Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 77.3% → 97.1%

Time: 16.8s

Alternatives: 11

Speedup: 1.9×

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: 77.3% 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: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - (l + pow(((0.5 * (n + m)) - M), 2.0)))) * 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 = exp((abs((n - m)) - (l + (((0.5d0 * (n + m)) - m_1) ** 2.0d0)))) * cos(m_1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in K around 0 95.7%

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

5. Simplified 95.7%

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

6. Final simplification 95.7%

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

Alternative 2: 92.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}\\ \mathbf{if}\;K \leq -8 \cdot 10^{+149} \lor \neg \left(K \leq 1.2 \cdot 10^{+69}\right):\\ \;\;\;\;\cos M \cdot e^{\left(n - m\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot e^{\left(\left(m - n\right) - \ell\right) - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (pow (- (* 0.5 (+ n m)) M) 2.0)))
   (if (or (<= K -8e+149) (not (<= K 1.2e+69)))
     (* (cos M) (exp (- (- n m) t_0)))
     (* (cos (- (/ (* (+ n m) K) 2.0) M)) (exp (- (- (- m n) l) t_0))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = pow(((0.5 * (n + m)) - M), 2.0);
	double tmp;
	if ((K <= -8e+149) || !(K <= 1.2e+69)) {
		tmp = cos(M) * exp(((n - m) - t_0));
	} else {
		tmp = cos(((((n + m) * K) / 2.0) - M)) * exp((((m - n) - l) - 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 = ((0.5d0 * (n + m)) - m_1) ** 2.0d0
    if ((k <= (-8d+149)) .or. (.not. (k <= 1.2d+69))) then
        tmp = cos(m_1) * exp(((n - m) - t_0))
    else
        tmp = cos(((((n + m) * k) / 2.0d0) - m_1)) * exp((((m - n) - l) - 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.pow(((0.5 * (n + m)) - M), 2.0);
	double tmp;
	if ((K <= -8e+149) || !(K <= 1.2e+69)) {
		tmp = Math.cos(M) * Math.exp(((n - m) - t_0));
	} else {
		tmp = Math.cos(((((n + m) * K) / 2.0) - M)) * Math.exp((((m - n) - l) - t_0));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.pow(((0.5 * (n + m)) - M), 2.0)
	tmp = 0
	if (K <= -8e+149) or not (K <= 1.2e+69):
		tmp = math.cos(M) * math.exp(((n - m) - t_0))
	else:
		tmp = math.cos(((((n + m) * K) / 2.0) - M)) * math.exp((((m - n) - l) - t_0))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * Float64(n + m)) - M) ^ 2.0
	tmp = 0.0
	if ((K <= -8e+149) || !(K <= 1.2e+69))
		tmp = Float64(cos(M) * exp(Float64(Float64(n - m) - t_0)));
	else
		tmp = Float64(cos(Float64(Float64(Float64(Float64(n + m) * K) / 2.0) - M)) * exp(Float64(Float64(Float64(m - n) - l) - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = ((0.5 * (n + m)) - M) ^ 2.0;
	tmp = 0.0;
	if ((K <= -8e+149) || ~((K <= 1.2e+69)))
		tmp = cos(M) * exp(((n - m) - t_0));
	else
		tmp = cos(((((n + m) * K) / 2.0) - M)) * exp((((m - n) - l) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Power[N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[K, -8e+149], N[Not[LessEqual[K, 1.2e+69]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n - m), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[(N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if K < -8.00000000000000039e149 or 1.2000000000000001e69 < K

   1. Initial program 33.6%

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

   3. Taylor expanded in K around 0 89.5%

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

   5. Simplified 89.5%

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

   6. Taylor expanded in l around 0 85.8%

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

      1. fabs-sub 85.8%

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

      2. fabs-sub 85.8%

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

      3. rem-square-sqrt 44.9%

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

      4. fabs-sqr 44.9%

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

      5. rem-square-sqrt 85.8%

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

   8. Simplified 85.8%

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

   if -8.00000000000000039e149 < K < 1.2000000000000001e69

   1. Initial program 97.8%

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

      1. sub-neg 97.8%

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

      2. distribute-neg-out 97.8%

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

      3. div-inv 97.8%

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

      4. fmm-def 97.8%

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

      5. metadata-eval 97.8%

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

      6. add-sqr-sqrt 47.5%

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

      7. fabs-sqr 47.5%

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

      8. add-sqr-sqrt 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. +-commutative 97.8%

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

      2. distribute-neg-in 97.8%

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

      3. sub-neg 97.8%

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

      4. sub-neg 97.8%

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

      5. distribute-neg-in 97.8%

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

      6. sub-neg 97.8%

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

      7. mul-1-neg 97.8%

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

      8. distribute-neg-in 97.8%

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

      9. mul-1-neg 97.8%

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

      10. mul-1-neg 97.8%

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

      11. remove-double-neg 97.8%

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

      12. distribute-neg-in 97.8%

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

      13. mul-1-neg 97.8%

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

      14. remove-double-neg 97.8%

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

      15. sub-neg 97.8%

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

      16. fmm-undef 97.8%

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

      17. *-commutative 97.8%

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

   6. Simplified 97.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq -8 \cdot 10^{+149} \lor \neg \left(K \leq 1.2 \cdot 10^{+69}\right):\\ \;\;\;\;\cos M \cdot e^{\left(n - m\right) - {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot e^{\left(\left(m - n\right) - \ell\right) - {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 730:\\ \;\;\;\;\cos M \cdot e^{\left(n - m\right) - {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 730.0)
   (* (cos M) (exp (- (- n m) (pow (- (* 0.5 (+ n m)) M) 2.0))))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 730.0) {
		tmp = cos(M) * exp(((n - m) - pow(((0.5 * (n + m)) - M), 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 (l <= 730.0d0) then
        tmp = cos(m_1) * exp(((n - m) - (((0.5d0 * (n + m)) - m_1) ** 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 (l <= 730.0) {
		tmp = Math.cos(M) * Math.exp(((n - m) - Math.pow(((0.5 * (n + m)) - M), 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 730.0:
		tmp = math.cos(M) * math.exp(((n - m) - math.pow(((0.5 * (n + m)) - M), 2.0)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 730.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(n - m) - (Float64(Float64(0.5 * Float64(n + m)) - M) ^ 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 (l <= 730.0)
		tmp = cos(M) * exp(((n - m) - (((0.5 * (n + m)) - M) ^ 2.0)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 730.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n - m), $MachinePrecision] - N[Power[N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 730

   1. Initial program 77.3%

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

   3. Taylor expanded in K around 0 94.0%

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

   5. Simplified 94.0%

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

   6. Taylor expanded in l around 0 84.1%

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

      1. fabs-sub 84.1%

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

      2. fabs-sub 84.1%

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

      3. rem-square-sqrt 42.9%

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

      4. fabs-sqr 42.9%

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

      5. rem-square-sqrt 83.7%

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

   8. Simplified 83.7%

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

   if 730 < l

   1. Initial program 81.4%

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

   3. Taylor expanded in l around inf 81.4%

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

      1. mul-1-neg 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 730:\\ \;\;\;\;\cos M \cdot e^{\left(n - m\right) - {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 730:\\ \;\;\;\;\cos M \cdot e^{\left(n - m\right) - 0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 730.0)
   (* (cos M) (exp (- (- n m) (* 0.25 (* (+ n m) (+ n m))))))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 730.0) {
		tmp = cos(M) * exp(((n - m) - (0.25 * ((n + m) * (n + m)))));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 730.0d0) then
        tmp = cos(m_1) * exp(((n - m) - (0.25d0 * ((n + m) * (n + m)))))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 730.0) {
		tmp = Math.cos(M) * Math.exp(((n - m) - (0.25 * ((n + m) * (n + m)))));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 730.0:
		tmp = math.cos(M) * math.exp(((n - m) - (0.25 * ((n + m) * (n + m)))))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 730.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(n - m) - Float64(0.25 * Float64(Float64(n + m) * Float64(n + m))))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 730.0)
		tmp = cos(M) * exp(((n - m) - (0.25 * ((n + m) * (n + m)))));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 730.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n - m), $MachinePrecision] - N[(0.25 * N[(N[(n + m), $MachinePrecision] * N[(n + m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 730

   1. Initial program 77.3%

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

   3. Taylor expanded in K around 0 94.0%

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

   5. Simplified 94.0%

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

   6. Taylor expanded in l around 0 84.1%

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

      1. fabs-sub 84.1%

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

      2. fabs-sub 84.1%

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

      3. rem-square-sqrt 42.9%

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

      4. fabs-sqr 42.9%

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

      5. rem-square-sqrt 83.7%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in M around 0 76.0%

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

       1. unpow2 76.0%

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

   11. Applied egg-rr 76.0%

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

   if 730 < l

   1. Initial program 81.4%

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

   3. Taylor expanded in l around inf 81.4%

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

      1. mul-1-neg 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 730:\\ \;\;\;\;\cos M \cdot e^{\left(n - m\right) - 0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(\left(M + n \cdot -0.25\right) - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 720.0)
   (* (cos M) (exp (* n (- (+ M (* n -0.25)) (* m 0.5)))))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = cos(M) * exp((n * ((M + (n * -0.25)) - (m * 0.5))));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 720.0d0) then
        tmp = cos(m_1) * exp((n * ((m_1 + (n * (-0.25d0))) - (m * 0.5d0))))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = Math.cos(M) * Math.exp((n * ((M + (n * -0.25)) - (m * 0.5))));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 720.0:
		tmp = math.cos(M) * math.exp((n * ((M + (n * -0.25)) - (m * 0.5))))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 720.0)
		tmp = Float64(cos(M) * exp(Float64(n * Float64(Float64(M + Float64(n * -0.25)) - Float64(m * 0.5)))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 720.0)
		tmp = cos(M) * exp((n * ((M + (n * -0.25)) - (m * 0.5))));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 720.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(N[(M + N[(n * -0.25), $MachinePrecision]), $MachinePrecision] - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 720

   1. Initial program 77.3%

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

   3. Taylor expanded in n around inf 37.3%

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

   4. Taylor expanded in K around 0 47.6%

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

      1. cos-neg 47.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in n around 0 51.2%

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

   if 720 < l

   1. Initial program 81.4%

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

   3. Taylor expanded in l around inf 81.4%

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

      1. mul-1-neg 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(\left(M + n \cdot -0.25\right) - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(m \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 720.0) (* (cos M) (exp (* n (* m -0.5)))) (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = cos(M) * exp((n * (m * -0.5)));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 720.0d0) then
        tmp = cos(m_1) * exp((n * (m * (-0.5d0))))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = Math.cos(M) * Math.exp((n * (m * -0.5)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 720.0:
		tmp = math.cos(M) * math.exp((n * (m * -0.5)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 720.0)
		tmp = Float64(cos(M) * exp(Float64(n * Float64(m * -0.5))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 720.0)
		tmp = cos(M) * exp((n * (m * -0.5)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 720.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(m * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 720

   1. Initial program 77.3%

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

   3. Taylor expanded in n around inf 37.3%

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

   4. Taylor expanded in K around 0 47.6%

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

      1. cos-neg 47.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in m around inf 32.4%

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

      1. associate-*r* 32.4%

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

      2. *-commutative 32.4%

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

   9. Simplified 32.4%

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

   if 720 < l

   1. Initial program 81.4%

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

   3. Taylor expanded in l around inf 81.4%

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

      1. mul-1-neg 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(m \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;e^{-0.5 \cdot \left(n \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 720.0) (exp (* -0.5 (* n m))) (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = exp((-0.5 * (n * m)));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 720.0d0) then
        tmp = exp(((-0.5d0) * (n * m)))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = Math.exp((-0.5 * (n * m)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 720.0:
		tmp = math.exp((-0.5 * (n * m)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 720.0)
		tmp = exp(Float64(-0.5 * Float64(n * m)));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 720.0)
		tmp = exp((-0.5 * (n * m)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 720.0], N[Exp[N[(-0.5 * N[(n * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 720

   1. Initial program 77.3%

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

   3. Taylor expanded in n around inf 37.3%

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

   4. Taylor expanded in K around 0 47.6%

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

      1. cos-neg 47.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in m around inf 32.4%

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

      1. associate-*r* 32.4%

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

      2. *-commutative 32.4%

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

   9. Simplified 32.4%

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

   10. Taylor expanded in M around 0 32.4%

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

   if 720 < l

   1. Initial program 81.4%

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

   3. Taylor expanded in l around inf 81.4%

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

      1. mul-1-neg 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;e^{-0.5 \cdot \left(n \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{+57} \lor \neg \left(m \leq 1.15 \cdot 10^{-256}\right):\\ \;\;\;\;e^{-0.5 \cdot \left(n \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -3e+57) (not (<= m 1.15e-256)))
   (exp (* -0.5 (* n m)))
   (*
    (cos (- (/ (* (+ n m) K) 2.0) M))
    (+ 1.0 (* l (+ (* l (+ 0.5 (* l -0.16666666666666666))) -1.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -3e+57) || !(m <= 1.15e-256)) {
		tmp = exp((-0.5 * (n * m)));
	} else {
		tmp = cos(((((n + m) * K) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	}
	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 <= (-3d+57)) .or. (.not. (m <= 1.15d-256))) then
        tmp = exp(((-0.5d0) * (n * m)))
    else
        tmp = cos(((((n + m) * k) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * (0.5d0 + (l * (-0.16666666666666666d0)))) + (-1.0d0))))
    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 <= -3e+57) || !(m <= 1.15e-256)) {
		tmp = Math.exp((-0.5 * (n * m)));
	} else {
		tmp = Math.cos(((((n + m) * K) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -3e+57) or not (m <= 1.15e-256):
		tmp = math.exp((-0.5 * (n * m)))
	else:
		tmp = math.cos(((((n + m) * K) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -3e+57) || !(m <= 1.15e-256))
		tmp = exp(Float64(-0.5 * Float64(n * m)));
	else
		tmp = Float64(cos(Float64(Float64(Float64(Float64(n + m) * K) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * Float64(0.5 + Float64(l * -0.16666666666666666))) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -3e+57) || ~((m <= 1.15e-256)))
		tmp = exp((-0.5 * (n * m)));
	else
		tmp = cos(((((n + m) * K) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -3e+57], N[Not[LessEqual[m, 1.15e-256]], $MachinePrecision]], N[Exp[N[(-0.5 * N[(n * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(N[(N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * N[(0.5 + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3e57 or 1.15e-256 < m

   1. Initial program 76.0%

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

   3. Taylor expanded in n around inf 33.1%

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

   4. Taylor expanded in K around 0 45.7%

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

      1. cos-neg 45.7%

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

   6. Simplified 45.7%

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

   7. Taylor expanded in m around inf 36.4%

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

      1. associate-*r* 36.4%

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

      2. *-commutative 36.4%

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

   9. Simplified 36.4%

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

   10. Taylor expanded in M around 0 37.0%

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

   if -3e57 < m < 1.15e-256

   1. Initial program 83.2%

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

   3. Taylor expanded in l around inf 44.0%

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

      1. mul-1-neg 44.0%

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

   5. Simplified 44.0%

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

   6. Taylor expanded in l around 0 14.9%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + -0.16666666666666666 \cdot \ell\right) - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{+57} \lor \neg \left(m \leq 1.15 \cdot 10^{-256}\right):\\ \;\;\;\;e^{-0.5 \cdot \left(n \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.2 \cdot 10^{+41} \lor \neg \left(m \leq 2.4 \cdot 10^{-256}\right):\\ \;\;\;\;e^{-0.5 \cdot \left(n \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -6.2e+41) (not (<= m 2.4e-256)))
   (exp (* -0.5 (* n m)))
   (* (cos (- (/ (* (+ n m) K) 2.0) M)) (+ 1.0 (* l (+ (* l 0.5) -1.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -6.2e+41) || !(m <= 2.4e-256)) {
		tmp = exp((-0.5 * (n * m)));
	} else {
		tmp = cos(((((n + m) * K) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
	}
	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 <= (-6.2d+41)) .or. (.not. (m <= 2.4d-256))) then
        tmp = exp(((-0.5d0) * (n * m)))
    else
        tmp = cos(((((n + m) * k) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * 0.5d0) + (-1.0d0))))
    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 <= -6.2e+41) || !(m <= 2.4e-256)) {
		tmp = Math.exp((-0.5 * (n * m)));
	} else {
		tmp = Math.cos(((((n + m) * K) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -6.2e+41) or not (m <= 2.4e-256):
		tmp = math.exp((-0.5 * (n * m)))
	else:
		tmp = math.cos(((((n + m) * K) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -6.2e+41) || !(m <= 2.4e-256))
		tmp = exp(Float64(-0.5 * Float64(n * m)));
	else
		tmp = Float64(cos(Float64(Float64(Float64(Float64(n + m) * K) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * 0.5) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -6.2e+41) || ~((m <= 2.4e-256)))
		tmp = exp((-0.5 * (n * m)));
	else
		tmp = cos(((((n + m) * K) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -6.2e+41], N[Not[LessEqual[m, 2.4e-256]], $MachinePrecision]], N[Exp[N[(-0.5 * N[(n * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(N[(N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -6.2e41 or 2.3999999999999999e-256 < m

   1. Initial program 76.2%

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

   3. Taylor expanded in n around inf 33.5%

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

   4. Taylor expanded in K around 0 46.0%

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

      1. cos-neg 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in m around inf 36.2%

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

      1. associate-*r* 36.2%

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

      2. *-commutative 36.2%

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

   9. Simplified 36.2%

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

   10. Taylor expanded in M around 0 36.8%

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

   if -6.2e41 < m < 2.3999999999999999e-256

   1. Initial program 83.0%

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

   3. Taylor expanded in l around inf 44.5%

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

      1. mul-1-neg 44.5%

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

   5. Simplified 44.5%

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

   6. Taylor expanded in l around 0 15.0%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.2 \cdot 10^{+41} \lor \neg \left(m \leq 2.4 \cdot 10^{-256}\right):\\ \;\;\;\;e^{-0.5 \cdot \left(n \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in n around inf 36.3%

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

4. Taylor expanded in K around 0 48.1%

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

   1. cos-neg 48.1%

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

6. Simplified 48.1%

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

7. Taylor expanded in m around inf 31.9%

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

   1. associate-*r* 31.9%

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

   2. *-commutative 31.9%

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

9. Simplified 31.9%

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

10. Taylor expanded in M around 0 31.9%

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

11. Final simplification 31.9%

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

Alternative 11: 7.4% 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 78.5%

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

3. Taylor expanded in l around inf 33.7%

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

   1. mul-1-neg 33.7%

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

5. Simplified 33.7%

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

6. Taylor expanded in l around 0 4.8%

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

7. Taylor expanded in K around 0 5.4%

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

   1. cos-neg 5.4%

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

9. Simplified 5.4%

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

Reproduce

herbie shell --seed 2024155 
(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)))))))