Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.9% → 96.7%

Time: 9.5s

Alternatives: 12

Speedup: 2.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \cdot \frac{1}{\frac{1}{\cos M}} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (exp (- (- (fabs (- n m)) l) (pow (- (/ (+ n m) 2.0) M) 2.0)))
  (/ 1.0 (/ 1.0 (cos M)))))

C

double code(double K, double m, double n, double M, double l) {
	return exp(((fabs((n - m)) - l) - pow((((n + m) / 2.0) - M), 2.0))) * (1.0 / (1.0 / 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) - ((((n + m) / 2.0d0) - m_1) ** 2.0d0))) * (1.0d0 / (1.0d0 / 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((((n + m) / 2.0) - M), 2.0))) * (1.0 / (1.0 / Math.cos(M)));
}

Python

def code(K, m, n, M, l):
	return math.exp(((math.fabs((n - m)) - l) - math.pow((((n + m) / 2.0) - M), 2.0))) * (1.0 / (1.0 / math.cos(M)))

Julia

function code(K, m, n, M, l)
	return Float64(exp(Float64(Float64(abs(Float64(n - m)) - l) - (Float64(Float64(Float64(n + m) / 2.0) - M) ^ 2.0))) * Float64(1.0 / Float64(1.0 / cos(M))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((abs((n - m)) - l) - ((((n + m) / 2.0) - M) ^ 2.0))) * (1.0 / (1.0 / cos(M)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(1.0 / N[Cos[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. sin-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. cos-neg N/A

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

   13. lower-cos.f64 81.6

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

5. Applied rewrites 81.6%

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

7. Applied rewrites 59.0%

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

8. Taylor expanded in K around 0

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

10. Applied rewrites 95.9%

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

11. Final simplification 95.9%

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

Alternative 2: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 95.9%

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

6. Final simplification 95.9%

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

Alternative 3: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M}\\ \mathbf{if}\;M \leq -9:\\ \;\;\;\;t\_0 \cdot \cos M\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* (- M) M))))
   (if (<= M -9.0)
     (* t_0 (cos M))
     (if (<= M 26.5)
       (*
        (fma (* (* (+ n m) M) K) 0.5 1.0)
        (exp (- (- (fabs (- n m)) l) (pow (- (/ (+ n m) 2.0) M) 2.0))))
       (* t_0 1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M));
	double tmp;
	if (M <= -9.0) {
		tmp = t_0 * cos(M);
	} else if (M <= 26.5) {
		tmp = fma((((n + m) * M) * K), 0.5, 1.0) * exp(((fabs((n - m)) - l) - pow((((n + m) / 2.0) - M), 2.0)));
	} else {
		tmp = t_0 * 1.0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(-M) * M))
	tmp = 0.0
	if (M <= -9.0)
		tmp = Float64(t_0 * cos(M));
	elseif (M <= 26.5)
		tmp = Float64(fma(Float64(Float64(Float64(n + m) * M) * K), 0.5, 1.0) * exp(Float64(Float64(abs(Float64(n - m)) - l) - (Float64(Float64(Float64(n + m) / 2.0) - M) ^ 2.0))));
	else
		tmp = Float64(t_0 * 1.0);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -9.0], N[(t$95$0 * N[Cos[M], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 26.5], N[(N[(N[(N[(N[(n + m), $MachinePrecision] * M), $MachinePrecision] * K), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < -9

   1. Initial program 78.6%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 98.3%

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

   if -9 < M < 26.5

   1. Initial program 68.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 86.2%

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

   if 26.5 < M

   1. Initial program 75.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 43.1%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

       3. distribute-lft-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 41.7

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

   11. Applied rewrites 41.7%

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

   12. Taylor expanded in K around 0

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]
   13. Step-by-step derivation

   14. Applied rewrites 98.6%

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -9:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, -0.5, 1\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (/ 1.0 (fma (* (* (+ n m) M) K) -0.5 1.0))
  (exp (- (- (fabs (- n m)) l) (pow (- (/ (+ n m) 2.0) M) 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return (1.0 / fma((((n + m) * M) * K), -0.5, 1.0)) * exp(((fabs((n - m)) - l) - pow((((n + m) / 2.0) - M), 2.0)));
}

Julia

function code(K, m, n, M, l)
	return Float64(Float64(1.0 / fma(Float64(Float64(Float64(n + m) * M) * K), -0.5, 1.0)) * exp(Float64(Float64(abs(Float64(n - m)) - l) - (Float64(Float64(Float64(n + m) / 2.0) - M) ^ 2.0))))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[(1.0 / N[(N[(N[(N[(n + m), $MachinePrecision] * M), $MachinePrecision] * K), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. sin-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. cos-neg N/A

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

   13. lower-cos.f64 81.6

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

5. Applied rewrites 81.6%

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

7. Applied rewrites 59.0%

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

8. Taylor expanded in M around 0

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

10. Applied rewrites 94.9%

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

11. Final simplification 94.9%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, -0.5, 1\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \]
12. Add Preprocessing

Alternative 5: 80.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right| - \ell\\ t_1 := e^{\left(-M\right) \cdot M}\\ t_2 := \mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right)\\ \mathbf{if}\;M \leq -7.6:\\ \;\;\;\;t\_1 \cdot \cos M\\ \mathbf{elif}\;M \leq -2 \cdot 10^{-292}:\\ \;\;\;\;e^{t\_0 - 0.25 \cdot \left(m \cdot m\right)} \cdot t\_2\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;e^{t\_0 - \left(n \cdot n\right) \cdot 0.25} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l))
        (t_1 (exp (* (- M) M)))
        (t_2 (fma (* (* (+ n m) M) K) 0.5 1.0)))
   (if (<= M -7.6)
     (* t_1 (cos M))
     (if (<= M -2e-292)
       (* (exp (- t_0 (* 0.25 (* m m)))) t_2)
       (if (<= M 26.5) (* (exp (- t_0 (* (* n n) 0.25))) t_2) (* t_1 1.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m)) - l;
	double t_1 = exp((-M * M));
	double t_2 = fma((((n + m) * M) * K), 0.5, 1.0);
	double tmp;
	if (M <= -7.6) {
		tmp = t_1 * cos(M);
	} else if (M <= -2e-292) {
		tmp = exp((t_0 - (0.25 * (m * m)))) * t_2;
	} else if (M <= 26.5) {
		tmp = exp((t_0 - ((n * n) * 0.25))) * t_2;
	} else {
		tmp = t_1 * 1.0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(n - m)) - l)
	t_1 = exp(Float64(Float64(-M) * M))
	t_2 = fma(Float64(Float64(Float64(n + m) * M) * K), 0.5, 1.0)
	tmp = 0.0
	if (M <= -7.6)
		tmp = Float64(t_1 * cos(M));
	elseif (M <= -2e-292)
		tmp = Float64(exp(Float64(t_0 - Float64(0.25 * Float64(m * m)))) * t_2);
	elseif (M <= 26.5)
		tmp = Float64(exp(Float64(t_0 - Float64(Float64(n * n) * 0.25))) * t_2);
	else
		tmp = Float64(t_1 * 1.0);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(n + m), $MachinePrecision] * M), $MachinePrecision] * K), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[M, -7.6], N[(t$95$1 * N[Cos[M], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, -2e-292], N[(N[Exp[N[(t$95$0 - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[M, 26.5], N[(N[Exp[N[(t$95$0 - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], N[(t$95$1 * 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -7.5999999999999996

   1. Initial program 78.6%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 98.3%

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

   if -7.5999999999999996 < M < -2.0000000000000001e-292

   1. Initial program 70.8%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in m around inf

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

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{{m}^{2} \cdot \frac{1}{4}}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{{m}^{2} \cdot \frac{1}{4}}\right) - \left(\ell - \left|m - n\right|\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{\left(m \cdot m\right)} \cdot \frac{1}{4}\right) - \left(\ell - \left|m - n\right|\right)} \]

       4. lower-*.f64 63.0

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\left(-\color{blue}{\left(m \cdot m\right)} \cdot 0.25\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Applied rewrites 63.0%

       \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\left(-\color{blue}{\left(m \cdot m\right) \cdot 0.25}\right) - \left(\ell - \left|m - n\right|\right)} \]

   if -2.0000000000000001e-292 < M < 26.5

   1. Initial program 65.4%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 83.3%

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

   9. Taylor expanded in n around inf

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

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{{n}^{2} \cdot \frac{1}{4}}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{{n}^{2} \cdot \frac{1}{4}}\right) - \left(\ell - \left|m - n\right|\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{\left(n \cdot n\right)} \cdot \frac{1}{4}\right) - \left(\ell - \left|m - n\right|\right)} \]

       4. lower-*.f64 58.3

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\left(-\color{blue}{\left(n \cdot n\right)} \cdot 0.25\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Applied rewrites 58.3%

       \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\left(-\color{blue}{\left(n \cdot n\right) \cdot 0.25}\right) - \left(\ell - \left|m - n\right|\right)} \]

   if 26.5 < M

   1. Initial program 75.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 43.1%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

       3. distribute-lft-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 41.7

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

   11. Applied rewrites 41.7%

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

   12. Taylor expanded in K around 0

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]
   13. Step-by-step derivation

   14. Applied rewrites 98.6%

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7.6:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{elif}\;M \leq -2 \cdot 10^{-292}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - 0.25 \cdot \left(m \cdot m\right)} \cdot \mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right)\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25} \cdot \mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right| - \ell\\ t_1 := e^{\left(-M\right) \cdot M} \cdot 1\\ t_2 := \mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right)\\ \mathbf{if}\;M \leq -9:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq -1.15 \cdot 10^{-271}:\\ \;\;\;\;e^{t\_0 - 0.25 \cdot \left(m \cdot m\right)} \cdot t\_2\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;e^{t\_0 - \left(n \cdot n\right) \cdot 0.25} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l))
        (t_1 (* (exp (* (- M) M)) 1.0))
        (t_2 (fma (* (* (+ n m) M) K) 0.5 1.0)))
   (if (<= M -9.0)
     t_1
     (if (<= M -1.15e-271)
       (* (exp (- t_0 (* 0.25 (* m m)))) t_2)
       (if (<= M 26.5) (* (exp (- t_0 (* (* n n) 0.25))) t_2) t_1)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m)) - l;
	double t_1 = exp((-M * M)) * 1.0;
	double t_2 = fma((((n + m) * M) * K), 0.5, 1.0);
	double tmp;
	if (M <= -9.0) {
		tmp = t_1;
	} else if (M <= -1.15e-271) {
		tmp = exp((t_0 - (0.25 * (m * m)))) * t_2;
	} else if (M <= 26.5) {
		tmp = exp((t_0 - ((n * n) * 0.25))) * t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(n - m)) - l)
	t_1 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	t_2 = fma(Float64(Float64(Float64(n + m) * M) * K), 0.5, 1.0)
	tmp = 0.0
	if (M <= -9.0)
		tmp = t_1;
	elseif (M <= -1.15e-271)
		tmp = Float64(exp(Float64(t_0 - Float64(0.25 * Float64(m * m)))) * t_2);
	elseif (M <= 26.5)
		tmp = Float64(exp(Float64(t_0 - Float64(Float64(n * n) * 0.25))) * t_2);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(n + m), $MachinePrecision] * M), $MachinePrecision] * K), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[M, -9.0], t$95$1, If[LessEqual[M, -1.15e-271], N[(N[Exp[N[(t$95$0 - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[M, 26.5], N[(N[Exp[N[(t$95$0 - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -9 or 26.5 < M

   1. Initial program 76.6%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 41.4%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

       3. distribute-lft-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 40.6

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

   11. Applied rewrites 40.6%

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

   12. Taylor expanded in K around 0

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]
   13. Step-by-step derivation

   14. Applied rewrites 97.8%

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]

   if -9 < M < -1.15000000000000004e-271

   1. Initial program 72.8%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 90.2%

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

   9. Taylor expanded in m around inf

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

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{{m}^{2} \cdot \frac{1}{4}}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{{m}^{2} \cdot \frac{1}{4}}\right) - \left(\ell - \left|m - n\right|\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{\left(m \cdot m\right)} \cdot \frac{1}{4}\right) - \left(\ell - \left|m - n\right|\right)} \]

       4. lower-*.f64 63.1

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\left(-\color{blue}{\left(m \cdot m\right)} \cdot 0.25\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Applied rewrites 63.1%

       \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\left(-\color{blue}{\left(m \cdot m\right) \cdot 0.25}\right) - \left(\ell - \left|m - n\right|\right)} \]

   if -1.15000000000000004e-271 < M < 26.5

   1. Initial program 63.7%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in n around inf

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

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{{n}^{2} \cdot \frac{1}{4}}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{{n}^{2} \cdot \frac{1}{4}}\right) - \left(\ell - \left|m - n\right|\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{\left(n \cdot n\right)} \cdot \frac{1}{4}\right) - \left(\ell - \left|m - n\right|\right)} \]

       4. lower-*.f64 58.4

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\left(-\color{blue}{\left(n \cdot n\right)} \cdot 0.25\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Applied rewrites 58.4%

       \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\left(-\color{blue}{\left(n \cdot n\right) \cdot 0.25}\right) - \left(\ell - \left|m - n\right|\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -9:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq -1.15 \cdot 10^{-271}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - 0.25 \cdot \left(m \cdot m\right)} \cdot \mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right)\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25} \cdot \mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq -7.5 \cdot 10^{-268}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - 0.25 \cdot \left(m \cdot m\right)} \cdot \mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right)\\ \mathbf{elif}\;M \leq 3.35 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -9.0)
     t_0
     (if (<= M -7.5e-268)
       (*
        (exp (- (- (fabs (- n m)) l) (* 0.25 (* m m))))
        (fma (* (* (+ n m) M) K) 0.5 1.0))
       (if (<= M 3.35e+16)
         (* (fma (* M M) -0.5 1.0) (exp (* -0.25 (* n n))))
         t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -9.0) {
		tmp = t_0;
	} else if (M <= -7.5e-268) {
		tmp = exp(((fabs((n - m)) - l) - (0.25 * (m * m)))) * fma((((n + m) * M) * K), 0.5, 1.0);
	} else if (M <= 3.35e+16) {
		tmp = fma((M * M), -0.5, 1.0) * exp((-0.25 * (n * n)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -9.0)
		tmp = t_0;
	elseif (M <= -7.5e-268)
		tmp = Float64(exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(0.25 * Float64(m * m)))) * fma(Float64(Float64(Float64(n + m) * M) * K), 0.5, 1.0));
	elseif (M <= 3.35e+16)
		tmp = Float64(fma(Float64(M * M), -0.5, 1.0) * exp(Float64(-0.25 * Float64(n * n))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -9.0], t$95$0, If[LessEqual[M, -7.5e-268], N[(N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(n + m), $MachinePrecision] * M), $MachinePrecision] * K), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 3.35e+16], N[(N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if M < -9 or 3.35e16 < M

   1. Initial program 77.8%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 41.2%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

       3. distribute-lft-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 41.2

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

   11. Applied rewrites 41.2%

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

   12. Taylor expanded in K around 0

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]

   if -9 < M < -7.4999999999999999e-268

   1. Initial program 74.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in m around inf

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

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{{m}^{2} \cdot \frac{1}{4}}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{{m}^{2} \cdot \frac{1}{4}}\right) - \left(\ell - \left|m - n\right|\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\left(-\color{blue}{\left(m \cdot m\right)} \cdot \frac{1}{4}\right) - \left(\ell - \left|m - n\right|\right)} \]

       4. lower-*.f64 62.2

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\left(-\color{blue}{\left(m \cdot m\right)} \cdot 0.25\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Applied rewrites 62.2%

       \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\left(-\color{blue}{\left(m \cdot m\right) \cdot 0.25}\right) - \left(\ell - \left|m - n\right|\right)} \]

   if -7.4999999999999999e-268 < M < 3.35e16

   1. Initial program 61.2%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in n around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {n}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 53.5%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 53.5%

       \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -9:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq -7.5 \cdot 10^{-268}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - 0.25 \cdot \left(m \cdot m\right)} \cdot \mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right)\\ \mathbf{elif}\;M \leq 3.35 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -1.35 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq -7.5 \cdot 10^{-268}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right)\\ \mathbf{elif}\;M \leq 3.35 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -1.35e-15)
     t_0
     (if (<= M -7.5e-268)
       (* (exp (* -0.25 (* m m))) (fma (* (* (+ n m) M) K) 0.5 1.0))
       (if (<= M 3.35e+16)
         (* (fma (* M M) -0.5 1.0) (exp (* -0.25 (* n n))))
         t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -1.35e-15) {
		tmp = t_0;
	} else if (M <= -7.5e-268) {
		tmp = exp((-0.25 * (m * m))) * fma((((n + m) * M) * K), 0.5, 1.0);
	} else if (M <= 3.35e+16) {
		tmp = fma((M * M), -0.5, 1.0) * exp((-0.25 * (n * n)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -1.35e-15)
		tmp = t_0;
	elseif (M <= -7.5e-268)
		tmp = Float64(exp(Float64(-0.25 * Float64(m * m))) * fma(Float64(Float64(Float64(n + m) * M) * K), 0.5, 1.0));
	elseif (M <= 3.35e+16)
		tmp = Float64(fma(Float64(M * M), -0.5, 1.0) * exp(Float64(-0.25 * Float64(n * n))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -1.35e-15], t$95$0, If[LessEqual[M, -7.5e-268], N[(N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(n + m), $MachinePrecision] * M), $MachinePrecision] * K), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 3.35e+16], N[(N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if M < -1.35000000000000005e-15 or 3.35e16 < M

   1. Initial program 77.4%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 41.4%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

       3. distribute-lft-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 40.7

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

   11. Applied rewrites 40.7%

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

   12. Taylor expanded in K around 0

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]
   13. Step-by-step derivation

   14. Applied rewrites 97.1%

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]

   if -1.35000000000000005e-15 < M < -7.4999999999999999e-268

   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 K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 91.0%

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

   9. Taylor expanded in m around inf

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

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, \frac{1}{2}, 1\right) \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 49.8

          \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   11. Applied rewrites 49.8%

       \[\leadsto \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right) \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if -7.4999999999999999e-268 < M < 3.35e16

   1. Initial program 61.2%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in n around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {n}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 53.5%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 53.5%

       \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.35 \cdot 10^{-15}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq -7.5 \cdot 10^{-268}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \mathsf{fma}\left(\left(\left(n + m\right) \cdot M\right) \cdot K, 0.5, 1\right)\\ \mathbf{elif}\;M \leq 3.35 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -27:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 3.35 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -27.0)
     t_0
     (if (<= M 3.35e+16)
       (* (fma (* M M) -0.5 1.0) (exp (* -0.25 (* n n))))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -27.0) {
		tmp = t_0;
	} else if (M <= 3.35e+16) {
		tmp = fma((M * M), -0.5, 1.0) * exp((-0.25 * (n * n)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -27.0)
		tmp = t_0;
	elseif (M <= 3.35e+16)
		tmp = Float64(fma(Float64(M * M), -0.5, 1.0) * exp(Float64(-0.25 * Float64(n * n))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -27.0], t$95$0, If[LessEqual[M, 3.35e+16], N[(N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -27 or 3.35e16 < M

   1. Initial program 77.7%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 41.5%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

       3. distribute-lft-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 41.6

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

   11. Applied rewrites 41.6%

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

   12. Taylor expanded in K around 0

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]
   13. Step-by-step derivation

   14. Applied rewrites 99.2%

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]

   if -27 < M < 3.35e16

   1. Initial program 67.1%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in n around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {n}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 54.7%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 54.7%

       \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -27:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq 3.35 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -2.8 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq -4.9 \cdot 10^{-206}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;M \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -2.8e+14)
     t_1
     (if (<= M -4.9e-206)
       (* t_0 (fma (* M M) -0.5 1.0))
       (if (<= M 3.2e-15) (* (* (* M M) -0.5) t_0) t_1)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double t_1 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -2.8e+14) {
		tmp = t_1;
	} else if (M <= -4.9e-206) {
		tmp = t_0 * fma((M * M), -0.5, 1.0);
	} else if (M <= 3.2e-15) {
		tmp = ((M * M) * -0.5) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	t_1 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -2.8e+14)
		tmp = t_1;
	elseif (M <= -4.9e-206)
		tmp = Float64(t_0 * fma(Float64(M * M), -0.5, 1.0));
	elseif (M <= 3.2e-15)
		tmp = Float64(Float64(Float64(M * M) * -0.5) * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -2.8e+14], t$95$1, If[LessEqual[M, -4.9e-206], N[(t$95$0 * N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 3.2e-15], N[(N[(N[(M * M), $MachinePrecision] * -0.5), $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -2.8e14 or 3.1999999999999999e-15 < M

   1. Initial program 76.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 42.4%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

       3. distribute-lft-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 40.2

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

   11. Applied rewrites 40.2%

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

   12. Taylor expanded in K around 0

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]
   13. Step-by-step derivation

   14. Applied rewrites 97.8%

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]

   if -2.8e14 < M < -4.9e-206

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 50.9

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

   8. Applied rewrites 50.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 50.9%

       \[\leadsto \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \cdot e^{-\ell} \]

   if -4.9e-206 < M < 3.1999999999999999e-15

   1. Initial program 62.3%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 31.0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 42.2

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

   8. Applied rewrites 42.2%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 42.2%

       \[\leadsto \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \cdot e^{-\ell} \]

   12. Taylor expanded in M around inf

       \[\leadsto \left(\frac{-1}{2} \cdot {M}^{\color{blue}{2}}\right) \cdot e^{-\ell} \]
   13. Step-by-step derivation

   14. Applied rewrites 56.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.8 \cdot 10^{+14}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq -4.9 \cdot 10^{-206}:\\ \;\;\;\;e^{-\ell} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;M \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -2.8 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -2.8e+14)
     t_0
     (if (<= M 3.2e-15) (* (* (* M M) -0.5) (exp (- l))) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -2.8e+14) {
		tmp = t_0;
	} else if (M <= 3.2e-15) {
		tmp = ((M * M) * -0.5) * 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((-m_1 * m_1)) * 1.0d0
    if (m_1 <= (-2.8d+14)) then
        tmp = t_0
    else if (m_1 <= 3.2d-15) then
        tmp = ((m_1 * m_1) * (-0.5d0)) * 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((-M * M)) * 1.0;
	double tmp;
	if (M <= -2.8e+14) {
		tmp = t_0;
	} else if (M <= 3.2e-15) {
		tmp = ((M * M) * -0.5) * Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-M * M)) * 1.0
	tmp = 0
	if M <= -2.8e+14:
		tmp = t_0
	elif M <= 3.2e-15:
		tmp = ((M * M) * -0.5) * math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -2.8e+14)
		tmp = t_0;
	elseif (M <= 3.2e-15)
		tmp = Float64(Float64(Float64(M * M) * -0.5) * exp(Float64(-l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-M * M)) * 1.0;
	tmp = 0.0;
	if (M <= -2.8e+14)
		tmp = t_0;
	elseif (M <= 3.2e-15)
		tmp = ((M * M) * -0.5) * exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -2.8e+14], t$95$0, If[LessEqual[M, 3.2e-15], N[(N[(N[(M * M), $MachinePrecision] * -0.5), $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -2.8e14 or 3.1999999999999999e-15 < M

   1. Initial program 76.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 42.4%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

       3. distribute-lft-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 40.2

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

   11. Applied rewrites 40.2%

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

   12. Taylor expanded in K around 0

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]
   13. Step-by-step derivation

   14. Applied rewrites 97.8%

       \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]

   if -2.8e14 < M < 3.1999999999999999e-15

   1. Initial program 68.2%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 37.6

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

   5. Applied rewrites 37.6%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 45.2

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

   8. Applied rewrites 45.2%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 45.2%

       \[\leadsto \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \cdot e^{-\ell} \]

   12. Taylor expanded in M around inf

       \[\leadsto \left(\frac{-1}{2} \cdot {M}^{\color{blue}{2}}\right) \cdot e^{-\ell} \]
   13. Step-by-step derivation

   14. Applied rewrites 47.7%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.8 \cdot 10^{+14}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ e^{\left(-M\right) \cdot M} \cdot 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((-M * M)) * 1.0;
}

Python

def code(K, m, n, M, l):
	return math.exp((-M * M)) * 1.0

Julia

function code(K, m, n, M, l)
	return Float64(exp(Float64(Float64(-M) * M)) * 1.0)
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. sin-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. cos-neg N/A

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

   13. lower-cos.f64 81.6

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

5. Applied rewrites 81.6%

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

6. Taylor expanded in M around 0

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

8. Applied rewrites 62.9%

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

9. Taylor expanded in M around inf

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

    1. mul-1-neg N/A

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

    2. unpow2 N/A

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

    3. distribute-lft-neg-in N/A

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

    4. lower-*.f64 N/A

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

    5. lower-neg.f64 28.4

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

11. Applied rewrites 28.4%

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

12. Taylor expanded in K around 0

    \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]
13. Step-by-step derivation

14. Applied rewrites 58.4%

    \[\leadsto 1 \cdot e^{\left(-M\right) \cdot M} \]

15. Final simplification 58.4%

    \[\leadsto e^{\left(-M\right) \cdot M} \cdot 1 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(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)))))))