Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.9%

Time: 10.5s

Alternatives: 13

Speedup: 1.5×

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: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

function code(K, m, n, M, l)
	return Float64(exp(Float64(abs(Float64(n - m)) - Float64((fma(Float64(n + m), 0.5, 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[(N[(n + m), $MachinePrecision] * 0.5 + (-M)), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.1%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 96.3%

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

6. Final simplification 96.3%

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

Alternative 2: 96.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.1%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 96.3%

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

7. Applied rewrites 96.3%

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

8. Taylor expanded in M around 0

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

10. Applied rewrites 95.9%

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

11. Final simplification 95.9%

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

Alternative 3: 75.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ t_1 := \left|n - m\right|\\ t_2 := {\left(e^{-\left(\left(t\_1 - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ t_3 := e^{t\_1 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot t\_0\\ \mathbf{if}\;M \leq -6.5 \cdot 10^{+109}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;M \leq -7.5 \cdot 10^{-194}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;M \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot t\_0\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;M \leq 5.4 \cdot 10^{+106}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma (* M M) -0.5 1.0))
        (t_1 (fabs (- n m)))
        (t_2 (* (pow (exp (- (- (- t_1 l) (* M M)))) -1.0) 1.0))
        (t_3 (* (exp (- t_1 (+ (* (* m m) 0.25) l))) t_0)))
   (if (<= M -6.5e+109)
     t_2
     (if (<= M -7.5e-194)
       t_3
       (if (<= M 1.35e-145)
         (* (exp (* (* n n) -0.25)) t_0)
         (if (<= M 27.0)
           t_3
           (if (<= M 5.4e+106) (* (exp (* (- M) M)) t_0) t_2)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma((M * M), -0.5, 1.0);
	double t_1 = fabs((n - m));
	double t_2 = pow(exp(-((t_1 - l) - (M * M))), -1.0) * 1.0;
	double t_3 = exp((t_1 - (((m * m) * 0.25) + l))) * t_0;
	double tmp;
	if (M <= -6.5e+109) {
		tmp = t_2;
	} else if (M <= -7.5e-194) {
		tmp = t_3;
	} else if (M <= 1.35e-145) {
		tmp = exp(((n * n) * -0.25)) * t_0;
	} else if (M <= 27.0) {
		tmp = t_3;
	} else if (M <= 5.4e+106) {
		tmp = exp((-M * M)) * t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(Float64(M * M), -0.5, 1.0)
	t_1 = abs(Float64(n - m))
	t_2 = Float64((exp(Float64(-Float64(Float64(t_1 - l) - Float64(M * M)))) ^ -1.0) * 1.0)
	t_3 = Float64(exp(Float64(t_1 - Float64(Float64(Float64(m * m) * 0.25) + l))) * t_0)
	tmp = 0.0
	if (M <= -6.5e+109)
		tmp = t_2;
	elseif (M <= -7.5e-194)
		tmp = t_3;
	elseif (M <= 1.35e-145)
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * t_0);
	elseif (M <= 27.0)
		tmp = t_3;
	elseif (M <= 5.4e+106)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Exp[(-N[(N[(t$95$1 - l), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision])], $MachinePrecision], -1.0], $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[N[(t$95$1 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[M, -6.5e+109], t$95$2, If[LessEqual[M, -7.5e-194], t$95$3, If[LessEqual[M, 1.35e-145], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[M, 27.0], t$95$3, If[LessEqual[M, 5.4e+106], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -6.5e109 or 5.40000000000000012e106 < M

   1. Initial program 71.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in M around inf

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

   13. Applied rewrites 100.0%

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

   if -6.5e109 < M < -7.4999999999999998e-194 or 1.35e-145 < M < 27

   1. Initial program 76.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 91.9%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 94.0%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 59.7%

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

   if -7.4999999999999998e-194 < M < 1.35e-145

   1. Initial program 61.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 96.6%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 61.3%

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

   if 27 < M < 5.40000000000000012e106

   1. Initial program 71.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 95.3%

       \[\leadsto e^{\left(-M\right) \cdot M} \cdot \mathsf{fma}\left(\color{blue}{M} \cdot M, -0.5, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6.5 \cdot 10^{+109}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ \mathbf{elif}\;M \leq -7.5 \cdot 10^{-194}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;M \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;M \leq 5.4 \cdot 10^{+106}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ t_1 := \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ t_2 := e^{\left(-M\right) \cdot M} \cdot t\_1\\ \mathbf{if}\;M \leq -2 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq -0.0017:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;M \leq -1 \cdot 10^{-183}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot t\_1\\ \mathbf{elif}\;M \leq 0.0082:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot t\_1\\ \mathbf{elif}\;M \leq 5.4 \cdot 10^{+106}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (pow (exp (- (- (- (fabs (- n m)) l) (* M M)))) -1.0) 1.0))
        (t_1 (fma (* M M) -0.5 1.0))
        (t_2 (* (exp (* (- M) M)) t_1)))
   (if (<= M -2e+151)
     t_0
     (if (<= M -0.0017)
       t_2
       (if (<= M -1e-183)
         (* (exp (* (* m m) -0.25)) t_1)
         (if (<= M 0.0082)
           (* (exp (* (* n n) -0.25)) t_1)
           (if (<= M 5.4e+106) t_2 t_0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = pow(exp(-((fabs((n - m)) - l) - (M * M))), -1.0) * 1.0;
	double t_1 = fma((M * M), -0.5, 1.0);
	double t_2 = exp((-M * M)) * t_1;
	double tmp;
	if (M <= -2e+151) {
		tmp = t_0;
	} else if (M <= -0.0017) {
		tmp = t_2;
	} else if (M <= -1e-183) {
		tmp = exp(((m * m) * -0.25)) * t_1;
	} else if (M <= 0.0082) {
		tmp = exp(((n * n) * -0.25)) * t_1;
	} else if (M <= 5.4e+106) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64((exp(Float64(-Float64(Float64(abs(Float64(n - m)) - l) - Float64(M * M)))) ^ -1.0) * 1.0)
	t_1 = fma(Float64(M * M), -0.5, 1.0)
	t_2 = Float64(exp(Float64(Float64(-M) * M)) * t_1)
	tmp = 0.0
	if (M <= -2e+151)
		tmp = t_0;
	elseif (M <= -0.0017)
		tmp = t_2;
	elseif (M <= -1e-183)
		tmp = Float64(exp(Float64(Float64(m * m) * -0.25)) * t_1);
	elseif (M <= 0.0082)
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * t_1);
	elseif (M <= 5.4e+106)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Power[N[Exp[(-N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision])], $MachinePrecision], -1.0], $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[M, -2e+151], t$95$0, If[LessEqual[M, -0.0017], t$95$2, If[LessEqual[M, -1e-183], N[(N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[M, 0.0082], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[M, 5.4e+106], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -2.00000000000000003e151 or 5.40000000000000012e106 < M

   1. Initial program 73.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in M around inf

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

   13. Applied rewrites 100.0%

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

   if -2.00000000000000003e151 < M < -0.00169999999999999991 or 0.00820000000000000069 < M < 5.40000000000000012e106

   1. Initial program 68.3%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 96.7%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 91.8%

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

   if -0.00169999999999999991 < M < -1.00000000000000001e-183

   1. Initial program 85.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 91.3%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 91.3%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 47.6%

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

   if -1.00000000000000001e-183 < M < 0.00820000000000000069

   1. Initial program 64.0%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 95.3%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 95.3%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 61.4%

       \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(\color{blue}{M} \cdot M, -0.5, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2 \cdot 10^{+151}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ \mathbf{elif}\;M \leq -0.0017:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;M \leq -1 \cdot 10^{-183}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;M \leq 0.0082:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;M \leq 5.4 \cdot 10^{+106}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ t_1 := 0.5 \cdot n - M\\ \mathbf{if}\;m \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;e^{t\_0 - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-\left(\left(t\_0 - \ell\right) - t\_1 \cdot \left(m + t\_1\right)\right)}\right)}^{-1} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))) (t_1 (- (* 0.5 n) M)))
   (if (<= m -1.55e-6)
     (exp (- t_0 (fma 0.25 (pow (+ n m) 2.0) l)))
     (* (pow (exp (- (- (- t_0 l) (* t_1 (+ m t_1))))) -1.0) 1.0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double t_1 = (0.5 * n) - M;
	double tmp;
	if (m <= -1.55e-6) {
		tmp = exp((t_0 - fma(0.25, pow((n + m), 2.0), l)));
	} else {
		tmp = pow(exp(-((t_0 - l) - (t_1 * (m + t_1)))), -1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	t_1 = Float64(Float64(0.5 * n) - M)
	tmp = 0.0
	if (m <= -1.55e-6)
		tmp = exp(Float64(t_0 - fma(0.25, (Float64(n + m) ^ 2.0), l)));
	else
		tmp = Float64((exp(Float64(-Float64(Float64(t_0 - l) - Float64(t_1 * Float64(m + t_1))))) ^ -1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * n), $MachinePrecision] - M), $MachinePrecision]}, If[LessEqual[m, -1.55e-6], N[Exp[N[(t$95$0 - N[(0.25 * N[Power[N[(n + m), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[Exp[(-N[(N[(t$95$0 - l), $MachinePrecision] - N[(t$95$1 * N[(m + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision], -1.0], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.55e-6

   1. Initial program 63.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 97.2%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 95.8%

      \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \]

   if -1.55e-6 < m

   1. Initial program 74.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 96.0%

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

   7. Applied rewrites 96.0%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 96.0%

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

   11. Taylor expanded in m around 0

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

   13. Applied rewrites 84.8%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - \left(0.5 \cdot n - M\right) \cdot \left(m + \left(0.5 \cdot n - M\right)\right)\right)}\right)}^{-1} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot n - M\\ \mathbf{if}\;m \leq -610:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - t\_0 \cdot \left(m + t\_0\right)\right)}\right)}^{-1} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 n) M)))
   (if (<= m -610.0)
     (* (exp (* (* m m) -0.25)) (cos M))
     (* (pow (exp (- (- (- (fabs (- n m)) l) (* t_0 (+ m t_0))))) -1.0) 1.0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * n) - M;
	double tmp;
	if (m <= -610.0) {
		tmp = exp(((m * m) * -0.25)) * cos(M);
	} else {
		tmp = pow(exp(-((fabs((n - m)) - l) - (t_0 * (m + t_0)))), -1.0) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * n) - m_1
    if (m <= (-610.0d0)) then
        tmp = exp(((m * m) * (-0.25d0))) * cos(m_1)
    else
        tmp = (exp(-((abs((n - m)) - l) - (t_0 * (m + t_0)))) ** (-1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * n) - M;
	double tmp;
	if (m <= -610.0) {
		tmp = Math.exp(((m * m) * -0.25)) * Math.cos(M);
	} else {
		tmp = Math.pow(Math.exp(-((Math.abs((n - m)) - l) - (t_0 * (m + t_0)))), -1.0) * 1.0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (0.5 * n) - M
	tmp = 0
	if m <= -610.0:
		tmp = math.exp(((m * m) * -0.25)) * math.cos(M)
	else:
		tmp = math.pow(math.exp(-((math.fabs((n - m)) - l) - (t_0 * (m + t_0)))), -1.0) * 1.0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * n) - M)
	tmp = 0.0
	if (m <= -610.0)
		tmp = Float64(exp(Float64(Float64(m * m) * -0.25)) * cos(M));
	else
		tmp = Float64((exp(Float64(-Float64(Float64(abs(Float64(n - m)) - l) - Float64(t_0 * Float64(m + t_0))))) ^ -1.0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (0.5 * n) - M;
	tmp = 0.0;
	if (m <= -610.0)
		tmp = exp(((m * m) * -0.25)) * cos(M);
	else
		tmp = (exp(-((abs((n - m)) - l) - (t_0 * (m + t_0)))) ^ -1.0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(0.5 * n), $MachinePrecision] - M), $MachinePrecision]}, If[LessEqual[m, -610.0], N[(N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Exp[(-N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(t$95$0 * N[(m + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision], -1.0], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -610

   1. Initial program 63.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 98.5%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 95.7%

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

   if -610 < m

   1. Initial program 74.0%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 95.5%

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

   7. Applied rewrites 95.5%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 95.5%

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

   11. Taylor expanded in m around 0

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

   13. Applied rewrites 84.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -610:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - \left(0.5 \cdot n - M\right) \cdot \left(m + \left(0.5 \cdot n - M\right)\right)\right)}\right)}^{-1} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot n - M\\ t_1 := \left|n - m\right|\\ \mathbf{if}\;m \leq -8.6 \cdot 10^{+95}:\\ \;\;\;\;e^{t\_1 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-\left(\left(t\_1 - \ell\right) - t\_0 \cdot \left(m + t\_0\right)\right)}\right)}^{-1} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 n) M)) (t_1 (fabs (- n m))))
   (if (<= m -8.6e+95)
     (* (exp (- t_1 (+ (* (* m m) 0.25) l))) (fma (* M M) -0.5 1.0))
     (* (pow (exp (- (- (- t_1 l) (* t_0 (+ m t_0))))) -1.0) 1.0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * n) - M;
	double t_1 = fabs((n - m));
	double tmp;
	if (m <= -8.6e+95) {
		tmp = exp((t_1 - (((m * m) * 0.25) + l))) * fma((M * M), -0.5, 1.0);
	} else {
		tmp = pow(exp(-((t_1 - l) - (t_0 * (m + t_0)))), -1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * n) - M)
	t_1 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -8.6e+95)
		tmp = Float64(exp(Float64(t_1 - Float64(Float64(Float64(m * m) * 0.25) + l))) * fma(Float64(M * M), -0.5, 1.0));
	else
		tmp = Float64((exp(Float64(-Float64(Float64(t_1 - l) - Float64(t_0 * Float64(m + t_0))))) ^ -1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(0.5 * n), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -8.6e+95], N[(N[Exp[N[(t$95$1 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Exp[(-N[(N[(t$95$1 - l), $MachinePrecision] - N[(t$95$0 * N[(m + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision], -1.0], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if m < -8.6e95

   1. Initial program 58.7%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 71.8%

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

   if -8.6e95 < m

   1. Initial program 73.9%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 95.5%

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

   7. Applied rewrites 95.5%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 95.0%

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

   11. Taylor expanded in m around 0

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

   13. Applied rewrites 83.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.6 \cdot 10^{+95}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - \left(0.5 \cdot n - M\right) \cdot \left(m + \left(0.5 \cdot n - M\right)\right)\right)}\right)}^{-1} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ t_1 := \left|n - m\right|\\ t_2 := {\left(e^{-\left(\left(t\_1 - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ \mathbf{if}\;M \leq -6.5 \cdot 10^{+109}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;M \leq -2 \cdot 10^{-211}:\\ \;\;\;\;e^{t\_1 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot t\_0\\ \mathbf{elif}\;M \leq 0.0082:\\ \;\;\;\;e^{t\_1 - \left(\left(n \cdot n\right) \cdot 0.25 + \ell\right)} \cdot t\_0\\ \mathbf{elif}\;M \leq 5.4 \cdot 10^{+106}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma (* M M) -0.5 1.0))
        (t_1 (fabs (- n m)))
        (t_2 (* (pow (exp (- (- (- t_1 l) (* M M)))) -1.0) 1.0)))
   (if (<= M -6.5e+109)
     t_2
     (if (<= M -2e-211)
       (* (exp (- t_1 (+ (* (* m m) 0.25) l))) t_0)
       (if (<= M 0.0082)
         (* (exp (- t_1 (+ (* (* n n) 0.25) l))) t_0)
         (if (<= M 5.4e+106) (* (exp (* (- M) M)) t_0) t_2))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma((M * M), -0.5, 1.0);
	double t_1 = fabs((n - m));
	double t_2 = pow(exp(-((t_1 - l) - (M * M))), -1.0) * 1.0;
	double tmp;
	if (M <= -6.5e+109) {
		tmp = t_2;
	} else if (M <= -2e-211) {
		tmp = exp((t_1 - (((m * m) * 0.25) + l))) * t_0;
	} else if (M <= 0.0082) {
		tmp = exp((t_1 - (((n * n) * 0.25) + l))) * t_0;
	} else if (M <= 5.4e+106) {
		tmp = exp((-M * M)) * t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(Float64(M * M), -0.5, 1.0)
	t_1 = abs(Float64(n - m))
	t_2 = Float64((exp(Float64(-Float64(Float64(t_1 - l) - Float64(M * M)))) ^ -1.0) * 1.0)
	tmp = 0.0
	if (M <= -6.5e+109)
		tmp = t_2;
	elseif (M <= -2e-211)
		tmp = Float64(exp(Float64(t_1 - Float64(Float64(Float64(m * m) * 0.25) + l))) * t_0);
	elseif (M <= 0.0082)
		tmp = Float64(exp(Float64(t_1 - Float64(Float64(Float64(n * n) * 0.25) + l))) * t_0);
	elseif (M <= 5.4e+106)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Exp[(-N[(N[(t$95$1 - l), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision])], $MachinePrecision], -1.0], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -6.5e+109], t$95$2, If[LessEqual[M, -2e-211], N[(N[Exp[N[(t$95$1 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[M, 0.0082], N[(N[Exp[N[(t$95$1 - N[(N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[M, 5.4e+106], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -6.5e109 or 5.40000000000000012e106 < M

   1. Initial program 71.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in M around inf

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

   13. Applied rewrites 100.0%

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

   if -6.5e109 < M < -2.00000000000000017e-211

   1. Initial program 78.0%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 91.8%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 94.8%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 61.5%

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

   if -2.00000000000000017e-211 < M < 0.00820000000000000069

   1. Initial program 64.3%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 95.2%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 68.2%

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

   if 0.00820000000000000069 < M < 5.40000000000000012e106

   1. Initial program 72.7%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 91.1%

       \[\leadsto e^{\left(-M\right) \cdot M} \cdot \mathsf{fma}\left(\color{blue}{M} \cdot M, -0.5, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6.5 \cdot 10^{+109}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ \mathbf{elif}\;M \leq -2 \cdot 10^{-211}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;M \leq 0.0082:\\ \;\;\;\;e^{\left|n - m\right| - \left(\left(n \cdot n\right) \cdot 0.25 + \ell\right)} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;M \leq 5.4 \cdot 10^{+106}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ t_1 := 0.5 \cdot n - M\\ t_2 := \left|n - m\right|\\ t_3 := {\left(e^{-\left(\left(t\_2 - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ \mathbf{if}\;M \leq -6.5 \cdot 10^{+109}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;M \leq -5 \cdot 10^{-184}:\\ \;\;\;\;e^{t\_2 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot t\_0\\ \mathbf{elif}\;M \leq 4.9 \cdot 10^{+106}:\\ \;\;\;\;e^{t\_2 - \left(t\_1 \cdot \left(m + t\_1\right) + \ell\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma (* M M) -0.5 1.0))
        (t_1 (- (* 0.5 n) M))
        (t_2 (fabs (- n m)))
        (t_3 (* (pow (exp (- (- (- t_2 l) (* M M)))) -1.0) 1.0)))
   (if (<= M -6.5e+109)
     t_3
     (if (<= M -5e-184)
       (* (exp (- t_2 (+ (* (* m m) 0.25) l))) t_0)
       (if (<= M 4.9e+106)
         (* (exp (- t_2 (+ (* t_1 (+ m t_1)) l))) t_0)
         t_3)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma((M * M), -0.5, 1.0);
	double t_1 = (0.5 * n) - M;
	double t_2 = fabs((n - m));
	double t_3 = pow(exp(-((t_2 - l) - (M * M))), -1.0) * 1.0;
	double tmp;
	if (M <= -6.5e+109) {
		tmp = t_3;
	} else if (M <= -5e-184) {
		tmp = exp((t_2 - (((m * m) * 0.25) + l))) * t_0;
	} else if (M <= 4.9e+106) {
		tmp = exp((t_2 - ((t_1 * (m + t_1)) + l))) * t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(Float64(M * M), -0.5, 1.0)
	t_1 = Float64(Float64(0.5 * n) - M)
	t_2 = abs(Float64(n - m))
	t_3 = Float64((exp(Float64(-Float64(Float64(t_2 - l) - Float64(M * M)))) ^ -1.0) * 1.0)
	tmp = 0.0
	if (M <= -6.5e+109)
		tmp = t_3;
	elseif (M <= -5e-184)
		tmp = Float64(exp(Float64(t_2 - Float64(Float64(Float64(m * m) * 0.25) + l))) * t_0);
	elseif (M <= 4.9e+106)
		tmp = Float64(exp(Float64(t_2 - Float64(Float64(t_1 * Float64(m + t_1)) + l))) * t_0);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * n), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Exp[(-N[(N[(t$95$2 - l), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision])], $MachinePrecision], -1.0], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -6.5e+109], t$95$3, If[LessEqual[M, -5e-184], N[(N[Exp[N[(t$95$2 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[M, 4.9e+106], N[(N[Exp[N[(t$95$2 - N[(N[(t$95$1 * N[(m + t$95$1), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -6.5e109 or 4.89999999999999998e106 < M

   1. Initial program 71.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in M around inf

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

   13. Applied rewrites 100.0%

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

   if -6.5e109 < M < -5.00000000000000003e-184

   1. Initial program 78.8%

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

   3. Taylor expanded in K around 0

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

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 94.6%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 61.8%

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

   if -5.00000000000000003e-184 < M < 4.89999999999999998e106

   1. Initial program 65.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 96.3%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 96.3%

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

   9. Taylor expanded in m around 0

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

   11. Applied rewrites 72.9%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6.5 \cdot 10^{+109}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ \mathbf{elif}\;M \leq -5 \cdot 10^{-184}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;M \leq 4.9 \cdot 10^{+106}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\left(0.5 \cdot n - M\right) \cdot \left(m + \left(0.5 \cdot n - M\right)\right) + \ell\right)} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-\left(\left(\left|n - m\right| - \ell\right) - M \cdot M\right)}\right)}^{-1} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ t_1 := e^{\left(m \cdot m\right) \cdot -0.25} \cdot t\_0\\ \mathbf{if}\;m \leq -2 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;m \leq -2.4 \cdot 10^{-219}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-22}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma (* M M) -0.5 1.0)) (t_1 (* (exp (* (* m m) -0.25)) t_0)))
   (if (<= m -2e+48)
     t_1
     (if (<= m -2.4e-219)
       (* 1.0 (exp (- l)))
       (if (<= m 8e-22) (* (exp (* (- M) M)) t_0) t_1)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma((M * M), -0.5, 1.0);
	double t_1 = exp(((m * m) * -0.25)) * t_0;
	double tmp;
	if (m <= -2e+48) {
		tmp = t_1;
	} else if (m <= -2.4e-219) {
		tmp = 1.0 * exp(-l);
	} else if (m <= 8e-22) {
		tmp = exp((-M * M)) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(Float64(M * M), -0.5, 1.0)
	t_1 = Float64(exp(Float64(Float64(m * m) * -0.25)) * t_0)
	tmp = 0.0
	if (m <= -2e+48)
		tmp = t_1;
	elseif (m <= -2.4e-219)
		tmp = Float64(1.0 * exp(Float64(-l)));
	elseif (m <= 8e-22)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[m, -2e+48], t$95$1, If[LessEqual[m, -2.4e-219], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 8e-22], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.00000000000000009e48 or 8.0000000000000004e-22 < m

   1. Initial program 64.3%

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

   3. Taylor expanded in K around 0

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

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 71.4%

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

   if -2.00000000000000009e48 < m < -2.40000000000000014e-219

   1. Initial program 73.0%

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

   3. Taylor expanded in l around inf

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   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 32.1

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

   5. Applied rewrites 32.1%

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

   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-rev N/A

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

      2. lower-cos.f64 37.6

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

   8. Applied rewrites 37.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 36.1%

       \[\leadsto 1 \cdot e^{-\ell} \]

   if -2.40000000000000014e-219 < m < 8.0000000000000004e-22

   1. Initial program 80.3%

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

   3. Taylor expanded in 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 93.9%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 71.2%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 36.2%

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

Alternative 11: 48.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{if}\;m \leq -2 \cdot 10^{+48}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot t\_0\\ \mathbf{elif}\;m \leq -2.1 \cdot 10^{-218}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma (* M M) -0.5 1.0)))
   (if (<= m -2e+48)
     (* (exp (* (* m m) -0.25)) t_0)
     (if (<= m -2.1e-218)
       (* 1.0 (exp (- l)))
       (* (exp (* (* n n) -0.25)) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma((M * M), -0.5, 1.0);
	double tmp;
	if (m <= -2e+48) {
		tmp = exp(((m * m) * -0.25)) * t_0;
	} else if (m <= -2.1e-218) {
		tmp = 1.0 * exp(-l);
	} else {
		tmp = exp(((n * n) * -0.25)) * t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(Float64(M * M), -0.5, 1.0)
	tmp = 0.0
	if (m <= -2e+48)
		tmp = Float64(exp(Float64(Float64(m * m) * -0.25)) * t_0);
	elseif (m <= -2.1e-218)
		tmp = Float64(1.0 * exp(Float64(-l)));
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * t_0);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[m, -2e+48], N[(N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[m, -2.1e-218], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.00000000000000009e48

   1. Initial program 59.3%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 77.8%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 76.0%

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

   if -2.00000000000000009e48 < m < -2.09999999999999994e-218

   1. Initial program 73.0%

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

   3. Taylor expanded in l around inf

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   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 32.1

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

   5. Applied rewrites 32.1%

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

   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-rev N/A

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

      2. lower-cos.f64 37.6

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

   8. Applied rewrites 37.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 36.1%

       \[\leadsto 1 \cdot e^{-\ell} \]

   if -2.09999999999999994e-218 < m

   1. Initial program 74.9%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 96.6%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 47.2%

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

Alternative 12: 46.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ t_1 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+66}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma (* M M) -0.5 1.0)) (t_1 (exp (- l))))
   (if (<= l -1.7e+66)
     (* t_0 t_1)
     (if (<= l 4.3e-7) (* (exp (* (- M) M)) t_0) (* 1.0 t_1)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma((M * M), -0.5, 1.0);
	double t_1 = exp(-l);
	double tmp;
	if (l <= -1.7e+66) {
		tmp = t_0 * t_1;
	} else if (l <= 4.3e-7) {
		tmp = exp((-M * M)) * t_0;
	} else {
		tmp = 1.0 * t_1;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(Float64(M * M), -0.5, 1.0)
	t_1 = exp(Float64(-l))
	tmp = 0.0
	if (l <= -1.7e+66)
		tmp = Float64(t_0 * t_1);
	elseif (l <= 4.3e-7)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * t_0);
	else
		tmp = Float64(1.0 * t_1);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[l, -1.7e+66], N[(t$95$0 * t$95$1), $MachinePrecision], If[LessEqual[l, 4.3e-7], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.70000000000000015e66

   1. Initial program 70.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 26.2

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

   5. Applied rewrites 26.2%

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

   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-rev N/A

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

      2. lower-cos.f64 26.5

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

   8. Applied rewrites 26.5%

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

   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 26.5%

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

   if -1.70000000000000015e66 < l < 4.3000000000000001e-7

   1. Initial program 68.3%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 97.7%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 69.9%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 33.3%

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

   if 4.3000000000000001e-7 < l

   1. Initial program 79.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 76.0

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

   5. Applied rewrites 76.0%

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

   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-rev N/A

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

      2. lower-cos.f64 95.0

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

   8. Applied rewrites 95.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 95.0%

       \[\leadsto 1 \cdot e^{-\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 35.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot e^{-\ell} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Taylor expanded in 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 26.7

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

5. Applied rewrites 26.7%

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

6. Taylor expanded in K around 0

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

   1. cos-neg-rev N/A

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

   2. lower-cos.f64 31.7

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

8. Applied rewrites 31.7%

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

9. Taylor expanded in M around 0

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

11. Applied rewrites 31.4%

    \[\leadsto 1 \cdot e^{-\ell} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(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)))))))