Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.2% → 96.9%

Time: 16.0s

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: 76.2% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right| - \ell\\ t_1 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{t\_0 - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t\_0 - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- m n)) l))
        (t_1
         (*
          (cos (- (/ (* K (+ m n)) 2.0) M))
          (exp (- t_0 (pow (- (/ (+ m n) 2.0) M) 2.0))))))
   (if (<= t_1 INFINITY)
     t_1
     (* (cos M) (exp (- t_0 (pow (- (* (+ m n) 0.5) M) 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n)) - l;
	double t_1 = cos((((K * (m + n)) / 2.0) - M)) * exp((t_0 - pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = cos(M) * exp((t_0 - pow((((m + n) * 0.5) - M), 2.0)));
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((m - n)) - l;
	double t_1 = Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((t_0 - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - Math.pow((((m + n) * 0.5) - M), 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n)) - l
	t_1 = math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((t_0 - math.pow((((m + n) / 2.0) - M), 2.0)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = math.cos(M) * math.exp((t_0 - math.pow((((m + n) * 0.5) - M), 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(m - n)) - l)
	t_1 = Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(t_0 - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n)) - l;
	t_1 = cos((((K * (m + n)) / 2.0) - M)) * exp((t_0 - ((((m + n) / 2.0) - M) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = cos(M) * exp((t_0 - ((((m + n) * 0.5) - M) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(t$95$0 - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 96.7%

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

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 0.0%

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

      1. add-cube-cbrt 0.0%

         \[\leadsto \cos \left(\frac{\color{blue}{\left(\sqrt[3]{K \cdot \left(m + n\right)} \cdot \sqrt[3]{K \cdot \left(m + n\right)}\right) \cdot \sqrt[3]{K \cdot \left(m + n\right)}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. pow3 0.0%

         \[\leadsto \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{K \cdot \left(m + n\right)}\right)}^{3}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   4. Applied egg-rr 0.0%

      \[\leadsto \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{K \cdot \left(m + n\right)}\right)}^{3}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Taylor expanded in K around 0 98.0%

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

   7. Simplified 98.0%

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

4. Final simplification 96.9%

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

Alternative 2: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 10000000000000:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - \left(\ell + {\left(m \cdot 0.5 - M\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 10000000000000.0)
   (* (cos M) (exp (- (fabs (- m n)) (+ l (pow (- (* m 0.5) M) 2.0)))))
   (exp (* -0.25 (pow n 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 10000000000000.0) {
		tmp = cos(M) * exp((fabs((m - n)) - (l + pow(((m * 0.5) - M), 2.0))));
	} else {
		tmp = 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 <= 10000000000000.0d0) then
        tmp = cos(m_1) * exp((abs((m - n)) - (l + (((m * 0.5d0) - m_1) ** 2.0d0))))
    else
        tmp = 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 <= 10000000000000.0) {
		tmp = Math.cos(M) * Math.exp((Math.abs((m - n)) - (l + Math.pow(((m * 0.5) - M), 2.0))));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 10000000000000.0:
		tmp = math.cos(M) * math.exp((math.fabs((m - n)) - (l + math.pow(((m * 0.5) - M), 2.0))))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 10000000000000.0)
		tmp = Float64(cos(M) * exp(Float64(abs(Float64(m - n)) - Float64(l + (Float64(Float64(m * 0.5) - M) ^ 2.0)))));
	else
		tmp = 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 <= 10000000000000.0)
		tmp = cos(M) * exp((abs((m - n)) - (l + (((m * 0.5) - M) ^ 2.0))));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 10000000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[Power[N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1e13

   1. Initial program 79.0%

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

      1. add-cube-cbrt 78.7%

         \[\leadsto \cos \left(\frac{\color{blue}{\left(\sqrt[3]{K \cdot \left(m + n\right)} \cdot \sqrt[3]{K \cdot \left(m + n\right)}\right) \cdot \sqrt[3]{K \cdot \left(m + n\right)}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. pow3 78.2%

         \[\leadsto \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{K \cdot \left(m + n\right)}\right)}^{3}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   4. Applied egg-rr 78.2%

      \[\leadsto \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{K \cdot \left(m + n\right)}\right)}^{3}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Taylor expanded in m around 0 90.1%

      \[\leadsto \cos \left(\frac{{\color{blue}{\left(\sqrt[3]{K \cdot n}\right)}}^{3}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in n around 0 82.5%

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

      1. cos-neg 82.5%

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

      2. fabs-sub 82.5%

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

      3. *-commutative 82.5%

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

   8. Simplified 82.5%

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

   if 1e13 < n

   1. Initial program 75.4%

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

   3. Taylor expanded in n around inf 73.8%

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

   4. Taylor expanded in M around 0 73.8%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 73.8%

         \[\leadsto \cos \color{blue}{\left(\left(K \cdot \left(m + n\right)\right) \cdot 0.5\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

      2. associate-*r* 73.8%

         \[\leadsto \cos \color{blue}{\left(K \cdot \left(\left(m + n\right) \cdot 0.5\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

   6. Simplified 73.8%

      \[\leadsto \color{blue}{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

   7. Taylor expanded in K around 0 98.4%

      \[\leadsto \color{blue}{1} \cdot e^{-0.25 \cdot {n}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 10000000000000:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - \left(\ell + {\left(m \cdot 0.5 - M\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

   1. add-cube-cbrt 77.6%

      \[\leadsto \cos \left(\frac{\color{blue}{\left(\sqrt[3]{K \cdot \left(m + n\right)} \cdot \sqrt[3]{K \cdot \left(m + n\right)}\right) \cdot \sqrt[3]{K \cdot \left(m + n\right)}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. pow3 77.2%

      \[\leadsto \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{K \cdot \left(m + n\right)}\right)}^{3}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

4. Applied egg-rr 77.2%

   \[\leadsto \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{K \cdot \left(m + n\right)}\right)}^{3}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

5. Taylor expanded in K around 0 95.1%

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

7. Simplified 95.1%

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

8. Final simplification 95.1%

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

Alternative 4: 72.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)\\ t_1 := e^{-0.25 \cdot {n}^{2}}\\ \mathbf{if}\;n \leq -54:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-206}:\\ \;\;\;\;t\_0 \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;t\_0 \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;n \leq 250000000:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cos (- (/ (* K (+ m n)) 2.0) M)))
        (t_1 (exp (* -0.25 (pow n 2.0)))))
   (if (<= n -54.0)
     t_1
     (if (<= n 1.2e-206)
       (* t_0 (exp (* -0.25 (pow m 2.0))))
       (if (<= n 5.8e-12)
         (* t_0 (exp (- (pow M 2.0))))
         (if (<= n 250000000.0) (* (cos M) (exp (- l))) t_1))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos((((K * (m + n)) / 2.0) - M));
	double t_1 = exp((-0.25 * pow(n, 2.0)));
	double tmp;
	if (n <= -54.0) {
		tmp = t_1;
	} else if (n <= 1.2e-206) {
		tmp = t_0 * exp((-0.25 * pow(m, 2.0)));
	} else if (n <= 5.8e-12) {
		tmp = t_0 * exp(-pow(M, 2.0));
	} else if (n <= 250000000.0) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((((k * (m + n)) / 2.0d0) - m_1))
    t_1 = exp(((-0.25d0) * (n ** 2.0d0)))
    if (n <= (-54.0d0)) then
        tmp = t_1
    else if (n <= 1.2d-206) then
        tmp = t_0 * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (n <= 5.8d-12) then
        tmp = t_0 * exp(-(m_1 ** 2.0d0))
    else if (n <= 250000000.0d0) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos((((K * (m + n)) / 2.0) - M));
	double t_1 = Math.exp((-0.25 * Math.pow(n, 2.0)));
	double tmp;
	if (n <= -54.0) {
		tmp = t_1;
	} else if (n <= 1.2e-206) {
		tmp = t_0 * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (n <= 5.8e-12) {
		tmp = t_0 * Math.exp(-Math.pow(M, 2.0));
	} else if (n <= 250000000.0) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos((((K * (m + n)) / 2.0) - M))
	t_1 = math.exp((-0.25 * math.pow(n, 2.0)))
	tmp = 0
	if n <= -54.0:
		tmp = t_1
	elif n <= 1.2e-206:
		tmp = t_0 * math.exp((-0.25 * math.pow(m, 2.0)))
	elif n <= 5.8e-12:
		tmp = t_0 * math.exp(-math.pow(M, 2.0))
	elif n <= 250000000.0:
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = t_1
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M))
	t_1 = exp(Float64(-0.25 * (n ^ 2.0)))
	tmp = 0.0
	if (n <= -54.0)
		tmp = t_1;
	elseif (n <= 1.2e-206)
		tmp = Float64(t_0 * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (n <= 5.8e-12)
		tmp = Float64(t_0 * exp(Float64(-(M ^ 2.0))));
	elseif (n <= 250000000.0)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos((((K * (m + n)) / 2.0) - M));
	t_1 = exp((-0.25 * (n ^ 2.0)));
	tmp = 0.0;
	if (n <= -54.0)
		tmp = t_1;
	elseif (n <= 1.2e-206)
		tmp = t_0 * exp((-0.25 * (m ^ 2.0)));
	elseif (n <= 5.8e-12)
		tmp = t_0 * exp(-(M ^ 2.0));
	elseif (n <= 250000000.0)
		tmp = cos(M) * exp(-l);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -54.0], t$95$1, If[LessEqual[n, 1.2e-206], N[(t$95$0 * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-12], N[(t$95$0 * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 250000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -54 or 2.5e8 < n

   1. Initial program 77.5%

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

   3. Taylor expanded in n around inf 74.5%

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

   4. Taylor expanded in M around 0 74.5%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 74.5%

         \[\leadsto \cos \color{blue}{\left(\left(K \cdot \left(m + n\right)\right) \cdot 0.5\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

      2. associate-*r* 74.5%

         \[\leadsto \cos \color{blue}{\left(K \cdot \left(\left(m + n\right) \cdot 0.5\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

   6. Simplified 74.5%

      \[\leadsto \color{blue}{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

   7. Taylor expanded in K around 0 96.2%

      \[\leadsto \color{blue}{1} \cdot e^{-0.25 \cdot {n}^{2}} \]

   if -54 < n < 1.2e-206

   1. Initial program 77.1%

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

   3. Taylor expanded in m around inf 46.6%

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

   if 1.2e-206 < n < 5.8000000000000003e-12

   1. Initial program 80.9%

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

   3. Taylor expanded in M around inf 45.5%

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

      1. mul-1-neg 45.5%

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

   5. Simplified 45.5%

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

   if 5.8000000000000003e-12 < n < 2.5e8

   1. Initial program 85.7%

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

   3. Taylor expanded in l around inf 71.9%

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

      1. mul-1-neg 71.9%

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

   5. Simplified 71.9%

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

   6. Taylor expanded in K around 0 86.2%

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

      1. cos-neg 100.0%

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

   8. Simplified 86.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -54:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-206}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;n \leq 250000000:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{+206}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + \left(\left|m - n\right| - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1.45e+206)
   (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (* -0.25 (pow m 2.0))))
   (*
    (cos M)
    (exp (+ (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) (- (fabs (- m n)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.45e+206) {
		tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (fabs((m - n)) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-1.45d+206)) then
        tmp = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) + (abs((m - n)) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.45e+206) {
		tmp = Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (Math.abs((m - n)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1.45e+206:
		tmp = math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (math.fabs((m - n)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1.45e+206)
		tmp = Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) + Float64(abs(Float64(m - n)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1.45e+206)
		tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (abs((m - n)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1.45e+206], N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.45e206

   1. Initial program 57.1%

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

   3. Taylor expanded in m around inf 57.1%

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

   if -1.45e206 < 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 m around 0 65.5%

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

      1. +-commutative 65.5%

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

      2. unpow2 65.5%

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

      3. distribute-rgt-out 70.2%

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

      4. *-commutative 70.2%

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

      5. *-commutative 70.2%

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

   5. Simplified 70.2%

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

   6. Taylor expanded in K around 0 80.7%

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

      1. cos-neg 80.7%

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

   8. Simplified 80.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{+206}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + \left(\left|m - n\right| - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot {n}^{2}}\\ \mathbf{if}\;n \leq -54:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-12}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;n \leq 250000000:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (pow n 2.0)))))
   (if (<= n -54.0)
     t_0
     (if (<= n 8e-12)
       (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (pow M 2.0))))
       (if (<= n 250000000.0) (* (cos M) (exp (- l))) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * pow(n, 2.0)));
	double tmp;
	if (n <= -54.0) {
		tmp = t_0;
	} else if (n <= 8e-12) {
		tmp = cos((((K * (m + n)) / 2.0) - M)) * exp(-pow(M, 2.0));
	} else if (n <= 250000000.0) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((-0.25d0) * (n ** 2.0d0)))
    if (n <= (-54.0d0)) then
        tmp = t_0
    else if (n <= 8d-12) then
        tmp = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp(-(m_1 ** 2.0d0))
    else if (n <= 250000000.0d0) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((-0.25 * Math.pow(n, 2.0)));
	double tmp;
	if (n <= -54.0) {
		tmp = t_0;
	} else if (n <= 8e-12) {
		tmp = Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp(-Math.pow(M, 2.0));
	} else if (n <= 250000000.0) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * math.pow(n, 2.0)))
	tmp = 0
	if n <= -54.0:
		tmp = t_0
	elif n <= 8e-12:
		tmp = math.cos((((K * (m + n)) / 2.0) - M)) * math.exp(-math.pow(M, 2.0))
	elif n <= 250000000.0:
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * (n ^ 2.0)))
	tmp = 0.0
	if (n <= -54.0)
		tmp = t_0;
	elseif (n <= 8e-12)
		tmp = Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(-(M ^ 2.0))));
	elseif (n <= 250000000.0)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (n ^ 2.0)));
	tmp = 0.0;
	if (n <= -54.0)
		tmp = t_0;
	elseif (n <= 8e-12)
		tmp = cos((((K * (m + n)) / 2.0) - M)) * exp(-(M ^ 2.0));
	elseif (n <= 250000000.0)
		tmp = cos(M) * exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -54.0], t$95$0, If[LessEqual[n, 8e-12], N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 250000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -54 or 2.5e8 < n

   1. Initial program 77.5%

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

   3. Taylor expanded in n around inf 74.5%

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

   4. Taylor expanded in M around 0 74.5%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 74.5%

         \[\leadsto \cos \color{blue}{\left(\left(K \cdot \left(m + n\right)\right) \cdot 0.5\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

      2. associate-*r* 74.5%

         \[\leadsto \cos \color{blue}{\left(K \cdot \left(\left(m + n\right) \cdot 0.5\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

   6. Simplified 74.5%

      \[\leadsto \color{blue}{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

   7. Taylor expanded in K around 0 96.2%

      \[\leadsto \color{blue}{1} \cdot e^{-0.25 \cdot {n}^{2}} \]

   if -54 < n < 7.99999999999999984e-12

   1. Initial program 78.4%

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

   3. Taylor expanded in M around inf 42.2%

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

      1. mul-1-neg 42.2%

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

   5. Simplified 42.2%

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

   if 7.99999999999999984e-12 < n < 2.5e8

   1. Initial program 85.7%

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

   3. Taylor expanded in l around inf 71.9%

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

      1. mul-1-neg 71.9%

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

   5. Simplified 71.9%

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

   6. Taylor expanded in K around 0 86.2%

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

      1. cos-neg 100.0%

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

   8. Simplified 86.2%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -54:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-12}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;n \leq 250000000:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -53 \lor \neg \left(n \leq 55\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell} \cdot \cos \left(K \cdot \left(n \cdot 0.5\right) - M\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -53.0) (not (<= n 55.0)))
   (exp (* -0.25 (pow n 2.0)))
   (* (exp (- l)) (cos (- (* K (* n 0.5)) M)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -53.0) || !(n <= 55.0)) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = exp(-l) * cos(((K * (n * 0.5)) - M));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((n <= (-53.0d0)) .or. (.not. (n <= 55.0d0))) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = exp(-l) * cos(((k * (n * 0.5d0)) - m_1))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -53.0) || !(n <= 55.0)) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.exp(-l) * Math.cos(((K * (n * 0.5)) - M));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -53.0) or not (n <= 55.0):
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.exp(-l) * math.cos(((K * (n * 0.5)) - M))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -53.0) || !(n <= 55.0))
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = Float64(exp(Float64(-l)) * cos(Float64(Float64(K * Float64(n * 0.5)) - M)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -53.0) || ~((n <= 55.0)))
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = exp(-l) * cos(((K * (n * 0.5)) - M));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -53.0], N[Not[LessEqual[n, 55.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Exp[(-l)], $MachinePrecision] * N[Cos[N[(N[(K * N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -53 or 55 < n

   1. Initial program 76.9%

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

   3. Taylor expanded in n around inf 73.9%

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

   4. Taylor expanded in M around 0 73.9%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 73.9%

         \[\leadsto \cos \color{blue}{\left(\left(K \cdot \left(m + n\right)\right) \cdot 0.5\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

      2. associate-*r* 73.9%

         \[\leadsto \cos \color{blue}{\left(K \cdot \left(\left(m + n\right) \cdot 0.5\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

   6. Simplified 73.9%

      \[\leadsto \color{blue}{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

   7. Taylor expanded in K around 0 95.5%

      \[\leadsto \color{blue}{1} \cdot e^{-0.25 \cdot {n}^{2}} \]

   if -53 < n < 55

   1. Initial program 79.4%

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

   3. Taylor expanded in l around inf 38.0%

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

      1. mul-1-neg 38.0%

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

   5. Simplified 38.0%

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

   6. Taylor expanded in m around 0 41.2%

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

      1. *-commutative 41.2%

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

      2. associate-*l* 41.2%

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

   8. Simplified 41.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -53 \lor \neg \left(n \leq 55\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell} \cdot \cos \left(K \cdot \left(n \cdot 0.5\right) - M\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -54 \lor \neg \left(n \leq 250000000\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -54.0) (not (<= n 250000000.0)))
   (exp (* -0.25 (pow n 2.0)))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -54.0) || !(n <= 250000000.0)) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((n <= (-54.0d0)) .or. (.not. (n <= 250000000.0d0))) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -54.0) || !(n <= 250000000.0)) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -54.0) or not (n <= 250000000.0):
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -54.0) || !(n <= 250000000.0))
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -54.0) || ~((n <= 250000000.0)))
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -54.0], N[Not[LessEqual[n, 250000000.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -54 or 2.5e8 < n

   1. Initial program 77.5%

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

   3. Taylor expanded in n around inf 74.5%

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

   4. Taylor expanded in M around 0 74.5%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 74.5%

         \[\leadsto \cos \color{blue}{\left(\left(K \cdot \left(m + n\right)\right) \cdot 0.5\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

      2. associate-*r* 74.5%

         \[\leadsto \cos \color{blue}{\left(K \cdot \left(\left(m + n\right) \cdot 0.5\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

   6. Simplified 74.5%

      \[\leadsto \color{blue}{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right)\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]

   7. Taylor expanded in K around 0 96.2%

      \[\leadsto \color{blue}{1} \cdot e^{-0.25 \cdot {n}^{2}} \]

   if -54 < n < 2.5e8

   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 l around inf 37.7%

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

      1. mul-1-neg 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in K around 0 42.2%

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

      1. cos-neg 71.2%

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

   8. Simplified 42.2%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -54 \lor \neg \left(n \leq 250000000\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in l around inf 29.6%

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

   1. mul-1-neg 29.6%

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

5. Simplified 29.6%

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

6. Taylor expanded in K around 0 34.8%

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

   1. cos-neg 77.6%

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

8. Simplified 34.8%

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

Reproduce

herbie shell --seed 2024111 
(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)))))))