Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.3% → 96.7%

Time: 13.0s

Alternatives: 11

Speedup: 2.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

3. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. lower-cos.f64 97.4

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

5. Simplified 97.4%

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

6. Final simplification 97.4%

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

Alternative 2: 95.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -6200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 28:\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, \left(m + n\right) \cdot \left(m + n\right), \left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* M (- M)))))
   (if (<= M -6200.0)
     t_0
     (if (<= M 28.0)
       (exp (fma -0.25 (* (+ m n) (+ m n)) (- (fabs (- m n)) 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 <= -6200.0) {
		tmp = t_0;
	} else if (M <= 28.0) {
		tmp = exp(fma(-0.25, ((m + n) * (m + n)), (fabs((m - n)) - l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(M * Float64(-M)))
	tmp = 0.0
	if (M <= -6200.0)
		tmp = t_0;
	elseif (M <= 28.0)
		tmp = exp(fma(-0.25, Float64(Float64(m + n) * Float64(m + n)), Float64(abs(Float64(m - n)) - 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, -6200.0], t$95$0, If[LessEqual[M, 28.0], N[Exp[N[(-0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -6200 or 28 < M

   1. Initial program 80.5%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.7

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

   8. Simplified 83.7%

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

   9. Taylor expanded in M around 0

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

   11. Simplified 83.0%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. lower-neg.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       4. lower-*.f64 99.3

          \[\leadsto 1 \cdot e^{-\color{blue}{M \cdot M}} \]

   14. Simplified 99.3%

       \[\leadsto 1 \cdot e^{\color{blue}{-M \cdot M}} \]

   if -6200 < M < 28

   1. Initial program 72.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

   8. Simplified 31.8%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

   9. Taylor expanded in K around 0

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

       1. fabs-sub N/A

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

       2. associate--r+ N/A

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

       3. sub-neg N/A

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

       4. fabs-sub N/A

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

       5. neg-mul-1 N/A

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

       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. sub-neg N/A

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

       8. fabs-sub N/A

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

       9. neg-mul-1 N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

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

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 N/A

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

   11. Simplified 94.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6200:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 28:\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, \left(m + n\right) \cdot \left(m + n\right), \left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.00092:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq -7.2 \cdot 10^{-300}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -0.00092)
   (exp (* m (* m -0.25)))
   (if (<= m -7.2e-300)
     (exp (- (- (fabs (- m n)) l) (* M M)))
     (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.00092) {
		tmp = exp((m * (m * -0.25)));
	} else if (m <= -7.2e-300) {
		tmp = exp(((fabs((m - n)) - l) - (M * M)));
	} else {
		tmp = exp((-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 <= (-0.00092d0)) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (m <= (-7.2d-300)) then
        tmp = exp(((abs((m - n)) - l) - (m_1 * m_1)))
    else
        tmp = exp(((-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 <= -0.00092) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (m <= -7.2e-300) {
		tmp = Math.exp(((Math.abs((m - n)) - l) - (M * M)));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -0.00092:
		tmp = math.exp((m * (m * -0.25)))
	elif m <= -7.2e-300:
		tmp = math.exp(((math.fabs((m - n)) - l) - (M * M)))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -0.00092)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (m <= -7.2e-300)
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64(M * M)));
	else
		tmp = exp(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 <= -0.00092)
		tmp = exp((m * (m * -0.25)));
	elseif (m <= -7.2e-300)
		tmp = exp(((abs((m - n)) - l) - (M * M)));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -0.00092], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -7.2e-300], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9.2000000000000003e-4

   1. Initial program 62.7%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 16.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

   8. Simplified 16.9%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

   9. Taylor expanded in K around 0

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

       1. fabs-sub N/A

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

       2. associate--r+ N/A

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

       3. sub-neg N/A

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

       4. fabs-sub N/A

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

       5. neg-mul-1 N/A

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

       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. sub-neg N/A

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

       8. fabs-sub N/A

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

       9. neg-mul-1 N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

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

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 N/A

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

   11. Simplified 98.3%

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

   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-*r* 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 98.3

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

   14. Simplified 98.3%

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

   if -9.2000000000000003e-4 < m < -7.20000000000000031e-300

   1. Initial program 85.3%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 95.9

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

   5. Simplified 95.9%

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

   6. Taylor expanded in M around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.7

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

   8. Simplified 72.7%

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

   9. Taylor expanded in M around 0

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

   11. Simplified 72.7%

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

   if -7.20000000000000031e-300 < m

   1. Initial program 78.2%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 33.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

   8. Simplified 25.8%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

   9. Taylor expanded in K around 0

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

       1. fabs-sub N/A

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

       2. associate--r+ N/A

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

       3. sub-neg N/A

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

       4. fabs-sub N/A

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

       5. neg-mul-1 N/A

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

       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. sub-neg N/A

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

       8. fabs-sub N/A

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

       9. neg-mul-1 N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

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

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 N/A

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

   11. Simplified 87.6%

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

   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. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 56.7

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

   14. Simplified 56.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.00092:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq -7.2 \cdot 10^{-300}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.8 \cdot 10^{-177}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 2.8e-177)
   (exp (* m (* m -0.25)))
   (if (<= n 54.0) (exp (* M (- M))) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2.8e-177) {
		tmp = exp((m * (m * -0.25)));
	} else if (n <= 54.0) {
		tmp = exp((M * -M));
	} else {
		tmp = exp((-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 (n <= 2.8d-177) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (n <= 54.0d0) then
        tmp = exp((m_1 * -m_1))
    else
        tmp = exp(((-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 (n <= 2.8e-177) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (n <= 54.0) {
		tmp = Math.exp((M * -M));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 2.8e-177:
		tmp = math.exp((m * (m * -0.25)))
	elif n <= 54.0:
		tmp = math.exp((M * -M))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 2.8e-177)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (n <= 54.0)
		tmp = exp(Float64(M * Float64(-M)));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 2.8e-177)
		tmp = exp((m * (m * -0.25)));
	elseif (n <= 54.0)
		tmp = exp((M * -M));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 2.8e-177], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 54.0], N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 2.79999999999999987e-177

   1. Initial program 80.3%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 40.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

   8. Simplified 31.0%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

   9. Taylor expanded in K around 0

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

       1. fabs-sub N/A

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

       2. associate--r+ N/A

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

       3. sub-neg N/A

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

       4. fabs-sub N/A

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

       5. neg-mul-1 N/A

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

       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. sub-neg N/A

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

       8. fabs-sub N/A

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

       9. neg-mul-1 N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

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

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 N/A

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

   11. Simplified 89.6%

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

   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-*r* 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 59.3

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

   14. Simplified 59.3%

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

   if 2.79999999999999987e-177 < n < 54

   1. Initial program 76.8%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 92.0

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

   5. Simplified 92.0%

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

   6. Taylor expanded in M around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.4

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

   8. Simplified 65.4%

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

   9. Taylor expanded in M around 0

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

   11. Simplified 62.7%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. lower-neg.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       4. lower-*.f64 63.1

          \[\leadsto 1 \cdot e^{-\color{blue}{M \cdot M}} \]

   14. Simplified 63.1%

       \[\leadsto 1 \cdot e^{\color{blue}{-M \cdot M}} \]

   if 54 < n

   1. Initial program 66.7%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 15.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

   8. Simplified 15.0%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

   9. Taylor expanded in K around 0

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

       1. fabs-sub N/A

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

       2. associate--r+ N/A

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

       3. sub-neg N/A

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

       4. fabs-sub N/A

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

       5. neg-mul-1 N/A

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

       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. sub-neg N/A

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

       8. fabs-sub N/A

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

       9. neg-mul-1 N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

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

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 N/A

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

   11. Simplified 100.0%

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

   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. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 98.4

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

   14. Simplified 98.4%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.8 \cdot 10^{-177}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{if}\;m \leq -0.00092:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* m (* m -0.25)))))
   (if (<= m -0.00092) t_0 (if (<= m 1.75e-47) (exp (- l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((m * (m * -0.25)));
	double tmp;
	if (m <= -0.00092) {
		tmp = t_0;
	} else if (m <= 1.75e-47) {
		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 * (m * (-0.25d0))))
    if (m <= (-0.00092d0)) then
        tmp = t_0
    else if (m <= 1.75d-47) 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 * -0.25)));
	double tmp;
	if (m <= -0.00092) {
		tmp = t_0;
	} else if (m <= 1.75e-47) {
		tmp = Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((m * (m * -0.25)))
	tmp = 0
	if m <= -0.00092:
		tmp = t_0
	elif m <= 1.75e-47:
		tmp = math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(m * Float64(m * -0.25)))
	tmp = 0.0
	if (m <= -0.00092)
		tmp = t_0;
	elseif (m <= 1.75e-47)
		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 * -0.25)));
	tmp = 0.0;
	if (m <= -0.00092)
		tmp = t_0;
	elseif (m <= 1.75e-47)
		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 * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -0.00092], t$95$0, If[LessEqual[m, 1.75e-47], N[Exp[(-l)], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -9.2000000000000003e-4 or 1.7499999999999999e-47 < m

   1. Initial program 69.6%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 21.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

   8. Simplified 20.3%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

   9. Taylor expanded in K around 0

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

       1. fabs-sub N/A

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

       2. associate--r+ N/A

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

       3. sub-neg N/A

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

       4. fabs-sub N/A

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

       5. neg-mul-1 N/A

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

       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. sub-neg N/A

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

       8. fabs-sub N/A

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

       9. neg-mul-1 N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

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

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 N/A

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

   11. Simplified 95.0%

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

   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-*r* 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 92.9

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

   14. Simplified 92.9%

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

   if -9.2000000000000003e-4 < m < 1.7499999999999999e-47

   1. Initial program 84.9%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 51.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

   8. Simplified 32.6%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

   9. Taylor expanded in K around 0

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

       1. fabs-sub N/A

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

       2. associate--r+ N/A

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

       3. sub-neg N/A

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

       4. fabs-sub N/A

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

       5. neg-mul-1 N/A

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

       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. sub-neg N/A

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

       8. fabs-sub N/A

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

       9. neg-mul-1 N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

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

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 N/A

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

   11. Simplified 78.6%

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

   12. Taylor expanded in l around inf

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

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 47.3

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

   14. Simplified 47.3%

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

Alternative 6: 65.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.7 \cdot 10^{-16}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.7e-16) (exp (* m (* m -0.25))) (exp (* -0.25 (* n n)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.7e-16) {
		tmp = exp((m * (m * -0.25)));
	} else {
		tmp = exp((-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 (n <= 1.7d-16) then
        tmp = exp((m * (m * (-0.25d0))))
    else
        tmp = exp(((-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 (n <= 1.7e-16) {
		tmp = Math.exp((m * (m * -0.25)));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.7e-16:
		tmp = math.exp((m * (m * -0.25)))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.7e-16)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1.7e-16)
		tmp = exp((m * (m * -0.25)));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.7e-16], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.7e-16

   1. Initial program 79.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 41.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

   8. Simplified 29.3%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

   9. Taylor expanded in K around 0

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

       1. fabs-sub N/A

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

       2. associate--r+ N/A

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

       3. sub-neg N/A

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

       4. fabs-sub N/A

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

       5. neg-mul-1 N/A

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

       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. sub-neg N/A

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

       8. fabs-sub N/A

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

       9. neg-mul-1 N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

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

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 N/A

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

   11. Simplified 84.4%

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

   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-*r* 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 56.9

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

   14. Simplified 56.9%

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

   if 1.7e-16 < n

   1. Initial program 68.3%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 17.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

   8. Simplified 15.9%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

   9. Taylor expanded in K around 0

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

       1. fabs-sub N/A

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

       2. associate--r+ N/A

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

       3. sub-neg N/A

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

       4. fabs-sub N/A

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

       5. neg-mul-1 N/A

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

       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. sub-neg N/A

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

       8. fabs-sub N/A

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

       9. neg-mul-1 N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

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

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 N/A

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

   11. Simplified 96.9%

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

   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. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 95.3

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

   14. Simplified 95.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.7 \cdot 10^{-16}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 10.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(m + n\right) \cdot \left(m + n\right), 1\right)\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(\ell, t\_0 \cdot \mathsf{fma}\left(0.5, \ell, -1\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma (* -0.125 (* K K)) (* (+ m n) (+ m n)) 1.0)))
   (if (<= l -9e+116) (fma l (* t_0 (fma 0.5 l -1.0)) t_0) (cos M))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma((-0.125 * (K * K)), ((m + n) * (m + n)), 1.0);
	double tmp;
	if (l <= -9e+116) {
		tmp = fma(l, (t_0 * fma(0.5, l, -1.0)), t_0);
	} else {
		tmp = cos(M);
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(Float64(-0.125 * Float64(K * K)), Float64(Float64(m + n) * Float64(m + n)), 1.0)
	tmp = 0.0
	if (l <= -9e+116)
		tmp = fma(l, Float64(t_0 * fma(0.5, l, -1.0)), t_0);
	else
		tmp = cos(M);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[l, -9e+116], N[(l * N[(t$95$0 * N[(0.5 * l + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[Cos[M], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.00000000000000032e116

   1. Initial program 79.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 41.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

   8. Simplified 26.7%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

   9. Taylor expanded in l around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \ell}} \cdot \mathsf{fma}\left(\frac{-1}{8}, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]
   10. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \cdot \mathsf{fma}\left(\frac{-1}{8}, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

       2. lower-neg.f64 21.6

          \[\leadsto e^{\color{blue}{-\ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

   11. Simplified 21.6%

       \[\leadsto e^{\color{blue}{-\ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

   12. Taylor expanded in l around 0

       \[\leadsto \color{blue}{1 + \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right) + \ell \cdot \left(-1 \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right) + \ell \cdot \left(-1 \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\ell \cdot \left(-1 \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right)\right) + \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, -1 \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right), 1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

   14. Simplified 21.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(n + m\right) \cdot \left(n + m\right), 1\right) \cdot \mathsf{fma}\left(0.5, \ell, -1\right), \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(n + m\right) \cdot \left(n + m\right), 1\right)\right)} \]

   if -9.00000000000000032e116 < l

   1. Initial program 76.2%

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

   3. Taylor expanded in M around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 48.7

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

   5. Simplified 48.7%

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

   6. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 32.0

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

   8. Simplified 32.0%

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

   9. Taylor expanded in l around 0

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

       1. lower-cos.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 7.2

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

   11. Simplified 7.2%

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

   12. Taylor expanded in K around 0

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

       1. cos-neg N/A

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

       2. lower-cos.f64 7.5

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

   14. Simplified 7.5%

       \[\leadsto \color{blue}{\cos M} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(m + n\right) \cdot \left(m + n\right), 1\right) \cdot \mathsf{fma}\left(0.5, \ell, -1\right), \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(m + n\right) \cdot \left(m + n\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.6% 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 76.6%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

5. Simplified 35.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

6. Taylor expanded in M around 0

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

8. Simplified 26.0%

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

9. Taylor expanded in K around 0

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

    1. fabs-sub N/A

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

    2. associate--r+ N/A

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

    3. sub-neg N/A

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

    4. fabs-sub N/A

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

    5. neg-mul-1 N/A

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

    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. sub-neg N/A

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

    8. fabs-sub N/A

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

    9. neg-mul-1 N/A

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

    10. sub-neg N/A

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

    11. +-commutative N/A

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

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

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

    13. metadata-eval N/A

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

    14. lower-fma.f64 N/A

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

11. Simplified 87.5%

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

12. Taylor expanded in l around inf

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

    1. neg-mul-1 N/A

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

    2. lower-neg.f64 35.0

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

14. Simplified 35.0%

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

Alternative 9: 9.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(m + n\right) \cdot \left(m + n\right), 1\right)\\ \mathsf{fma}\left(\ell, t\_0 \cdot \mathsf{fma}\left(0.5, \ell, -1\right), t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma (* -0.125 (* K K)) (* (+ m n) (+ m n)) 1.0)))
   (fma l (* t_0 (fma 0.5 l -1.0)) t_0)))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma((-0.125 * (K * K)), ((m + n) * (m + n)), 1.0);
	return fma(l, (t_0 * fma(0.5, l, -1.0)), t_0);
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(Float64(-0.125 * Float64(K * K)), Float64(Float64(m + n) * Float64(m + n)), 1.0)
	return fma(l, Float64(t_0 * fma(0.5, l, -1.0)), t_0)
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(l * N[(t$95$0 * N[(0.5 * l + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]

Derivation

1. Initial program 76.6%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

5. Simplified 35.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

6. Taylor expanded in M around 0

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

8. Simplified 26.0%

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

9. Taylor expanded in l around inf

   \[\leadsto e^{\color{blue}{-1 \cdot \ell}} \cdot \mathsf{fma}\left(\frac{-1}{8}, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]
10. Step-by-step derivation

    1. neg-mul-1 N/A

       \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \cdot \mathsf{fma}\left(\frac{-1}{8}, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

    2. lower-neg.f64 15.1

       \[\leadsto e^{\color{blue}{-\ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

11. Simplified 15.1%

    \[\leadsto e^{\color{blue}{-\ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

12. Taylor expanded in l around 0

    \[\leadsto \color{blue}{1 + \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right) + \ell \cdot \left(-1 \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right)\right)\right)} \]
13. Step-by-step derivation

    1. associate-+r+ N/A

       \[\leadsto \color{blue}{\left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right) + \ell \cdot \left(-1 \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right)\right)} \]

    2. +-commutative N/A

       \[\leadsto \color{blue}{\ell \cdot \left(-1 \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right)\right) + \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

    3. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, -1 \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right), 1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

14. Simplified 7.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(n + m\right) \cdot \left(n + m\right), 1\right) \cdot \mathsf{fma}\left(0.5, \ell, -1\right), \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(n + m\right) \cdot \left(n + m\right), 1\right)\right)} \]

15. Final simplification 7.5%

    \[\leadsto \mathsf{fma}\left(\ell, \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(m + n\right) \cdot \left(m + n\right), 1\right) \cdot \mathsf{fma}\left(0.5, \ell, -1\right), \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(m + n\right) \cdot \left(m + n\right), 1\right)\right) \]
16. Add Preprocessing

Alternative 10: 6.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(m + n\right) \cdot \left(m + n\right)\\ \mathsf{fma}\left(\ell, \mathsf{fma}\left(0.125, \left(K \cdot K\right) \cdot t\_0, -1\right), \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), t\_0, 1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (+ m n) (+ m n))))
   (fma l (fma 0.125 (* (* K K) t_0) -1.0) (fma (* -0.125 (* K K)) t_0 1.0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (m + n) * (m + n);
	return fma(l, fma(0.125, ((K * K) * t_0), -1.0), fma((-0.125 * (K * K)), t_0, 1.0));
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(m + n) * Float64(m + n))
	return fma(l, fma(0.125, Float64(Float64(K * K) * t_0), -1.0), fma(Float64(-0.125 * Float64(K * K)), t_0, 1.0))
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]}, N[(l * N[(0.125 * N[(N[(K * K), $MachinePrecision] * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.6%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

5. Simplified 35.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

6. Taylor expanded in M around 0

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

8. Simplified 26.0%

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

9. Taylor expanded in l around inf

   \[\leadsto e^{\color{blue}{-1 \cdot \ell}} \cdot \mathsf{fma}\left(\frac{-1}{8}, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]
10. Step-by-step derivation

    1. neg-mul-1 N/A

       \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \cdot \mathsf{fma}\left(\frac{-1}{8}, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

    2. lower-neg.f64 15.1

       \[\leadsto e^{\color{blue}{-\ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

11. Simplified 15.1%

    \[\leadsto e^{\color{blue}{-\ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

12. Taylor expanded in l around 0

    \[\leadsto \color{blue}{1 + \left(-1 \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right) + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{\left(-1 \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right) + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right) + 1} \]

    2. associate-+l+ N/A

       \[\leadsto \color{blue}{-1 \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right) + \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right) + 1\right)} \]

    3. mul-1-neg N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\ell \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right)\right)} + \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right) + 1\right) \]

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

       \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{neg}\left(\left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right)\right)} + \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right) + 1\right) \]

    5. mul-1-neg N/A

       \[\leadsto \ell \cdot \color{blue}{\left(-1 \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right)} + \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right) + 1\right) \]

    6. +-commutative N/A

       \[\leadsto \ell \cdot \left(-1 \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)\right) + \color{blue}{\left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

    7. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, -1 \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right), 1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

14. Simplified 5.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{fma}\left(0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), -1\right), \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(n + m\right) \cdot \left(n + m\right), 1\right)\right)} \]

15. Final simplification 5.5%

    \[\leadsto \mathsf{fma}\left(\ell, \mathsf{fma}\left(0.125, \left(K \cdot K\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right), -1\right), \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(m + n\right) \cdot \left(m + n\right), 1\right)\right) \]
16. Add Preprocessing

Alternative 11: 6.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(m + n\right) \cdot \left(m + n\right), 1\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (fma (* -0.125 (* K K)) (* (+ m n) (+ m n)) 1.0))

C

double code(double K, double m, double n, double M, double l) {
	return fma((-0.125 * (K * K)), ((m + n) * (m + n)), 1.0);
}

Julia

function code(K, m, n, M, l)
	return fma(Float64(-0.125 * Float64(K * K)), Float64(Float64(m + n) * Float64(m + n)), 1.0)
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\left(\cos \left(\mathsf{neg}\left(M\right)\right) + K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(K \cdot \left(\frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right)\right) + \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(K, \frac{-1}{8} \cdot \left(K \cdot \left(\cos \left(\mathsf{neg}\left(M\right)\right) \cdot {\left(m + n\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left(\sin \left(\mathsf{neg}\left(M\right)\right) \cdot \left(m + n\right)\right), \cos \left(\mathsf{neg}\left(M\right)\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

5. Simplified 35.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(K, \mathsf{fma}\left(m + n, 0.5 \cdot \sin M, \left(\left(K \cdot -0.125\right) \cdot \cos M\right) \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \cos M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

6. Taylor expanded in M around 0

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \cdot \left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \]

8. Simplified 26.0%

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right)} \]

9. Taylor expanded in l around inf

   \[\leadsto e^{\color{blue}{-1 \cdot \ell}} \cdot \mathsf{fma}\left(\frac{-1}{8}, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]
10. Step-by-step derivation

    1. neg-mul-1 N/A

       \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \cdot \mathsf{fma}\left(\frac{-1}{8}, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

    2. lower-neg.f64 15.1

       \[\leadsto e^{\color{blue}{-\ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

11. Simplified 15.1%

    \[\leadsto e^{\color{blue}{-\ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(K \cdot K\right), 1\right) \]

12. Taylor expanded in l around 0

    \[\leadsto \color{blue}{1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right) + 1} \]

    2. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot {\left(m + n\right)}^{2}} + 1 \]

    3. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{8} \cdot {K}^{2}, {\left(m + n\right)}^{2}, 1\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot {K}^{2}}, {\left(m + n\right)}^{2}, 1\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(K \cdot K\right)}, {\left(m + n\right)}^{2}, 1\right) \]

    6. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(K \cdot K\right)}, {\left(m + n\right)}^{2}, 1\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \left(K \cdot K\right), \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, 1\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \left(K \cdot K\right), \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, 1\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \left(K \cdot K\right), \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), 1\right) \]

    10. lower-+.f64 N/A

        \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \left(K \cdot K\right), \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), 1\right) \]

    11. +-commutative N/A

        \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \left(K \cdot K\right), \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, 1\right) \]

    12. lower-+.f64 5.2

        \[\leadsto \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, 1\right) \]

14. Simplified 5.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(n + m\right) \cdot \left(n + m\right), 1\right)} \]

15. Final simplification 5.2%

    \[\leadsto \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \left(m + n\right) \cdot \left(m + n\right), 1\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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)))))))