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. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

   6. Taylor expanded in n around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

         \[\leadsto e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

   8. Simplified 100.0%

      \[\leadsto e^{\color{blue}{\left(n \cdot n\right) \cdot -0.25}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.65 \cdot 10^{-280}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-154}:\\ \;\;\;\;\cos M \cdot e^{-M \cdot M}\\ \mathbf{elif}\;n \leq 1.02 \cdot 10^{-109}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{-M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 4: 64.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;n \leq 4.4 \cdot 10^{-274}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-110}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (* M (- M))))))
   (if (<= n 4.4e-274)
     (* (cos M) (exp (* -0.25 (* m m))))
     (if (<= n 3.5e-154)
       t_0
       (if (<= n 1.15e-110)
         (* (cos M) (exp (- l)))
         (if (<= n 54.0) t_0 (exp (* (* n n) -0.25))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp((M * -M));
	double tmp;
	if (n <= 4.4e-274) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (n <= 3.5e-154) {
		tmp = t_0;
	} else if (n <= 1.15e-110) {
		tmp = cos(M) * exp(-l);
	} else if (n <= 54.0) {
		tmp = t_0;
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	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_1))
    if (n <= 4.4d-274) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (n <= 3.5d-154) then
        tmp = t_0
    else if (n <= 1.15d-110) then
        tmp = cos(m_1) * exp(-l)
    else if (n <= 54.0d0) then
        tmp = t_0
    else
        tmp = exp(((n * n) * (-0.25d0)))
    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));
	double tmp;
	if (n <= 4.4e-274) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (n <= 3.5e-154) {
		tmp = t_0;
	} else if (n <= 1.15e-110) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else if (n <= 54.0) {
		tmp = t_0;
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp((M * -M))
	tmp = 0
	if n <= 4.4e-274:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif n <= 3.5e-154:
		tmp = t_0
	elif n <= 1.15e-110:
		tmp = math.cos(M) * math.exp(-l)
	elif n <= 54.0:
		tmp = t_0
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(M * Float64(-M))))
	tmp = 0.0
	if (n <= 4.4e-274)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 3.5e-154)
		tmp = t_0;
	elseif (n <= 1.15e-110)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	elseif (n <= 54.0)
		tmp = t_0;
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp((M * -M));
	tmp = 0.0;
	if (n <= 4.4e-274)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (n <= 3.5e-154)
		tmp = t_0;
	elseif (n <= 1.15e-110)
		tmp = cos(M) * exp(-l);
	elseif (n <= 54.0)
		tmp = t_0;
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 4.4e-274], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.5e-154], t$95$0, If[LessEqual[n, 1.15e-110], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], t$95$0, N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < 4.3999999999999999e-274

   1. Initial program 81.8%

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

   2. Taylor expanded in K around 0 96.6%

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

      1. cos-neg 96.6%

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

   4. Simplified 96.6%

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

   5. Taylor expanded in m around inf 60.0%

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

      1. *-commutative 60.0%

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

      2. unpow2 60.0%

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

   7. Simplified 60.0%

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

   if 4.3999999999999999e-274 < n < 3.5000000000000001e-154 or 1.1500000000000001e-110 < n < 54

   1. Initial program 89.8%

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

   2. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in M around inf 68.8%

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

      1. mul-1-neg 68.8%

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

      2. unpow2 68.8%

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

      3. distribute-rgt-neg-in 68.8%

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

   7. Simplified 68.8%

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

   if 3.5000000000000001e-154 < n < 1.1500000000000001e-110

   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. Taylor expanded in l around inf 70.3%

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

      1. mul-1-neg 70.3%

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

   4. Simplified 70.3%

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

   5. Taylor expanded in K around 0 70.8%

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

      1. cos-neg 70.8%

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

   7. Simplified 70.8%

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

   if 54 < n

   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. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

   6. Taylor expanded in n around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

         \[\leadsto e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

   8. Simplified 100.0%

      \[\leadsto e^{\color{blue}{\left(n \cdot n\right) \cdot -0.25}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.4 \cdot 10^{-274}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-154}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-110}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 5: 62.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;n \leq 1.65 \cdot 10^{-280}:\\ \;\;\;\;e^{\left|m - n\right| - m \cdot \left(m \cdot 0.25\right)}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{-110}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (* M (- M))))))
   (if (<= n 1.65e-280)
     (exp (- (fabs (- m n)) (* m (* m 0.25))))
     (if (<= n 3.4e-154)
       t_0
       (if (<= n 2.35e-110)
         (* (cos M) (exp (- l)))
         (if (<= n 54.0) t_0 (exp (* (* n n) -0.25))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp((M * -M));
	double tmp;
	if (n <= 1.65e-280) {
		tmp = exp((fabs((m - n)) - (m * (m * 0.25))));
	} else if (n <= 3.4e-154) {
		tmp = t_0;
	} else if (n <= 2.35e-110) {
		tmp = cos(M) * exp(-l);
	} else if (n <= 54.0) {
		tmp = t_0;
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	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_1))
    if (n <= 1.65d-280) then
        tmp = exp((abs((m - n)) - (m * (m * 0.25d0))))
    else if (n <= 3.4d-154) then
        tmp = t_0
    else if (n <= 2.35d-110) then
        tmp = cos(m_1) * exp(-l)
    else if (n <= 54.0d0) then
        tmp = t_0
    else
        tmp = exp(((n * n) * (-0.25d0)))
    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));
	double tmp;
	if (n <= 1.65e-280) {
		tmp = Math.exp((Math.abs((m - n)) - (m * (m * 0.25))));
	} else if (n <= 3.4e-154) {
		tmp = t_0;
	} else if (n <= 2.35e-110) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else if (n <= 54.0) {
		tmp = t_0;
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp((M * -M))
	tmp = 0
	if n <= 1.65e-280:
		tmp = math.exp((math.fabs((m - n)) - (m * (m * 0.25))))
	elif n <= 3.4e-154:
		tmp = t_0
	elif n <= 2.35e-110:
		tmp = math.cos(M) * math.exp(-l)
	elif n <= 54.0:
		tmp = t_0
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(M * Float64(-M))))
	tmp = 0.0
	if (n <= 1.65e-280)
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(m * Float64(m * 0.25))));
	elseif (n <= 3.4e-154)
		tmp = t_0;
	elseif (n <= 2.35e-110)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	elseif (n <= 54.0)
		tmp = t_0;
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp((M * -M));
	tmp = 0.0;
	if (n <= 1.65e-280)
		tmp = exp((abs((m - n)) - (m * (m * 0.25))));
	elseif (n <= 3.4e-154)
		tmp = t_0;
	elseif (n <= 2.35e-110)
		tmp = cos(M) * exp(-l);
	elseif (n <= 54.0)
		tmp = t_0;
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 1.65e-280], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(m * N[(m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.4e-154], t$95$0, If[LessEqual[n, 2.35e-110], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], t$95$0, N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < 1.64999999999999995e-280

   1. Initial program 82.3%

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

   2. Taylor expanded in K around 0 97.2%

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

      1. cos-neg 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in M around 0 90.8%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

   6. Taylor expanded in m around inf 52.5%

      \[\leadsto e^{\left|m - n\right| - \color{blue}{0.25 \cdot {m}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 52.5%

         \[\leadsto e^{\left|m - n\right| - 0.25 \cdot \color{blue}{\left(m \cdot m\right)}} \]

      2. associate-*r* 52.5%

         \[\leadsto e^{\left|m - n\right| - \color{blue}{\left(0.25 \cdot m\right) \cdot m}} \]

   8. Simplified 52.5%

      \[\leadsto e^{\left|m - n\right| - \color{blue}{\left(0.25 \cdot m\right) \cdot m}} \]

   if 1.64999999999999995e-280 < n < 3.3999999999999998e-154 or 2.34999999999999996e-110 < n < 54

   1. Initial program 88.3%

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

   2. Taylor expanded in K around 0 97.0%

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

      1. cos-neg 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in M around inf 68.5%

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

      1. mul-1-neg 68.5%

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

      2. unpow2 68.5%

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

      3. distribute-rgt-neg-in 68.5%

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

   7. Simplified 68.5%

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

   if 3.3999999999999998e-154 < n < 2.34999999999999996e-110

   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. Taylor expanded in l around inf 70.3%

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

      1. mul-1-neg 70.3%

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

   4. Simplified 70.3%

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

   5. Taylor expanded in K around 0 70.8%

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

      1. cos-neg 70.8%

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

   7. Simplified 70.8%

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

   if 54 < n

   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. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

   6. Taylor expanded in n around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

         \[\leadsto e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

   8. Simplified 100.0%

      \[\leadsto e^{\color{blue}{\left(n \cdot n\right) \cdot -0.25}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.65 \cdot 10^{-280}:\\ \;\;\;\;e^{\left|m - n\right| - m \cdot \left(m \cdot 0.25\right)}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-154}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{-110}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 6: 65.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.2:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 4.2) (exp (* m (* m -0.25))) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 4.2) {
		tmp = exp((m * (m * -0.25)));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 4.2d0) then
        tmp = exp((m * (m * (-0.25d0))))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

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

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 4.2:
		tmp = math.exp((m * (m * -0.25)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 4.2)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 4.2)
		tmp = exp((m * (m * -0.25)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 4.2], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.20000000000000018

   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. Taylor expanded in K around 0 97.2%

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

      1. cos-neg 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in M around 0 85.3%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

   6. Taylor expanded in m around inf 58.9%

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

      1. *-commutative 58.9%

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

      2. unpow2 58.9%

         \[\leadsto e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

      3. associate-*l* 58.9%

         \[\leadsto e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   8. Simplified 58.9%

      \[\leadsto e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   if 4.20000000000000018 < 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. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

   6. Taylor expanded in l around inf 98.7%

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

      1. neg-mul-1 98.7%

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

   8. Simplified 98.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 7: 66.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3600000:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -3600000.0) (exp (* m (* m -0.25))) (exp (* (* n n) -0.25))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -3600000.0) {
		tmp = exp((m * (m * -0.25)));
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	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 <= (-3600000.0d0)) then
        tmp = exp((m * (m * (-0.25d0))))
    else
        tmp = exp(((n * n) * (-0.25d0)))
    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 <= -3600000.0) {
		tmp = Math.exp((m * (m * -0.25)));
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -3600000.0:
		tmp = math.exp((m * (m * -0.25)))
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -3600000.0)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -3600000.0)
		tmp = exp((m * (m * -0.25)));
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -3600000.0], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.6e6

   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. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

   6. Taylor expanded in m around inf 98.5%

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

      1. *-commutative 98.5%

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

      2. unpow2 98.5%

         \[\leadsto e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

      3. associate-*l* 98.5%

         \[\leadsto e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   8. Simplified 98.5%

      \[\leadsto e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   if -3.6e6 < 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. Taylor expanded in K around 0 97.4%

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

      1. cos-neg 97.4%

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

   4. Simplified 97.4%

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

   5. Taylor expanded in M around 0 86.2%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

   6. Taylor expanded in n around inf 58.7%

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

      1. *-commutative 58.7%

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

      2. unpow2 58.7%

         \[\leadsto e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

   8. Simplified 58.7%

      \[\leadsto e^{\color{blue}{\left(n \cdot n\right) \cdot -0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3600000:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 8: 34.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

2. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

4. Simplified 98.1%

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

5. Taylor expanded in M around 0 89.7%

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

6. Taylor expanded in l around inf 37.5%

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

   1. neg-mul-1 37.5%

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

8. Simplified 37.5%

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

9. Final simplification 37.5%

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

Alternative 9: 7.2% 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 80.2%

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

2. Taylor expanded in n around inf 41.9%

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

   1. *-commutative 41.9%

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

   2. unpow2 41.9%

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

4. Simplified 41.9%

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

5. Taylor expanded in n around 0 7.3%

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

   1. *-commutative 7.3%

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

7. Simplified 7.3%

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

8. Taylor expanded in m around 0 7.9%

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

   1. cos-neg 7.9%

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

10. Simplified 7.9%

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

11. Final simplification 7.9%

    \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023264 
(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)))))))