Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.6% → 96.4%

Time: 13.0s

Alternatives: 10

Speedup: 3.1×

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: 75.6% 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: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right)\\ t_1 := \left|m - n\right|\\ \mathbf{if}\;e^{\left(t\_1 - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot t\_0 \leq 0:\\ \;\;\;\;t\_0 \cdot e^{n \cdot \left(n \cdot \left(-0.25 + \frac{\mathsf{fma}\left(m, -0.5, M\right) + \frac{t\_1 - \mathsf{fma}\left(\mathsf{fma}\left(m, -0.5, M\right), \mathsf{fma}\left(m, -0.5, M\right), \ell\right)}{n}}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_1 - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \end{array} \end{array} \]

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 (cos (- (/ (* (+ m n) K) 2.0) M))) (t_1 (fabs (- m n))))
   (if (<= (* (exp (- (- t_1 l) (pow (- (/ (+ m n) 2.0) M) 2.0))) t_0) 0.0)
     (*
      t_0
      (exp
       (*
        n
        (*
         n
         (+
          -0.25
          (/
           (+
            (fma m -0.5 M)
            (/ (- t_1 (fma (fma m -0.5 M) (fma m -0.5 M) l)) n))
           n))))))
     (exp (- t_1 (fma 0.25 (* (+ m n) (+ m n)) l))))))

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 = cos(((((m + n) * K) / 2.0) - M));
	double t_1 = fabs((m - n));
	double tmp;
	if ((exp(((t_1 - l) - pow((((m + n) / 2.0) - M), 2.0))) * t_0) <= 0.0) {
		tmp = t_0 * exp((n * (n * (-0.25 + ((fma(m, -0.5, M) + ((t_1 - fma(fma(m, -0.5, M), fma(m, -0.5, M), l)) / n)) / n)))));
	} else {
		tmp = exp((t_1 - fma(0.25, ((m + n) * (m + n)), l)));
	}
	return tmp;
}

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = cos(Float64(Float64(Float64(Float64(m + n) * K) / 2.0) - M))
	t_1 = abs(Float64(m - n))
	tmp = 0.0
	if (Float64(exp(Float64(Float64(t_1 - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0))) * t_0) <= 0.0)
		tmp = Float64(t_0 * exp(Float64(n * Float64(n * Float64(-0.25 + Float64(Float64(fma(m, -0.5, M) + Float64(Float64(t_1 - fma(fma(m, -0.5, M), fma(m, -0.5, M), l)) / n)) / n))))));
	else
		tmp = exp(Float64(t_1 - fma(0.25, Float64(Float64(m + n) * Float64(m + n)), l)));
	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[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Exp[N[(N[(t$95$1 - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], 0.0], N[(t$95$0 * N[Exp[N[(n * N[(n * N[(-0.25 + N[(N[(N[(m * -0.5 + M), $MachinePrecision] + N[(N[(t$95$1 - N[(N[(m * -0.5 + M), $MachinePrecision] * N[(m * -0.5 + M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$1 - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < -0.0

   1. Initial program 98.9%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 98.9%

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

   if -0.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 34.2%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 96.4

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 95.7

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

   8. Applied rewrites 95.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \leq 0:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{n \cdot \left(n \cdot \left(-0.25 + \frac{\mathsf{fma}\left(m, -0.5, M\right) + \frac{\left|m - n\right| - \mathsf{fma}\left(\mathsf{fma}\left(m, -0.5, M\right), \mathsf{fma}\left(m, -0.5, M\right), \ell\right)}{n}}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 1.1× speedup

Math

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

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

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(((abs((m - n)) - l) - ((((m + n) / 2.0d0) - m_1) ** 2.0d0)))
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(((Math.abs((m - n)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
}

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(((math.fabs((m - n)) - l) - math.pow((((m + n) / 2.0) - M), 2.0)))

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(abs(Float64(m - n)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0))))
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(((abs((m - n)) - l) - ((((m + n) / 2.0) - M) ^ 2.0)));
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[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.7%

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

3. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. lower-cos.f64 97.8

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

5. Applied rewrites 97.8%

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

6. Final simplification 97.8%

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

Alternative 3: 95.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -5.5 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 170:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* (cos M) (exp (* M (- M))))))
   (if (<= M -5.5e+53)
     t_0
     (if (<= M 170.0)
       (exp (- (fabs (- m n)) (fma 0.25 (* (+ m n) (+ m n)) l)))
       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 = cos(M) * exp((M * -M));
	double tmp;
	if (M <= -5.5e+53) {
		tmp = t_0;
	} else if (M <= 170.0) {
		tmp = exp((fabs((m - n)) - fma(0.25, ((m + n) * (m + n)), l)));
	} else {
		tmp = 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(cos(M) * exp(Float64(M * Float64(-M))))
	tmp = 0.0
	if (M <= -5.5e+53)
		tmp = t_0;
	elseif (M <= 170.0)
		tmp = exp(Float64(abs(Float64(m - n)) - fma(0.25, Float64(Float64(m + n) * Float64(m + n)), l)));
	else
		tmp = t_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[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -5.5e+53], t$95$0, If[LessEqual[M, 170.0], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -5.49999999999999975e53 or 170 < M

   1. Initial program 81.7%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in M around inf

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

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \cos M \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

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

         \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(\mathsf{neg}\left(M\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(\mathsf{neg}\left(M\right)\right)}} \]

      5. lower-neg.f64 99.1

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

   8. Applied rewrites 99.1%

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

   if -5.49999999999999975e53 < M < 170

   1. Initial program 78.2%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 95.2

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

   8. Applied rewrites 95.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -5.5 \cdot 10^{+53}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 170:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;M \leq -7.6 \cdot 10^{+21}:\\ \;\;\;\;e^{t\_0 - \mathsf{fma}\left(0.25, n \cdot \left(n \cdot \mathsf{fma}\left(m, \frac{2}{n} + \frac{m}{n \cdot n}, 1\right)\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \end{array} \end{array} \]

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 (- m n))))
   (if (<= M -7.6e+21)
     (exp
      (- t_0 (fma 0.25 (* n (* n (fma m (+ (/ 2.0 n) (/ m (* n n))) 1.0))) l)))
     (exp (- t_0 (fma 0.25 (* (+ m n) (+ m n)) l))))))

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((m - n));
	double tmp;
	if (M <= -7.6e+21) {
		tmp = exp((t_0 - fma(0.25, (n * (n * fma(m, ((2.0 / n) + (m / (n * n))), 1.0))), l)));
	} else {
		tmp = exp((t_0 - fma(0.25, ((m + n) * (m + n)), l)));
	}
	return tmp;
}

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if (M <= -7.6e+21)
		tmp = exp(Float64(t_0 - fma(0.25, Float64(n * Float64(n * fma(m, Float64(Float64(2.0 / n) + Float64(m / Float64(n * n))), 1.0))), l)));
	else
		tmp = exp(Float64(t_0 - fma(0.25, Float64(Float64(m + n) * Float64(m + n)), l)));
	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[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -7.6e+21], N[Exp[N[(t$95$0 - N[(0.25 * N[(n * N[(n * N[(m * N[(N[(2.0 / n), $MachinePrecision] + N[(m / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -7.6e21

   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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 72.9

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in n around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. associate-+l+ N/A

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

       7. associate-*r/ N/A

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

       8. *-commutative N/A

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

       9. associate-/l* N/A

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

       10. metadata-eval N/A

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

       11. associate-*r/ N/A

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

       12. lower-fma.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 N/A

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

       16. unpow2 N/A

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

       17. associate-/l* N/A

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

       18. lower-fma.f64 N/A

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

   11. Applied rewrites 80.1%

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \color{blue}{n \cdot \left(n \cdot \mathsf{fma}\left(m, \frac{2}{n}, \mathsf{fma}\left(m, \frac{m}{n \cdot n}, 1\right)\right)\right)}, \ell\right)} \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-fma.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 80.1

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \color{blue}{\left(n \cdot \mathsf{fma}\left(m, \frac{2}{n}, \mathsf{fma}\left(m, \frac{m}{n \cdot n}, 1\right)\right)\right) \cdot n}, \ell\right)} \]

   13. Applied rewrites 80.1%

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \color{blue}{\left(n \cdot \mathsf{fma}\left(m, \frac{2}{n} + \frac{m}{n \cdot n}, 1\right)\right) \cdot n}, \ell\right)} \]

   if -7.6e21 < M

   1. Initial program 78.2%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 93.1

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

   8. Applied rewrites 93.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7.6 \cdot 10^{+21}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, n \cdot \left(n \cdot \mathsf{fma}\left(m, \frac{2}{n} + \frac{m}{n \cdot n}, 1\right)\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)} \end{array} \]

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
 (exp (- (fabs (- m n)) (fma 0.25 (* (+ m n) (+ m n)) l))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return exp((fabs((m - n)) - fma(0.25, ((m + n) * (m + n)), l)));
}

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return exp(Float64(abs(Float64(m - n)) - fma(0.25, Float64(Float64(m + n) * Float64(m + n)), 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[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.7%

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

3. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. lower-cos.f64 97.8

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in M around 0

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. fabs-sub N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. lower-fabs.f64 N/A

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

   7. mul-1-neg N/A

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

   8. sub-neg N/A

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

   9. lower--.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 88.9

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

8. Applied rewrites 88.9%

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

9. Final simplification 88.9%

   \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)} \]
10. Add Preprocessing

Alternative 6: 69.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := e^{n \cdot \left(n \cdot -0.25\right)}\\ \mathbf{if}\;n \leq -7.2 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 115000:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (exp (* n (* n -0.25)))))
   (if (<= n -7.2e-16) t_0 (if (<= n 115000.0) (exp (- l)) 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 = exp((n * (n * -0.25)));
	double tmp;
	if (n <= -7.2e-16) {
		tmp = t_0;
	} else if (n <= 115000.0) {
		tmp = exp(-l);
	} else {
		tmp = 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 = exp((n * (n * (-0.25d0))))
    if (n <= (-7.2d-16)) then
        tmp = t_0
    else if (n <= 115000.0d0) then
        tmp = exp(-l)
    else
        tmp = 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.exp((n * (n * -0.25)));
	double tmp;
	if (n <= -7.2e-16) {
		tmp = t_0;
	} else if (n <= 115000.0) {
		tmp = Math.exp(-l);
	} else {
		tmp = 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.exp((n * (n * -0.25)))
	tmp = 0
	if n <= -7.2e-16:
		tmp = t_0
	elif n <= 115000.0:
		tmp = math.exp(-l)
	else:
		tmp = 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 = exp(Float64(n * Float64(n * -0.25)))
	tmp = 0.0
	if (n <= -7.2e-16)
		tmp = t_0;
	elseif (n <= 115000.0)
		tmp = exp(Float64(-l));
	else
		tmp = 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 = exp((n * (n * -0.25)));
	tmp = 0.0;
	if (n <= -7.2e-16)
		tmp = t_0;
	elseif (n <= 115000.0)
		tmp = exp(-l);
	else
		tmp = 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[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -7.2e-16], t$95$0, If[LessEqual[n, 115000.0], N[Exp[(-l)], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -7.19999999999999965e-16 or 115000 < n

   1. Initial program 76.7%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 97.7

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in n around inf

      \[\leadsto e^{\color{blue}{\frac{-1}{4} \cdot {n}^{2}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

       2. unpow2 N/A

          \[\leadsto e^{\color{blue}{\left(n \cdot n\right)} \cdot \frac{-1}{4}} \]

       3. associate-*l* N/A

          \[\leadsto e^{\color{blue}{n \cdot \left(n \cdot \frac{-1}{4}\right)}} \]

       4. *-commutative N/A

          \[\leadsto e^{n \cdot \color{blue}{\left(\frac{-1}{4} \cdot n\right)}} \]

       5. lower-*.f64 N/A

          \[\leadsto e^{\color{blue}{n \cdot \left(\frac{-1}{4} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto e^{n \cdot \color{blue}{\left(n \cdot \frac{-1}{4}\right)}} \]

       7. lower-*.f64 95.4

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

   11. Applied rewrites 95.4%

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

   if -7.19999999999999965e-16 < n < 115000

   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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 79.9

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

   8. Applied rewrites 79.9%

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

   9. Taylor expanded in l around inf

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

       1. neg-mul-1 N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

       2. lower-neg.f64 50.8

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

   11. Applied rewrites 50.8%

       \[\leadsto e^{\color{blue}{-\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 115000:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

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
 (if (<= n 115000.0) (exp (* -0.25 (* m m))) (exp (* n (* n -0.25)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 115000.0) {
		tmp = exp((-0.25 * (m * m)));
	} else {
		tmp = exp((n * (n * -0.25)));
	}
	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) :: tmp
    if (n <= 115000.0d0) then
        tmp = exp(((-0.25d0) * (m * m)))
    else
        tmp = exp((n * (n * (-0.25d0))))
    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 tmp;
	if (n <= 115000.0) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if n <= 115000.0:
		tmp = math.exp((-0.25 * (m * m)))
	else:
		tmp = math.exp((n * (n * -0.25)))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 115000.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	else
		tmp = exp(Float64(n * Float64(n * -0.25)));
	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)
	tmp = 0.0;
	if (n <= 115000.0)
		tmp = exp((-0.25 * (m * m)));
	else
		tmp = exp((n * (n * -0.25)));
	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_] := If[LessEqual[n, 115000.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 115000

   1. Initial program 79.8%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 85.6

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in m around inf

      \[\leadsto e^{\color{blue}{\frac{-1}{4} \cdot {m}^{2}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto e^{\color{blue}{\left(m \cdot m\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 57.3

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

   11. Applied rewrites 57.3%

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

   if 115000 < n

   1. Initial program 79.3%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in n around inf

      \[\leadsto e^{\color{blue}{\frac{-1}{4} \cdot {n}^{2}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

       2. unpow2 N/A

          \[\leadsto e^{\color{blue}{\left(n \cdot n\right)} \cdot \frac{-1}{4}} \]

       3. associate-*l* N/A

          \[\leadsto e^{\color{blue}{n \cdot \left(n \cdot \frac{-1}{4}\right)}} \]

       4. *-commutative N/A

          \[\leadsto e^{n \cdot \color{blue}{\left(\frac{-1}{4} \cdot n\right)}} \]

       5. lower-*.f64 N/A

          \[\leadsto e^{\color{blue}{n \cdot \left(\frac{-1}{4} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto e^{n \cdot \color{blue}{\left(n \cdot \frac{-1}{4}\right)}} \]

       7. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 115000:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ e^{-\ell} \end{array} \]

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 (exp (- l)))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return exp(-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 = exp(-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.exp(-l);
}

Python

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

Julia

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

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = exp(-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[Exp[(-l)], $MachinePrecision]

Derivation

1. Initial program 79.7%

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

3. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. lower-cos.f64 97.8

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in M around 0

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. fabs-sub N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. lower-fabs.f64 N/A

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

   7. mul-1-neg N/A

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

   8. sub-neg N/A

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

   9. lower--.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 88.9

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

8. Applied rewrites 88.9%

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

9. Taylor expanded in l around inf

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

    1. neg-mul-1 N/A

       \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

    2. lower-neg.f64 35.3

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

11. Applied rewrites 35.3%

    \[\leadsto e^{\color{blue}{-\ell}} \]
12. Add Preprocessing

Alternative 9: 6.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \cos M \end{array} \]

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))

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);
}

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)
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);
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return math.cos(M)

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return cos(M)
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);
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[Cos[M], $MachinePrecision]

Derivation

1. Initial program 79.7%

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

3. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 44.7

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

5. Applied rewrites 44.7%

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

6. Taylor expanded in n around 0

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

   1. lower-cos.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-neg.f64 8.3

      \[\leadsto \cos \left(\mathsf{fma}\left(0.5, m \cdot K, \color{blue}{-M}\right)\right) \]

8. Applied rewrites 8.3%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(0.5, m \cdot K, -M\right)\right)} \]

9. Taylor expanded in m around 0

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

    1. cos-neg N/A

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

    2. lower-cos.f64 8.7

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

11. Applied rewrites 8.7%

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

Alternative 10: 6.8% accurate, 359.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ 1 \end{array} \]

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 1.0)

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

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 = 1.0d0
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 1.0;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return 1.0

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return 1.0
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = 1.0;
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_] := 1.0

Derivation

1. Initial program 79.7%

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

3. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 44.7

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

5. Applied rewrites 44.7%

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

6. Taylor expanded in n around 0

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

   1. lower-cos.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-neg.f64 8.3

      \[\leadsto \cos \left(\mathsf{fma}\left(0.5, m \cdot K, \color{blue}{-M}\right)\right) \]

8. Applied rewrites 8.3%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(0.5, m \cdot K, -M\right)\right)} \]

9. Taylor expanded in m around 0

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

    1. cos-neg N/A

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

    2. lower-cos.f64 8.7

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

11. Applied rewrites 8.7%

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

12. Taylor expanded in M around 0

    \[\leadsto \color{blue}{1} \]
13. Step-by-step derivation

14. Applied rewrites 8.7%

    \[\leadsto \color{blue}{1} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))))))