Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.0% → 96.7%

Time: 11.5s

Alternatives: 10

Speedup: 2.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right) - M\\ \cos M \cdot e^{\left|n - m\right| - \mathsf{fma}\left(t\_0, t\_0, \ell\right)} \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (+ n m)) M)))
   (* (cos M) (exp (- (fabs (- n m)) (fma t_0 t_0 l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (n + m)) - M;
	return cos(M) * exp((fabs((n - m)) - fma(t_0, t_0, l)));
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * Float64(n + m)) - M)
	return Float64(cos(M) * exp(Float64(abs(Float64(n - m)) - fma(t_0, t_0, l))))
end

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. cos-neg N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. fabs-sub N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-fabs.f64 N/A

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

   10. mul-1-neg N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 N/A

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

5. Simplified 96.4%

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

6. Final simplification 96.4%

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

Alternative 2: 95.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-M \cdot M}\\ \mathbf{if}\;M \leq -1.45 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 2350:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (* M M)))))
   (if (<= M -1.45e+25)
     t_0
     (if (<= M 2350.0)
       (exp (- (fabs (- n m)) (fma 0.25 (* (+ n m) (+ n m)) l)))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-(M * M));
	double tmp;
	if (M <= -1.45e+25) {
		tmp = t_0;
	} else if (M <= 2350.0) {
		tmp = exp((fabs((n - m)) - fma(0.25, ((n + m) * (n + m)), l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-Float64(M * M)))
	tmp = 0.0
	if (M <= -1.45e+25)
		tmp = t_0;
	elseif (M <= 2350.0)
		tmp = exp(Float64(abs(Float64(n - m)) - fma(0.25, Float64(Float64(n + m) * Float64(n + m)), l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-N[(M * M), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[M, -1.45e+25], t$95$0, If[LessEqual[M, 2350.0], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(N[(n + m), $MachinePrecision] * N[(n + m), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -1.44999999999999995e25 or 2350 < M

   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

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 97.5%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 97.5

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

   11. Simplified 97.5%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 97.5

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

   14. Simplified 97.5%

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

   if -1.44999999999999995e25 < M < 2350

   1. Initial program 75.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 93.3%

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

   6. Taylor expanded in M around 0

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

      1. associate--r+ N/A

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

      2. exp-diff N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. exp-diff N/A

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

      6. lower-exp.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. fabs-sub N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. mul-1-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 N/A

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

   8. Simplified 93.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.45 \cdot 10^{+25}:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{elif}\;M \leq 2350:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-M \cdot M}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(m, 0.5, -M\right)\\ \mathbf{if}\;n \leq 70:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(t\_0, t\_0, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma m 0.5 (- M))))
   (if (<= n 70.0)
     (exp (- (fabs (- n m)) (fma t_0 t_0 l)))
     (exp (* n (* n -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma(m, 0.5, -M);
	double tmp;
	if (n <= 70.0) {
		tmp = exp((fabs((n - m)) - fma(t_0, t_0, l)));
	} else {
		tmp = exp((n * (n * -0.25)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(m, 0.5, Float64(-M))
	tmp = 0.0
	if (n <= 70.0)
		tmp = exp(Float64(abs(Float64(n - m)) - fma(t_0, t_0, l)));
	else
		tmp = exp(Float64(n * Float64(n * -0.25)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(m * 0.5 + (-M)), $MachinePrecision]}, If[LessEqual[n, 70.0], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(t$95$0 * t$95$0 + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 70

   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

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 95.5%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 93.9%

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

   9. Taylor expanded in n around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. +-commutative N/A

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

       11. unpow2 N/A

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

       12. lower-fma.f64 N/A

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

       13. sub-neg N/A

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

       14. *-commutative N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. *-commutative N/A

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

       19. lower-fma.f64 N/A

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

       20. lower-neg.f64 80.4

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

   11. Simplified 80.4%

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

   if 70 < n

   1. Initial program 67.1%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 98.6%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 98.6%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 98.7

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

   11. Simplified 98.7%

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

   12. Taylor expanded in n around inf

       \[\leadsto e^{\color{blue}{\frac{-1}{4} \cdot {n}^{2}}} \]
   13. 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 98.7

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

   14. Simplified 98.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 70:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(\mathsf{fma}\left(m, 0.5, -M\right), \mathsf{fma}\left(m, 0.5, -M\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right) - M\\ e^{\left|n - m\right| - \mathsf{fma}\left(t\_0, t\_0, \ell\right)} \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (+ n m)) M))) (exp (- (fabs (- n m)) (fma t_0 t_0 l)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (n + m)) - M;
	return exp((fabs((n - m)) - fma(t_0, t_0, l)));
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * Float64(n + m)) - M)
	return exp(Float64(abs(Float64(n - m)) - fma(t_0, t_0, l)))
end

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. cos-neg N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. fabs-sub N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-fabs.f64 N/A

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

   10. mul-1-neg N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 N/A

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

5. Simplified 96.4%

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

6. Taylor expanded in M around 0

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

8. Simplified 95.2%

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

9. Final simplification 95.2%

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

Alternative 5: 62.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 8.5 \cdot 10^{-211}:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -53.0)
   (exp (* m (* m -0.25)))
   (if (<= m 8.5e-211)
     (exp (- (* M M)))
     (exp (- (fabs (- n m)) (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = exp((m * (m * -0.25)));
	} else if (m <= 8.5e-211) {
		tmp = exp(-(M * M));
	} else {
		tmp = exp((fabs((n - m)) - (0.25 * (n * n))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-53.0d0)) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (m <= 8.5d-211) then
        tmp = exp(-(m_1 * m_1))
    else
        tmp = exp((abs((n - m)) - (0.25d0 * (n * n))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (m <= 8.5e-211) {
		tmp = Math.exp(-(M * M));
	} else {
		tmp = Math.exp((Math.abs((n - m)) - (0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -53.0:
		tmp = math.exp((m * (m * -0.25)))
	elif m <= 8.5e-211:
		tmp = math.exp(-(M * M))
	else:
		tmp = math.exp((math.fabs((n - m)) - (0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -53.0)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (m <= 8.5e-211)
		tmp = exp(Float64(-Float64(M * M)));
	else
		tmp = exp(Float64(abs(Float64(n - m)) - Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -53.0)
		tmp = exp((m * (m * -0.25)));
	elseif (m <= 8.5e-211)
		tmp = exp(-(M * M));
	else
		tmp = exp((abs((n - m)) - (0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -53.0], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 8.5e-211], N[Exp[(-N[(M * M), $MachinePrecision])], $MachinePrecision], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -53

   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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 98.4%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 98.4%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 96.8

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

   11. Simplified 96.8%

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

   12. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 96.8

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

   14. Simplified 96.8%

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

   if -53 < m < 8.49999999999999936e-211

   1. Initial program 80.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 94.9%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 92.7%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 78.9

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

   11. Simplified 78.9%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 47.0

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

   14. Simplified 47.0%

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

   if 8.49999999999999936e-211 < m

   1. Initial program 75.9%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 96.5%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 95.5%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 89.0

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

   11. Simplified 89.0%

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

   12. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 50.7

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

   14. Simplified 50.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 8.5 \cdot 10^{-211}:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 55:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, m \cdot m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 55.0)
   (exp (- (fabs (- n m)) (fma 0.25 (* m m) l)))
   (exp (* n (* n -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 55.0) {
		tmp = exp((fabs((n - m)) - fma(0.25, (m * m), l)));
	} else {
		tmp = exp((n * (n * -0.25)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 55.0)
		tmp = exp(Float64(abs(Float64(n - m)) - fma(0.25, Float64(m * m), l)));
	else
		tmp = exp(Float64(n * Float64(n * -0.25)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 55.0], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(m * m), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 55

   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

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 95.5%

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

   6. Taylor expanded in M around 0

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

      1. associate--r+ N/A

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

      2. exp-diff N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. exp-diff N/A

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

      6. lower-exp.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. fabs-sub N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. mul-1-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 N/A

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

   8. Simplified 81.1%

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

   9. Taylor expanded in n around 0

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

       1. fabs-sub N/A

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

       2. lower-exp.f64 N/A

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

       3. lower--.f64 N/A

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

       4. fabs-sub N/A

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

       5. sub-neg N/A

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

       6. mul-1-neg N/A

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

       7. lower-fabs.f64 N/A

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

       8. mul-1-neg N/A

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

       9. sub-neg N/A

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

       10. lower--.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 60.2

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

   11. Simplified 60.2%

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

   if 55 < n

   1. Initial program 67.1%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 98.6%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 98.6%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 98.7

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

   11. Simplified 98.7%

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

   12. Taylor expanded in n around inf

       \[\leadsto e^{\color{blue}{\frac{-1}{4} \cdot {n}^{2}}} \]
   13. 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 98.7

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

   14. Simplified 98.7%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 55:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, m \cdot m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 8.5 \cdot 10^{-211}:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -53.0)
   (exp (* m (* m -0.25)))
   (if (<= m 8.5e-211) (exp (- (* M M))) (exp (* n (* n -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = exp((m * (m * -0.25)));
	} else if (m <= 8.5e-211) {
		tmp = exp(-(M * M));
	} else {
		tmp = exp((n * (n * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-53.0d0)) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (m <= 8.5d-211) then
        tmp = exp(-(m_1 * m_1))
    else
        tmp = exp((n * (n * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (m <= 8.5e-211) {
		tmp = Math.exp(-(M * M));
	} else {
		tmp = Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -53.0:
		tmp = math.exp((m * (m * -0.25)))
	elif m <= 8.5e-211:
		tmp = math.exp(-(M * M))
	else:
		tmp = math.exp((n * (n * -0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -53.0)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (m <= 8.5e-211)
		tmp = exp(Float64(-Float64(M * M)));
	else
		tmp = exp(Float64(n * Float64(n * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -53.0)
		tmp = exp((m * (m * -0.25)));
	elseif (m <= 8.5e-211)
		tmp = exp(-(M * M));
	else
		tmp = exp((n * (n * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -53.0], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 8.5e-211], N[Exp[(-N[(M * M), $MachinePrecision])], $MachinePrecision], N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -53

   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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 98.4%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 98.4%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 96.8

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

   11. Simplified 96.8%

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

   12. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 96.8

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

   14. Simplified 96.8%

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

   if -53 < m < 8.49999999999999936e-211

   1. Initial program 80.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 94.9%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 92.7%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 78.9

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

   11. Simplified 78.9%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 47.0

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

   14. Simplified 47.0%

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

   if 8.49999999999999936e-211 < m

   1. Initial program 75.9%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 96.5%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 95.5%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 89.0

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

   11. Simplified 89.0%

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

   12. Taylor expanded in n around inf

       \[\leadsto e^{\color{blue}{\frac{-1}{4} \cdot {n}^{2}}} \]
   13. 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 57.5

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

   14. Simplified 57.5%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 8.5 \cdot 10^{-211}:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-M \cdot M}\\ \mathbf{if}\;M \leq -27.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (* M M)))))
   (if (<= M -27.5) t_0 (if (<= M 26.5) (exp (* m (* m -0.25))) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-(M * M));
	double tmp;
	if (M <= -27.5) {
		tmp = t_0;
	} else if (M <= 26.5) {
		tmp = exp((m * (m * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-(m_1 * m_1))
    if (m_1 <= (-27.5d0)) then
        tmp = t_0
    else if (m_1 <= 26.5d0) then
        tmp = exp((m * (m * (-0.25d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(-(M * M));
	double tmp;
	if (M <= -27.5) {
		tmp = t_0;
	} else if (M <= 26.5) {
		tmp = Math.exp((m * (m * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-(M * M))
	tmp = 0
	if M <= -27.5:
		tmp = t_0
	elif M <= 26.5:
		tmp = math.exp((m * (m * -0.25)))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-Float64(M * M)))
	tmp = 0.0
	if (M <= -27.5)
		tmp = t_0;
	elseif (M <= 26.5)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-(M * M));
	tmp = 0.0;
	if (M <= -27.5)
		tmp = t_0;
	elseif (M <= 26.5)
		tmp = exp((m * (m * -0.25)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-N[(M * M), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[M, -27.5], t$95$0, If[LessEqual[M, 26.5], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -27.5 or 26.5 < M

   1. Initial program 79.5%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 97.5%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 97.6

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

   11. Simplified 97.6%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 97.6

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

   14. Simplified 97.6%

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

   if -27.5 < M < 26.5

   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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 93.1%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 93.1%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 78.0

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

   11. Simplified 78.0%

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

   12. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 46.5

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

   14. Simplified 46.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -27.5:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-M \cdot M}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-M \cdot M}\\ \mathbf{if}\;M \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (* M M)))))
   (if (<= M -3.2e-6) t_0 (if (<= M 26.5) (exp (- l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-(M * M));
	double tmp;
	if (M <= -3.2e-6) {
		tmp = t_0;
	} else if (M <= 26.5) {
		tmp = exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-(m_1 * m_1))
    if (m_1 <= (-3.2d-6)) then
        tmp = t_0
    else if (m_1 <= 26.5d0) then
        tmp = exp(-l)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(-(M * M));
	double tmp;
	if (M <= -3.2e-6) {
		tmp = t_0;
	} else if (M <= 26.5) {
		tmp = Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-(M * M))
	tmp = 0
	if M <= -3.2e-6:
		tmp = t_0
	elif M <= 26.5:
		tmp = math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-Float64(M * M)))
	tmp = 0.0
	if (M <= -3.2e-6)
		tmp = t_0;
	elseif (M <= 26.5)
		tmp = exp(Float64(-l));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-(M * M));
	tmp = 0.0;
	if (M <= -3.2e-6)
		tmp = t_0;
	elseif (M <= 26.5)
		tmp = exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-N[(M * M), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[M, -3.2e-6], t$95$0, If[LessEqual[M, 26.5], N[Exp[(-l)], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -3.1999999999999999e-6 or 26.5 < M

   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

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Simplified 97.6%

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

   9. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. lower--.f64 N/A

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

       3. fabs-sub N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. lower-fabs.f64 N/A

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

       7. mul-1-neg N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. lower-fma.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-neg.f64 N/A

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

       17. sub-neg N/A

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

       18. lower-fma.f64 N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 N/A

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

       21. lower-neg.f64 97.6

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

   11. Simplified 97.6%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 96.8

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

   14. Simplified 96.8%

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

   if -3.1999999999999999e-6 < M < 26.5

   1. Initial program 75.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Simplified 93.1%

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

   6. Taylor expanded in M around 0

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

      1. associate--r+ N/A

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

      2. exp-diff N/A

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

      3. sub-neg N/A

         \[\leadsto \frac{e^{\color{blue}{\left|n - m\right| + \left(\mathsf{neg}\left(\ell\right)\right)}}}{e^{\frac{1}{4} \cdot {\left(m + n\right)}^{2}}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{e^{\left|n - m\right| + \color{blue}{-1 \cdot \ell}}}{e^{\frac{1}{4} \cdot {\left(m + n\right)}^{2}}} \]

      5. exp-diff N/A

         \[\leadsto \color{blue}{e^{\left(\left|n - m\right| + -1 \cdot \ell\right) - \frac{1}{4} \cdot {\left(m + n\right)}^{2}}} \]

      6. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\left(\left|n - m\right| + -1 \cdot \ell\right) - \frac{1}{4} \cdot {\left(m + n\right)}^{2}}} \]

      7. mul-1-neg N/A

         \[\leadsto e^{\left(\left|n - m\right| + \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right) - \frac{1}{4} \cdot {\left(m + n\right)}^{2}} \]

      8. sub-neg N/A

         \[\leadsto e^{\color{blue}{\left(\left|n - m\right| - \ell\right)} - \frac{1}{4} \cdot {\left(m + n\right)}^{2}} \]

      9. associate--r+ N/A

         \[\leadsto e^{\color{blue}{\left|n - m\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)}} \]

      10. lower--.f64 N/A

          \[\leadsto e^{\color{blue}{\left|n - m\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)}} \]

      11. fabs-sub N/A

          \[\leadsto e^{\color{blue}{\left|m - n\right|} - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      12. sub-neg N/A

          \[\leadsto e^{\left|\color{blue}{m + \left(\mathsf{neg}\left(n\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      13. mul-1-neg N/A

          \[\leadsto e^{\left|m + \color{blue}{-1 \cdot n}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      14. lower-fabs.f64 N/A

          \[\leadsto e^{\color{blue}{\left|m + -1 \cdot n\right|} - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      15. mul-1-neg N/A

          \[\leadsto e^{\left|m + \color{blue}{\left(\mathsf{neg}\left(n\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      16. sub-neg N/A

          \[\leadsto e^{\left|\color{blue}{m - n}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      17. lower--.f64 N/A

          \[\leadsto e^{\left|\color{blue}{m - n}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

   8. Simplified 93.1%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)}} \]

   9. Taylor expanded in n around 0

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)}} \]
   10. Step-by-step derivation

       1. fabs-sub N/A

          \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)}} \]

       3. lower--.f64 N/A

          \[\leadsto e^{\color{blue}{\left|n - m\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)}} \]

       4. fabs-sub N/A

          \[\leadsto e^{\color{blue}{\left|m - n\right|} - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

       5. sub-neg N/A

          \[\leadsto e^{\left|\color{blue}{m + \left(\mathsf{neg}\left(n\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

       6. mul-1-neg N/A

          \[\leadsto e^{\left|m + \color{blue}{-1 \cdot n}\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

       7. lower-fabs.f64 N/A

          \[\leadsto e^{\color{blue}{\left|m + -1 \cdot n\right|} - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

       8. mul-1-neg N/A

          \[\leadsto e^{\left|m + \color{blue}{\left(\mathsf{neg}\left(n\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

       9. sub-neg N/A

          \[\leadsto e^{\left|\color{blue}{m - n}\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

       10. lower--.f64 N/A

           \[\leadsto e^{\left|\color{blue}{m - n}\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

       11. +-commutative N/A

           \[\leadsto e^{\left|m - n\right| - \color{blue}{\left(\frac{1}{4} \cdot {m}^{2} + \ell\right)}} \]

       12. lower-fma.f64 N/A

           \[\leadsto e^{\left|m - n\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {m}^{2}, \ell\right)}} \]

       13. unpow2 N/A

           \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{m \cdot m}, \ell\right)} \]

       14. lower-*.f64 62.7

           \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \color{blue}{m \cdot m}, \ell\right)} \]

   11. Simplified 62.7%

       \[\leadsto \color{blue}{e^{\left|m - n\right| - \mathsf{fma}\left(0.25, m \cdot m, \ell\right)}} \]

   12. Taylor expanded in l around inf

       \[\leadsto e^{\color{blue}{-1 \cdot \ell}} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

       2. lower-neg.f64 41.6

          \[\leadsto e^{\color{blue}{-\ell}} \]

   14. Simplified 41.6%

       \[\leadsto e^{\color{blue}{-\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-M \cdot M}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ e^{-\ell} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (exp (- l)))

C

double code(double K, double m, double n, double M, double l) {
	return exp(-l);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp(-l)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(-l);
}

Python

def code(K, m, n, M, l):
	return math.exp(-l)

Julia

function code(K, m, n, M, l)
	return exp(Float64(-l))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(-l);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[(-l)], $MachinePrecision]

Derivation

1. Initial program 77.6%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   2. cos-neg N/A

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \]

   3. lower-cos.f64 N/A

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \]

   4. lower-exp.f64 N/A

      \[\leadsto \cos M \cdot \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   5. lower--.f64 N/A

      \[\leadsto \cos M \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   6. fabs-sub N/A

      \[\leadsto \cos M \cdot e^{\color{blue}{\left|n - m\right|} - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \]

   7. sub-neg N/A

      \[\leadsto \cos M \cdot e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \cos M \cdot e^{\left|n + \color{blue}{-1 \cdot m}\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \]

   9. lower-fabs.f64 N/A

      \[\leadsto \cos M \cdot e^{\color{blue}{\left|n + -1 \cdot m\right|} - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \cos M \cdot e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \]

   11. sub-neg N/A

       \[\leadsto \cos M \cdot e^{\left|\color{blue}{n - m}\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \cos M \cdot e^{\left|\color{blue}{n - m}\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \]

   13. +-commutative N/A

       \[\leadsto \cos M \cdot e^{\left|n - m\right| - \color{blue}{\left({\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2} + \ell\right)}} \]

   14. unpow2 N/A

       \[\leadsto \cos M \cdot e^{\left|n - m\right| - \left(\color{blue}{\left(\frac{1}{2} \cdot \left(m + n\right) - M\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)} + \ell\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \cos M \cdot e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(m + n\right) - M, \frac{1}{2} \cdot \left(m + n\right) - M, \ell\right)}} \]

5. Simplified 96.4%

   \[\leadsto \color{blue}{\cos M \cdot e^{\left|n - m\right| - \mathsf{fma}\left(\left(m + n\right) \cdot 0.5 - M, \left(m + n\right) \cdot 0.5 - M, \ell\right)}} \]

6. Taylor expanded in M around 0

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)}} \]
7. Step-by-step derivation

   1. associate--r+ N/A

      \[\leadsto e^{\color{blue}{\left(\left|n - m\right| - \ell\right) - \frac{1}{4} \cdot {\left(m + n\right)}^{2}}} \]

   2. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\left|n - m\right| - \ell}}{e^{\frac{1}{4} \cdot {\left(m + n\right)}^{2}}}} \]

   3. sub-neg N/A

      \[\leadsto \frac{e^{\color{blue}{\left|n - m\right| + \left(\mathsf{neg}\left(\ell\right)\right)}}}{e^{\frac{1}{4} \cdot {\left(m + n\right)}^{2}}} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{e^{\left|n - m\right| + \color{blue}{-1 \cdot \ell}}}{e^{\frac{1}{4} \cdot {\left(m + n\right)}^{2}}} \]

   5. exp-diff N/A

      \[\leadsto \color{blue}{e^{\left(\left|n - m\right| + -1 \cdot \ell\right) - \frac{1}{4} \cdot {\left(m + n\right)}^{2}}} \]

   6. lower-exp.f64 N/A

      \[\leadsto \color{blue}{e^{\left(\left|n - m\right| + -1 \cdot \ell\right) - \frac{1}{4} \cdot {\left(m + n\right)}^{2}}} \]

   7. mul-1-neg N/A

      \[\leadsto e^{\left(\left|n - m\right| + \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right) - \frac{1}{4} \cdot {\left(m + n\right)}^{2}} \]

   8. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\left|n - m\right| - \ell\right)} - \frac{1}{4} \cdot {\left(m + n\right)}^{2}} \]

   9. associate--r+ N/A

      \[\leadsto e^{\color{blue}{\left|n - m\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)}} \]

   10. lower--.f64 N/A

       \[\leadsto e^{\color{blue}{\left|n - m\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)}} \]

   11. fabs-sub N/A

       \[\leadsto e^{\color{blue}{\left|m - n\right|} - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

   12. sub-neg N/A

       \[\leadsto e^{\left|\color{blue}{m + \left(\mathsf{neg}\left(n\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

   13. mul-1-neg N/A

       \[\leadsto e^{\left|m + \color{blue}{-1 \cdot n}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

   14. lower-fabs.f64 N/A

       \[\leadsto e^{\color{blue}{\left|m + -1 \cdot n\right|} - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

   15. mul-1-neg N/A

       \[\leadsto e^{\left|m + \color{blue}{\left(\mathsf{neg}\left(n\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

   16. sub-neg N/A

       \[\leadsto e^{\left|\color{blue}{m - n}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

   17. lower--.f64 N/A

       \[\leadsto e^{\left|\color{blue}{m - n}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

8. Simplified 86.1%

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)}} \]

9. Taylor expanded in n around 0

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)}} \]
10. Step-by-step derivation

    1. fabs-sub N/A

       \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

    2. lower-exp.f64 N/A

       \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)}} \]

    3. lower--.f64 N/A

       \[\leadsto e^{\color{blue}{\left|n - m\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)}} \]

    4. fabs-sub N/A

       \[\leadsto e^{\color{blue}{\left|m - n\right|} - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

    5. sub-neg N/A

       \[\leadsto e^{\left|\color{blue}{m + \left(\mathsf{neg}\left(n\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

    6. mul-1-neg N/A

       \[\leadsto e^{\left|m + \color{blue}{-1 \cdot n}\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

    7. lower-fabs.f64 N/A

       \[\leadsto e^{\color{blue}{\left|m + -1 \cdot n\right|} - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

    8. mul-1-neg N/A

       \[\leadsto e^{\left|m + \color{blue}{\left(\mathsf{neg}\left(n\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

    9. sub-neg N/A

       \[\leadsto e^{\left|\color{blue}{m - n}\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

    10. lower--.f64 N/A

        \[\leadsto e^{\left|\color{blue}{m - n}\right| - \left(\ell + \frac{1}{4} \cdot {m}^{2}\right)} \]

    11. +-commutative N/A

        \[\leadsto e^{\left|m - n\right| - \color{blue}{\left(\frac{1}{4} \cdot {m}^{2} + \ell\right)}} \]

    12. lower-fma.f64 N/A

        \[\leadsto e^{\left|m - n\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {m}^{2}, \ell\right)}} \]

    13. unpow2 N/A

        \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{m \cdot m}, \ell\right)} \]

    14. lower-*.f64 57.3

        \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \color{blue}{m \cdot m}, \ell\right)} \]

11. Simplified 57.3%

    \[\leadsto \color{blue}{e^{\left|m - n\right| - \mathsf{fma}\left(0.25, m \cdot m, \ell\right)}} \]

12. Taylor expanded in l around inf

    \[\leadsto e^{\color{blue}{-1 \cdot \ell}} \]
13. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

    2. lower-neg.f64 30.5

       \[\leadsto e^{\color{blue}{-\ell}} \]

14. Simplified 30.5%

    \[\leadsto e^{\color{blue}{-\ell}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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)))))))