Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.4% → 96.5%

Time: 36.8s

Alternatives: 11

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% 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: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \left|n - m\right| - \ell\\ t_1 := e^{t_0 - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t_1 \leq \infty:\\ \;\;\;\;\cos \left(\frac{K}{-2 \cdot \frac{m}{{n}^{2}} + 2 \cdot \frac{1}{n}} - M\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - {\log \left(e^{\mathsf{fma}\left(m + n, 0.5, M\right)}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l))
        (t_1 (exp (- t_0 (pow (- (/ (+ m n) 2.0) M) 2.0)))))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_1) INFINITY)
     (* (cos (- (/ K (+ (* -2.0 (/ m (pow n 2.0))) (* 2.0 (/ 1.0 n)))) M)) t_1)
     (* (cos M) (exp (- t_0 (pow (log (exp (fma (+ m n) 0.5 M))) 2.0)))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m)) - l;
	double t_1 = exp((t_0 - pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_1) <= ((double) INFINITY)) {
		tmp = cos(((K / ((-2.0 * (m / pow(n, 2.0))) + (2.0 * (1.0 / n)))) - M)) * t_1;
	} else {
		tmp = cos(M) * exp((t_0 - pow(log(exp(fma((m + n), 0.5, M))), 2.0)));
	}
	return tmp;
}

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(n - m)) - l)
	t_1 = exp(Float64(t_0 - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_1) <= Inf)
		tmp = Float64(cos(Float64(Float64(K / Float64(Float64(-2.0 * Float64(m / (n ^ 2.0))) + Float64(2.0 * Float64(1.0 / n)))) - M)) * t_1);
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - (log(exp(fma(Float64(m + n), 0.5, M))) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(t$95$0 - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], Infinity], N[(N[Cos[N[(N[(K / N[(N[(-2.0 * N[(m / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[Power[N[Log[N[Exp[N[(N[(m + n), $MachinePrecision] * 0.5 + M), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.7%

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

      1. associate-/l* 96.0%

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

      2. +-commutative 96.0%

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

      3. fabs-sub 96.0%

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

      4. +-commutative 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in m around 0 95.3%

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

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

   1. Initial program 0.0%

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

      1. associate-/l* 0.0%

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

      2. +-commutative 0.0%

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

      3. fabs-sub 0.0%

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

      4. +-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in K around 0 96.0%

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

      1. cos-neg 96.0%

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

   7. Simplified 96.0%

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

      1. log1p-expm1-u 96.1%

         \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{m + n}{2} - M\right)\right)\right)}}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      2. div-inv 96.1%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      3. metadata-eval 96.1%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      4. fma-neg 96.1%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(m + n, 0.5, -M\right)}\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

   9. Applied egg-rr 96.1%

      \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)\right)\right)}}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 96.0%

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

       2. add-sqr-sqrt 40.0%

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

       3. sqrt-prod 96.0%

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

       4. unpow2 96.0%

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

       5. add-log-exp 96.1%

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

       6. unpow2 96.1%

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

       7. sqrt-prod 40.1%

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

       8. add-sqr-sqrt 96.1%

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

       9. add-sqr-sqrt 46.0%

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

       10. sqrt-unprod 96.1%

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

       11. sqr-neg 96.1%

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

       12. sqrt-unprod 50.1%

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

       13. add-sqr-sqrt 96.1%

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

   11. Applied egg-rr 96.1%

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

4. Final simplification 95.5%

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

Alternative 2: 96.6% accurate, 0.4× speedup

Math

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

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (- (fabs (- n m)) l) (pow (- (/ (+ m n) 2.0) M) 2.0)))))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_0) INFINITY)
     (* (cos (- (/ K (+ (* -2.0 (/ m (pow n 2.0))) (* 2.0 (/ 1.0 n)))) M)) t_0)
     (* t_0 (cos M)))))

C

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

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((Math.abs((n - m)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= Double.POSITIVE_INFINITY) {
		tmp = Math.cos(((K / ((-2.0 * (m / Math.pow(n, 2.0))) + (2.0 * (1.0 / n)))) - M)) * t_0;
	} else {
		tmp = t_0 * Math.cos(M);
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.exp(((math.fabs((n - m)) - l) - math.pow((((m + n) / 2.0) - M), 2.0)))
	tmp = 0
	if (math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= math.inf:
		tmp = math.cos(((K / ((-2.0 * (m / math.pow(n, 2.0))) + (2.0 * (1.0 / n)))) - M)) * t_0
	else:
		tmp = t_0 * math.cos(M)
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(abs(Float64(n - m)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_0) <= Inf)
		tmp = Float64(cos(Float64(Float64(K / Float64(Float64(-2.0 * Float64(m / (n ^ 2.0))) + Float64(2.0 * Float64(1.0 / n)))) - M)) * t_0);
	else
		tmp = Float64(t_0 * cos(M));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((abs((n - m)) - l) - ((((m + n) / 2.0) - M) ^ 2.0)));
	tmp = 0.0;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_0) <= Inf)
		tmp = cos(((K / ((-2.0 * (m / (n ^ 2.0))) + (2.0 * (1.0 / n)))) - M)) * t_0;
	else
		tmp = t_0 * cos(M);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], Infinity], N[(N[Cos[N[(N[(K / N[(N[(-2.0 * N[(m / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.7%

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

      1. associate-/l* 96.0%

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

      2. +-commutative 96.0%

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

      3. fabs-sub 96.0%

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

      4. +-commutative 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in m around 0 95.3%

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

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

   1. Initial program 0.0%

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

      1. associate-/l* 0.0%

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

      2. +-commutative 0.0%

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

      3. fabs-sub 0.0%

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

      4. +-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in K around 0 96.0%

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

      1. cos-neg 96.0%

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

   7. Simplified 96.0%

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

4. Final simplification 95.5%

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

Alternative 3: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \left|n - m\right| - \ell\\ t_1 := e^{t_0 - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t_1 \leq -0.5:\\ \;\;\;\;\cos \left(\frac{K}{-2 \cdot \frac{m}{{n}^{2}} + 2 \cdot \frac{1}{n}} - M\right) \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \cos M\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l))
        (t_1 (exp (- t_0 (pow (- (/ (+ m n) 2.0) M) 2.0)))))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_1) -0.5)
     (*
      (cos (- (/ K (+ (* -2.0 (/ m (pow n 2.0))) (* 2.0 (/ 1.0 n)))) M))
      (exp (+ (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) t_0)))
     (* t_1 (cos M)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m)) - l;
	double t_1 = exp((t_0 - pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_1) <= -0.5) {
		tmp = cos(((K / ((-2.0 * (m / pow(n, 2.0))) + (2.0 * (1.0 / n)))) - M)) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0));
	} else {
		tmp = t_1 * cos(M);
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((n - m)) - l
    t_1 = exp((t_0 - ((((m + n) / 2.0d0) - m_1) ** 2.0d0)))
    if ((cos((((k * (m + n)) / 2.0d0) - m_1)) * t_1) <= (-0.5d0)) then
        tmp = cos(((k / (((-2.0d0) * (m / (n ** 2.0d0))) + (2.0d0 * (1.0d0 / n)))) - m_1)) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) + t_0))
    else
        tmp = t_1 * cos(m_1)
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m)) - l;
	double t_1 = Math.exp((t_0 - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * t_1) <= -0.5) {
		tmp = Math.cos(((K / ((-2.0 * (m / Math.pow(n, 2.0))) + (2.0 * (1.0 / n)))) - M)) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0));
	} else {
		tmp = t_1 * Math.cos(M);
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.fabs((n - m)) - l
	t_1 = math.exp((t_0 - math.pow((((m + n) / 2.0) - M), 2.0)))
	tmp = 0
	if (math.cos((((K * (m + n)) / 2.0) - M)) * t_1) <= -0.5:
		tmp = math.cos(((K / ((-2.0 * (m / math.pow(n, 2.0))) + (2.0 * (1.0 / n)))) - M)) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0))
	else:
		tmp = t_1 * math.cos(M)
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(n - m)) - l)
	t_1 = exp(Float64(t_0 - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_1) <= -0.5)
		tmp = Float64(cos(Float64(Float64(K / Float64(Float64(-2.0 * Float64(m / (n ^ 2.0))) + Float64(2.0 * Float64(1.0 / n)))) - M)) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) + t_0)));
	else
		tmp = Float64(t_1 * cos(M));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m)) - l;
	t_1 = exp((t_0 - ((((m + n) / 2.0) - M) ^ 2.0)));
	tmp = 0.0;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_1) <= -0.5)
		tmp = cos(((K / ((-2.0 * (m / (n ^ 2.0))) + (2.0 * (1.0 / n)))) - M)) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0));
	else
		tmp = t_1 * cos(M);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(t$95$0 - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], -0.5], N[(N[Cos[N[(N[(K / N[(N[(-2.0 * N[(m / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) (-.f64 l (fabs.f64 (-.f64 m n)))))) < -0.5

   1. Initial program 53.9%

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

      1. associate-/l* 60.1%

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

      2. +-commutative 60.1%

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

      3. fabs-sub 60.1%

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

      4. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in m around 0 38.4%

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

   6. Taylor expanded in m around 0 38.4%

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

      1. +-commutative 9.7%

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

      2. unpow2 9.7%

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

      3. distribute-rgt-out 9.7%

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

      4. *-commutative 9.7%

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

      5. *-commutative 9.7%

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

   8. Simplified 38.4%

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

   if -0.5 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 78.1%

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

      1. associate-/l* 78.1%

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

      2. +-commutative 78.1%

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

      3. fabs-sub 78.1%

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

      4. +-commutative 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in K around 0 97.6%

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

      1. cos-neg 97.6%

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

   7. Simplified 97.6%

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

4. Final simplification 95.1%

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

Alternative 4: 96.3% accurate, 0.5× speedup

Math

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

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l))
        (t_1 (exp (- t_0 (pow (- (/ (+ m n) 2.0) M) 2.0)))))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_1) -0.5)
     (*
      (cos (* 0.5 (* K n)))
      (exp (+ (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) t_0)))
     (* t_1 (cos M)))))

C

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

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((n - m)) - l
    t_1 = exp((t_0 - ((((m + n) / 2.0d0) - m_1) ** 2.0d0)))
    if ((cos((((k * (m + n)) / 2.0d0) - m_1)) * t_1) <= (-0.5d0)) then
        tmp = cos((0.5d0 * (k * n))) * exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) + t_0))
    else
        tmp = t_1 * cos(m_1)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m)) - l;
	t_1 = exp((t_0 - ((((m + n) / 2.0) - M) ^ 2.0)));
	tmp = 0.0;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_1) <= -0.5)
		tmp = cos((0.5 * (K * n))) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + t_0));
	else
		tmp = t_1 * cos(M);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(t$95$0 - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], -0.5], N[(N[Cos[N[(0.5 * N[(K * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) (-.f64 l (fabs.f64 (-.f64 m n)))))) < -0.5

   1. Initial program 53.9%

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

      1. associate-/l* 60.1%

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

      2. +-commutative 60.1%

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

      3. fabs-sub 60.1%

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

      4. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in n around 0 60.1%

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

      1. +-commutative 9.7%

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

      2. unpow2 9.7%

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

      3. distribute-rgt-out 9.7%

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

      4. *-commutative 9.7%

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

      5. *-commutative 9.7%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in n around inf 47.5%

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

      1. *-commutative 47.5%

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

   10. Simplified 47.5%

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

   if -0.5 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 78.1%

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

      1. associate-/l* 78.1%

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

      2. +-commutative 78.1%

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

      3. fabs-sub 78.1%

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

      4. +-commutative 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in K around 0 97.6%

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

      1. cos-neg 97.6%

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

   7. Simplified 97.6%

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

4. Final simplification 95.5%

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

Alternative 5: 75.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \left|n - m\right| - \ell\\ t_1 := \cos M \cdot e^{t_0 - 0.5 \cdot \left(m \cdot \left(n + m \cdot 0.5\right)\right)}\\ \mathbf{if}\;m \leq -1 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;m \leq -1.2 \cdot 10^{-307}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot \left(\left(m + n\right) + M\right)}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{M \cdot \left(n - M\right) + t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l))
        (t_1 (* (cos M) (exp (- t_0 (* 0.5 (* m (+ n (* m 0.5)))))))))
   (if (<= m -1e+49)
     t_1
     (if (<= m -1.2e-307)
       (* (cos M) (exp (- t_0 (* M (+ (+ m n) M)))))
       (if (<= m 4.2e+17)
         (* (cos (* (+ m n) (* K 0.5))) (exp (+ (* M (- n M)) t_0)))
         t_1)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m)) - l;
	double t_1 = cos(M) * exp((t_0 - (0.5 * (m * (n + (m * 0.5))))));
	double tmp;
	if (m <= -1e+49) {
		tmp = t_1;
	} else if (m <= -1.2e-307) {
		tmp = cos(M) * exp((t_0 - (M * ((m + n) + M))));
	} else if (m <= 4.2e+17) {
		tmp = cos(((m + n) * (K * 0.5))) * exp(((M * (n - M)) + t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((n - m)) - l
    t_1 = cos(m_1) * exp((t_0 - (0.5d0 * (m * (n + (m * 0.5d0))))))
    if (m <= (-1d+49)) then
        tmp = t_1
    else if (m <= (-1.2d-307)) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * ((m + n) + m_1))))
    else if (m <= 4.2d+17) then
        tmp = cos(((m + n) * (k * 0.5d0))) * exp(((m_1 * (n - m_1)) + t_0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m)) - l;
	double t_1 = Math.cos(M) * Math.exp((t_0 - (0.5 * (m * (n + (m * 0.5))))));
	double tmp;
	if (m <= -1e+49) {
		tmp = t_1;
	} else if (m <= -1.2e-307) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * ((m + n) + M))));
	} else if (m <= 4.2e+17) {
		tmp = Math.cos(((m + n) * (K * 0.5))) * Math.exp(((M * (n - M)) + t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.fabs((n - m)) - l
	t_1 = math.cos(M) * math.exp((t_0 - (0.5 * (m * (n + (m * 0.5))))))
	tmp = 0
	if m <= -1e+49:
		tmp = t_1
	elif m <= -1.2e-307:
		tmp = math.cos(M) * math.exp((t_0 - (M * ((m + n) + M))))
	elif m <= 4.2e+17:
		tmp = math.cos(((m + n) * (K * 0.5))) * math.exp(((M * (n - M)) + t_0))
	else:
		tmp = t_1
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(n - m)) - l)
	t_1 = Float64(cos(M) * exp(Float64(t_0 - Float64(0.5 * Float64(m * Float64(n + Float64(m * 0.5)))))))
	tmp = 0.0
	if (m <= -1e+49)
		tmp = t_1;
	elseif (m <= -1.2e-307)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * Float64(Float64(m + n) + M)))));
	elseif (m <= 4.2e+17)
		tmp = Float64(cos(Float64(Float64(m + n) * Float64(K * 0.5))) * exp(Float64(Float64(M * Float64(n - M)) + t_0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m)) - l;
	t_1 = cos(M) * exp((t_0 - (0.5 * (m * (n + (m * 0.5))))));
	tmp = 0.0;
	if (m <= -1e+49)
		tmp = t_1;
	elseif (m <= -1.2e-307)
		tmp = cos(M) * exp((t_0 - (M * ((m + n) + M))));
	elseif (m <= 4.2e+17)
		tmp = cos(((m + n) * (K * 0.5))) * exp(((M * (n - M)) + t_0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(0.5 * N[(m * N[(n + N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1e+49], t$95$1, If[LessEqual[m, -1.2e-307], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * N[(N[(m + n), $MachinePrecision] + M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.2e+17], N[(N[Cos[N[(N[(m + n), $MachinePrecision] * N[(K * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -9.99999999999999946e48 or 4.2e17 < m

   1. Initial program 69.2%

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

      1. associate-/l* 70.1%

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

      2. +-commutative 70.1%

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

      3. fabs-sub 70.1%

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

      4. +-commutative 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in K around 0 98.1%

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

      1. cos-neg 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in n around 0 77.7%

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

      1. +-commutative 77.7%

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

      2. unpow2 77.7%

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

      3. distribute-rgt-out 84.3%

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

      4. *-commutative 84.3%

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

      5. *-commutative 84.3%

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

   10. Simplified 84.3%

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

   11. Taylor expanded in M around 0 84.3%

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

   if -9.99999999999999946e48 < m < -1.20000000000000009e-307

   1. Initial program 84.7%

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

      1. associate-/l* 84.7%

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

      2. +-commutative 84.7%

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

      3. fabs-sub 84.7%

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

      4. +-commutative 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in K around 0 94.2%

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

      1. cos-neg 94.2%

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

   7. Simplified 94.2%

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

      1. log1p-expm1-u 93.0%

         \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{m + n}{2} - M\right)\right)\right)}}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      2. div-inv 93.0%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      3. metadata-eval 93.0%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      4. fma-neg 93.0%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(m + n, 0.5, -M\right)}\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

   9. Applied egg-rr 93.0%

      \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)\right)\right)}}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 94.2%

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

       2. add-sqr-sqrt 43.9%

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

       3. sqrt-prod 94.2%

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

       4. unpow2 94.2%

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

       5. add-log-exp 93.0%

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

       6. unpow2 93.0%

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

       7. sqrt-prod 43.9%

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

       8. add-sqr-sqrt 93.0%

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

       9. add-sqr-sqrt 41.5%

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

       10. sqrt-unprod 93.0%

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

       11. sqr-neg 93.0%

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

       12. sqrt-unprod 51.5%

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

       13. add-sqr-sqrt 93.0%

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

   11. Applied egg-rr 93.0%

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

   12. Taylor expanded in M around inf 64.1%

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

       1. unpow2 64.1%

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

       2. distribute-lft-out 65.3%

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

       3. +-commutative 65.3%

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

   14. Simplified 65.3%

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

   if -1.20000000000000009e-307 < m < 4.2e17

   1. Initial program 80.2%

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

      1. associate-/l* 79.7%

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

      2. +-commutative 79.7%

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

      3. fabs-sub 79.7%

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

      4. +-commutative 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in n around 0 59.2%

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

      1. +-commutative 55.7%

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

      2. unpow2 55.7%

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

      3. distribute-rgt-out 55.8%

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

      4. *-commutative 55.8%

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

      5. *-commutative 55.8%

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

   7. Simplified 59.2%

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

      1. frac-2neg 59.2%

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

      2. div-inv 59.2%

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

      3. distribute-neg-frac 59.2%

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

      4. metadata-eval 59.2%

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

   9. Applied egg-rr 59.2%

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

   10. Taylor expanded in m around 0 56.3%

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

       1. mul-1-neg 52.8%

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

       2. *-commutative 52.8%

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

       3. distribute-rgt-neg-in 52.8%

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

   12. Simplified 56.3%

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

   13. Taylor expanded in M around 0 56.7%

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

       1. associate-*r* 56.7%

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

       2. *-commutative 56.7%

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

       3. +-commutative 56.7%

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

   15. Simplified 56.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+49}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) - 0.5 \cdot \left(m \cdot \left(n + m \cdot 0.5\right)\right)}\\ \mathbf{elif}\;m \leq -1.2 \cdot 10^{-307}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) - M \cdot \left(\left(m + n\right) + M\right)}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{M \cdot \left(n - M\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) - 0.5 \cdot \left(m \cdot \left(n + m \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \left|n - m\right| - \ell\\ t_1 := \cos M \cdot e^{M \cdot \left(n - M\right) + t_0}\\ \mathbf{if}\;M \leq -1.22 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \leq 4 \cdot 10^{-23}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right) + t_0}\\ \mathbf{elif}\;M \leq 4.8 \cdot 10^{+136}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot \left(\left(m + n\right) + M\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l))
        (t_1 (* (cos M) (exp (+ (* M (- n M)) t_0)))))
   (if (<= M -1.22e+32)
     t_1
     (if (<= M 4e-23)
       (* (cos M) (exp (+ (* n (- M (* m 0.5))) t_0)))
       (if (<= M 4.8e+136)
         (* (cos M) (exp (- t_0 (* M (+ (+ m n) M)))))
         t_1)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m)) - l;
	double t_1 = cos(M) * exp(((M * (n - M)) + t_0));
	double tmp;
	if (M <= -1.22e+32) {
		tmp = t_1;
	} else if (M <= 4e-23) {
		tmp = cos(M) * exp(((n * (M - (m * 0.5))) + t_0));
	} else if (M <= 4.8e+136) {
		tmp = cos(M) * exp((t_0 - (M * ((m + n) + M))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((n - m)) - l
    t_1 = cos(m_1) * exp(((m_1 * (n - m_1)) + t_0))
    if (m_1 <= (-1.22d+32)) then
        tmp = t_1
    else if (m_1 <= 4d-23) then
        tmp = cos(m_1) * exp(((n * (m_1 - (m * 0.5d0))) + t_0))
    else if (m_1 <= 4.8d+136) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * ((m + n) + m_1))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m)) - l;
	double t_1 = Math.cos(M) * Math.exp(((M * (n - M)) + t_0));
	double tmp;
	if (M <= -1.22e+32) {
		tmp = t_1;
	} else if (M <= 4e-23) {
		tmp = Math.cos(M) * Math.exp(((n * (M - (m * 0.5))) + t_0));
	} else if (M <= 4.8e+136) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * ((m + n) + M))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.fabs((n - m)) - l
	t_1 = math.cos(M) * math.exp(((M * (n - M)) + t_0))
	tmp = 0
	if M <= -1.22e+32:
		tmp = t_1
	elif M <= 4e-23:
		tmp = math.cos(M) * math.exp(((n * (M - (m * 0.5))) + t_0))
	elif M <= 4.8e+136:
		tmp = math.cos(M) * math.exp((t_0 - (M * ((m + n) + M))))
	else:
		tmp = t_1
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(n - m)) - l)
	t_1 = Float64(cos(M) * exp(Float64(Float64(M * Float64(n - M)) + t_0)))
	tmp = 0.0
	if (M <= -1.22e+32)
		tmp = t_1;
	elseif (M <= 4e-23)
		tmp = Float64(cos(M) * exp(Float64(Float64(n * Float64(M - Float64(m * 0.5))) + t_0)));
	elseif (M <= 4.8e+136)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * Float64(Float64(m + n) + M)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m)) - l;
	t_1 = cos(M) * exp(((M * (n - M)) + t_0));
	tmp = 0.0;
	if (M <= -1.22e+32)
		tmp = t_1;
	elseif (M <= 4e-23)
		tmp = cos(M) * exp(((n * (M - (m * 0.5))) + t_0));
	elseif (M <= 4.8e+136)
		tmp = cos(M) * exp((t_0 - (M * ((m + n) + M))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -1.22e+32], t$95$1, If[LessEqual[M, 4e-23], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 4.8e+136], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * N[(N[(m + n), $MachinePrecision] + M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -1.22000000000000002e32 or 4.8000000000000001e136 < M

   1. Initial program 76.5%

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

      1. associate-/l* 76.5%

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

      2. +-commutative 76.5%

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

      3. fabs-sub 76.5%

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

      4. +-commutative 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in K around 0 98.8%

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

      1. cos-neg 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in n around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. unpow2 78.9%

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

      3. distribute-rgt-out 84.9%

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

      4. *-commutative 84.9%

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

      5. *-commutative 84.9%

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

   10. Simplified 84.9%

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

   11. Taylor expanded in m around 0 80.3%

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

       1. mul-1-neg 80.3%

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

       2. *-commutative 80.3%

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

       3. distribute-rgt-neg-in 80.3%

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

   13. Simplified 80.3%

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

   if -1.22000000000000002e32 < M < 3.99999999999999984e-23

   1. Initial program 75.8%

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

      1. associate-/l* 76.3%

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

      2. +-commutative 76.3%

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

      3. fabs-sub 76.3%

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

      4. +-commutative 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in K around 0 89.8%

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

      1. cos-neg 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in n around 0 58.2%

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

      1. +-commutative 58.2%

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

      2. unpow2 58.2%

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

      3. distribute-rgt-out 61.1%

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

      4. *-commutative 61.1%

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

      5. *-commutative 61.1%

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

   10. Simplified 61.1%

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

   11. Taylor expanded in n around inf 39.2%

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

   if 3.99999999999999984e-23 < M < 4.8000000000000001e136

   1. Initial program 85.2%

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

      1. associate-/l* 85.2%

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

      2. +-commutative 85.2%

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

      3. fabs-sub 85.2%

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

      4. +-commutative 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

      1. log1p-expm1-u 96.4%

         \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{m + n}{2} - M\right)\right)\right)}}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      2. div-inv 96.4%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      3. metadata-eval 96.4%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      4. fma-neg 96.4%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(m + n, 0.5, -M\right)}\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

   9. Applied egg-rr 96.4%

      \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)\right)\right)}}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 100.0%

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

       2. add-sqr-sqrt 37.0%

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

       3. sqrt-prod 100.0%

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

       4. unpow2 100.0%

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

       5. add-log-exp 96.4%

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

       6. unpow2 96.4%

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

       7. sqrt-prod 37.0%

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

       8. add-sqr-sqrt 96.4%

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

       9. add-sqr-sqrt 0.0%

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

       10. sqrt-unprod 96.4%

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

       11. sqr-neg 96.4%

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

       12. sqrt-unprod 96.4%

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

       13. add-sqr-sqrt 96.4%

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

   11. Applied egg-rr 96.4%

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

   12. Taylor expanded in M around inf 81.8%

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

       1. unpow2 81.8%

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

       2. distribute-lft-out 81.8%

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

       3. +-commutative 81.8%

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

   14. Simplified 81.8%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.22 \cdot 10^{+32}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{elif}\;M \leq 4 \cdot 10^{-23}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{elif}\;M \leq 4.8 \cdot 10^{+136}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) - M \cdot \left(\left(m + n\right) + M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) + \left(\left|n - m\right| - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \left|n - m\right| - \ell\\ \mathbf{if}\;m \leq -1 \cdot 10^{+49} \lor \neg \left(m \leq 4.2 \cdot 10^{+17}\right):\\ \;\;\;\;\cos M \cdot e^{t_0 - 0.5 \cdot \left(m \cdot \left(n + m \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot \left(\left(m + n\right) + M\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l)))
   (if (or (<= m -1e+49) (not (<= m 4.2e+17)))
     (* (cos M) (exp (- t_0 (* 0.5 (* m (+ n (* m 0.5)))))))
     (* (cos M) (exp (- t_0 (* M (+ (+ m n) M))))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m)) - l;
	double tmp;
	if ((m <= -1e+49) || !(m <= 4.2e+17)) {
		tmp = cos(M) * exp((t_0 - (0.5 * (m * (n + (m * 0.5))))));
	} else {
		tmp = cos(M) * exp((t_0 - (M * ((m + n) + M))));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 = abs((n - m)) - l
    if ((m <= (-1d+49)) .or. (.not. (m <= 4.2d+17))) then
        tmp = cos(m_1) * exp((t_0 - (0.5d0 * (m * (n + (m * 0.5d0))))))
    else
        tmp = cos(m_1) * exp((t_0 - (m_1 * ((m + n) + m_1))))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m)) - l;
	double tmp;
	if ((m <= -1e+49) || !(m <= 4.2e+17)) {
		tmp = Math.cos(M) * Math.exp((t_0 - (0.5 * (m * (n + (m * 0.5))))));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * ((m + n) + M))));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.fabs((n - m)) - l
	tmp = 0
	if (m <= -1e+49) or not (m <= 4.2e+17):
		tmp = math.cos(M) * math.exp((t_0 - (0.5 * (m * (n + (m * 0.5))))))
	else:
		tmp = math.cos(M) * math.exp((t_0 - (M * ((m + n) + M))))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(n - m)) - l)
	tmp = 0.0
	if ((m <= -1e+49) || !(m <= 4.2e+17))
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(0.5 * Float64(m * Float64(n + Float64(m * 0.5)))))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * Float64(Float64(m + n) + M)))));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m)) - l;
	tmp = 0.0;
	if ((m <= -1e+49) || ~((m <= 4.2e+17)))
		tmp = cos(M) * exp((t_0 - (0.5 * (m * (n + (m * 0.5))))));
	else
		tmp = cos(M) * exp((t_0 - (M * ((m + n) + M))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, If[Or[LessEqual[m, -1e+49], N[Not[LessEqual[m, 4.2e+17]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(0.5 * N[(m * N[(n + N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * N[(N[(m + n), $MachinePrecision] + M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -9.99999999999999946e48 or 4.2e17 < m

   1. Initial program 69.2%

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

      1. associate-/l* 70.1%

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

      2. +-commutative 70.1%

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

      3. fabs-sub 70.1%

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

      4. +-commutative 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in K around 0 98.1%

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

      1. cos-neg 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in n around 0 77.7%

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

      1. +-commutative 77.7%

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

      2. unpow2 77.7%

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

      3. distribute-rgt-out 84.3%

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

      4. *-commutative 84.3%

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

      5. *-commutative 84.3%

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

   10. Simplified 84.3%

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

   11. Taylor expanded in M around 0 84.3%

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

   if -9.99999999999999946e48 < m < 4.2e17

   1. Initial program 82.7%

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

      1. associate-/l* 82.5%

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

      2. +-commutative 82.5%

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

      3. fabs-sub 82.5%

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

      4. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in K around 0 90.8%

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

      1. cos-neg 90.8%

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

   7. Simplified 90.8%

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

      1. log1p-expm1-u 88.9%

         \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{m + n}{2} - M\right)\right)\right)}}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      2. div-inv 88.9%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      3. metadata-eval 88.9%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      4. fma-neg 88.9%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(m + n, 0.5, -M\right)}\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

   9. Applied egg-rr 88.9%

      \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)\right)\right)}}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 90.8%

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

       2. add-sqr-sqrt 41.6%

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

       3. sqrt-prod 90.8%

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

       4. unpow2 90.8%

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

       5. add-log-exp 88.9%

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

       6. unpow2 88.9%

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

       7. sqrt-prod 41.7%

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

       8. add-sqr-sqrt 88.9%

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

       9. add-sqr-sqrt 40.3%

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

       10. sqrt-unprod 88.9%

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

       11. sqr-neg 88.9%

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

       12. sqrt-unprod 48.5%

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

       13. add-sqr-sqrt 88.9%

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

   11. Applied egg-rr 88.9%

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

   12. Taylor expanded in M around inf 61.0%

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

       1. unpow2 61.0%

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

       2. distribute-lft-out 63.0%

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

       3. +-commutative 63.0%

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

   14. Simplified 63.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+49} \lor \neg \left(m \leq 4.2 \cdot 10^{+17}\right):\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) - 0.5 \cdot \left(m \cdot \left(n + m \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) - M \cdot \left(\left(m + n\right) + M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \left|n - m\right| - \ell\\ \mathbf{if}\;m \leq -6.5 \cdot 10^{+43}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + t_0}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l)))
   (if (<= m -6.5e+43)
     (* (cos M) (exp (+ (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) t_0)))
     (* (cos M) (exp (+ (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) t_0))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m)) - l;
	double tmp;
	if (m <= -6.5e+43) {
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + t_0));
	} else {
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 = abs((n - m)) - l
    if (m <= (-6.5d+43)) then
        tmp = cos(m_1) * exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) + t_0))
    else
        tmp = cos(m_1) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) + t_0))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m)) - l;
	double tmp;
	if (m <= -6.5e+43) {
		tmp = Math.cos(M) * Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + t_0));
	} else {
		tmp = Math.cos(M) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.fabs((n - m)) - l
	tmp = 0
	if m <= -6.5e+43:
		tmp = math.cos(M) * math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + t_0))
	else:
		tmp = math.cos(M) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(n - m)) - l)
	tmp = 0.0
	if (m <= -6.5e+43)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) + t_0)));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) + t_0)));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m)) - l;
	tmp = 0.0;
	if (m <= -6.5e+43)
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + t_0));
	else
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, If[LessEqual[m, -6.5e+43], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -6.4999999999999998e43

   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. Step-by-step derivation

      1. associate-/l* 71.4%

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

      2. +-commutative 71.4%

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

      3. fabs-sub 71.4%

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

      4. +-commutative 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around 0 85.8%

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

      1. +-commutative 85.8%

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

      2. unpow2 85.8%

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

      3. distribute-rgt-out 94.0%

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

      4. *-commutative 94.0%

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

      5. *-commutative 94.0%

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

   10. Simplified 94.0%

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

   if -6.4999999999999998e43 < m

   1. Initial program 78.3%

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

      1. associate-/l* 78.7%

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

      2. +-commutative 78.7%

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

      3. fabs-sub 78.7%

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

      4. +-commutative 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in K around 0 92.4%

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

      1. cos-neg 92.4%

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

   7. Simplified 92.4%

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

   8. Taylor expanded in m around 0 77.1%

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

      1. +-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. distribute-rgt-out 82.4%

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

      4. *-commutative 82.4%

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

      5. *-commutative 82.4%

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

   10. Simplified 82.4%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{+43}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + \left(\left|n - m\right| - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.1% accurate, 1.3× speedup

Math

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

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (*
  (cos M)
  (exp (+ (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) (- (fabs (- n m)) l)))))

C

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

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) + (abs((n - m)) - l)))
end function

Java

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

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return math.cos(M) * math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (math.fabs((n - m)) - l)))

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) + Float64(abs(Float64(n - m)) - l))))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (abs((n - m)) - l)));
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

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

   1. associate-/l* 77.3%

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

   2. +-commutative 77.3%

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

   3. fabs-sub 77.3%

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

   4. +-commutative 77.3%

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

3. Simplified 77.3%

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

5. Taylor expanded in K around 0 93.9%

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

   1. cos-neg 93.9%

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

7. Simplified 93.9%

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

8. Taylor expanded in n around 0 66.4%

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

   1. +-commutative 66.4%

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

   2. unpow2 66.4%

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

   3. distribute-rgt-out 70.8%

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

   4. *-commutative 70.8%

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

   5. *-commutative 70.8%

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

10. Simplified 70.8%

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

11. Final simplification 70.8%

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

Alternative 10: 56.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \left|n - m\right| - \ell\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{-20}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot \left(\left(m + n\right) + M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) + t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l)))
   (if (<= l -8e-20)
     (* (cos M) (exp (- t_0 (* M (+ (+ m n) M)))))
     (* (cos M) (exp (+ (* M (- n M)) t_0))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m)) - l;
	double tmp;
	if (l <= -8e-20) {
		tmp = cos(M) * exp((t_0 - (M * ((m + n) + M))));
	} else {
		tmp = cos(M) * exp(((M * (n - M)) + t_0));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 = abs((n - m)) - l
    if (l <= (-8d-20)) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * ((m + n) + m_1))))
    else
        tmp = cos(m_1) * exp(((m_1 * (n - m_1)) + t_0))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m)) - l;
	double tmp;
	if (l <= -8e-20) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * ((m + n) + M))));
	} else {
		tmp = Math.cos(M) * Math.exp(((M * (n - M)) + t_0));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.fabs((n - m)) - l
	tmp = 0
	if l <= -8e-20:
		tmp = math.cos(M) * math.exp((t_0 - (M * ((m + n) + M))))
	else:
		tmp = math.cos(M) * math.exp(((M * (n - M)) + t_0))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(n - m)) - l)
	tmp = 0.0
	if (l <= -8e-20)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * Float64(Float64(m + n) + M)))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(n - M)) + t_0)));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m)) - l;
	tmp = 0.0;
	if (l <= -8e-20)
		tmp = cos(M) * exp((t_0 - (M * ((m + n) + M))));
	else
		tmp = cos(M) * exp(((M * (n - M)) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, If[LessEqual[l, -8e-20], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * N[(N[(m + n), $MachinePrecision] + M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -7.99999999999999956e-20

   1. Initial program 75.0%

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

      1. associate-/l* 76.6%

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

      2. +-commutative 76.6%

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

      3. fabs-sub 76.6%

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

      4. +-commutative 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in K around 0 84.4%

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

      1. cos-neg 84.4%

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

   7. Simplified 84.4%

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

      1. log1p-expm1-u 78.4%

         \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{m + n}{2} - M\right)\right)\right)}}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      2. div-inv 78.4%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      3. metadata-eval 78.4%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

      4. fma-neg 78.4%

         \[\leadsto \cos M \cdot e^{\left(-{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(m + n, 0.5, -M\right)}\right)\right)\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

   9. Applied egg-rr 78.4%

      \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)\right)\right)}}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 84.4%

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

       2. add-sqr-sqrt 35.9%

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

       3. sqrt-prod 84.4%

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

       4. unpow2 84.4%

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

       5. add-log-exp 78.4%

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

       6. unpow2 78.4%

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

       7. sqrt-prod 34.5%

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

       8. add-sqr-sqrt 78.4%

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

       9. add-sqr-sqrt 43.9%

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

       10. sqrt-unprod 78.4%

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

       11. sqr-neg 78.4%

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

       12. sqrt-unprod 34.5%

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

       13. add-sqr-sqrt 78.4%

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

   11. Applied egg-rr 78.4%

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

   12. Taylor expanded in M around inf 42.8%

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

       1. unpow2 42.8%

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

       2. distribute-lft-out 44.4%

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

       3. +-commutative 44.4%

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

   14. Simplified 44.4%

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

   if -7.99999999999999956e-20 < l

   1. Initial program 77.7%

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

      1. associate-/l* 77.5%

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

      2. +-commutative 77.5%

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

      3. fabs-sub 77.5%

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

      4. +-commutative 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in K around 0 97.0%

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

      1. cos-neg 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in n around 0 68.2%

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

      1. +-commutative 68.2%

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

      2. unpow2 68.2%

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

      3. distribute-rgt-out 72.9%

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

      4. *-commutative 72.9%

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

      5. *-commutative 72.9%

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

   10. Simplified 72.9%

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

   11. Taylor expanded in m around 0 55.0%

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

       1. mul-1-neg 55.0%

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

       2. *-commutative 55.0%

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

       3. distribute-rgt-neg-in 55.0%

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

   13. Simplified 55.0%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-20}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) - M \cdot \left(\left(m + n\right) + M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) + \left(\left|n - m\right| - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.4% accurate, 1.4× speedup

Math

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

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (+ (* M (- n M)) (- (fabs (- n m)) l)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp(((M * (n - M)) + (fabs((n - m)) - l)));
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp(((m_1 * (n - m_1)) + (abs((n - m)) - l)))
end function

Java

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

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return math.cos(M) * math.exp(((M * (n - M)) + (math.fabs((n - m)) - l)))

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(Float64(M * Float64(n - M)) + Float64(abs(Float64(n - m)) - l))))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp(((M * (n - M)) + (abs((n - m)) - l)));
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

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

   1. associate-/l* 77.3%

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

   2. +-commutative 77.3%

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

   3. fabs-sub 77.3%

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

   4. +-commutative 77.3%

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

3. Simplified 77.3%

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

5. Taylor expanded in K around 0 93.9%

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

   1. cos-neg 93.9%

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

7. Simplified 93.9%

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

8. Taylor expanded in n around 0 66.4%

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

   1. +-commutative 66.4%

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

   2. unpow2 66.4%

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

   3. distribute-rgt-out 70.8%

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

   4. *-commutative 70.8%

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

   5. *-commutative 70.8%

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

10. Simplified 70.8%

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

11. Taylor expanded in m around 0 50.4%

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

    1. mul-1-neg 50.4%

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

    2. *-commutative 50.4%

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

    3. distribute-rgt-neg-in 50.4%

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

13. Simplified 50.4%

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

14. Final simplification 50.4%

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

Reproduce

herbie shell --seed 2024014 
(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)))))))