Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.0% → 96.7%

Time: 13.8s

Alternatives: 11

Speedup: 2.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. cos-neg N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. fabs-sub N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-fabs.f64 N/A

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

   10. mul-1-neg N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 N/A

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

5. Applied rewrites 96.4%

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

Alternative 2: 95.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-M \cdot M}\\ \mathbf{if}\;M \leq -1.65 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 31.5:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (* M M)))))
   (if (<= M -1.65e+25)
     t_0
     (if (<= M 31.5)
       (exp (- (fabs (- n m)) (fma 0.25 (* (+ n m) (+ n m)) l)))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-(M * M));
	double tmp;
	if (M <= -1.65e+25) {
		tmp = t_0;
	} else if (M <= 31.5) {
		tmp = exp((fabs((n - m)) - fma(0.25, ((n + m) * (n + m)), l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-Float64(M * M)))
	tmp = 0.0
	if (M <= -1.65e+25)
		tmp = t_0;
	elseif (M <= 31.5)
		tmp = exp(Float64(abs(Float64(n - m)) - fma(0.25, Float64(Float64(n + m) * Float64(n + m)), l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-N[(M * M), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[M, -1.65e+25], t$95$0, If[LessEqual[M, 31.5], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(N[(n + m), $MachinePrecision] * N[(n + m), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -1.6500000000000001e25 or 31.5 < M

   1. Initial program 79.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 97.5%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 97.5

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

   11. Applied rewrites 97.5%

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

   if -1.6500000000000001e25 < M < 31.5

   1. Initial program 75.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in M around 0

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

      1. 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. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 93.3

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

   8. Applied rewrites 93.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.65 \cdot 10^{+25}:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{elif}\;M \leq 31.5:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-M \cdot M}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. cos-neg N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. fabs-sub N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-fabs.f64 N/A

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

   10. mul-1-neg N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 N/A

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

5. Applied rewrites 96.4%

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

6. Taylor expanded in M around 0

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

8. Applied rewrites 95.2%

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

9. Final simplification 95.2%

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

Alternative 4: 62.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{-210}:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(n, n \cdot -0.25, \left|n - m\right|\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -55.0)
   (exp (* -0.25 (* m m)))
   (if (<= m 1.02e-210)
     (exp (- (* M M)))
     (exp (fma n (* n -0.25) (fabs (- n m)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -55.0) {
		tmp = exp((-0.25 * (m * m)));
	} else if (m <= 1.02e-210) {
		tmp = exp(-(M * M));
	} else {
		tmp = exp(fma(n, (n * -0.25), fabs((n - m))));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -55.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (m <= 1.02e-210)
		tmp = exp(Float64(-Float64(M * M)));
	else
		tmp = exp(fma(n, Float64(n * -0.25), abs(Float64(n - m))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -55.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 1.02e-210], N[Exp[(-N[(M * M), $MachinePrecision])], $MachinePrecision], N[Exp[N[(n * N[(n * -0.25), $MachinePrecision] + N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -55

   1. Initial program 75.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in m around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 96.8

          \[\leadsto 1 \cdot e^{-0.25 \cdot \color{blue}{\left(m \cdot m\right)}} \]

   11. Applied rewrites 96.8%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(m \cdot m\right)}} \]

   if -55 < m < 1.02000000000000002e-210

   1. Initial program 80.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 92.7%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 47.0

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

   11. Applied rewrites 47.0%

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

   if 1.02000000000000002e-210 < m

   1. Initial program 75.9%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 95.5%

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

   9. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 50.7

          \[\leadsto 1 \cdot e^{\left|n - m\right| - 0.25 \cdot \color{blue}{\left(n \cdot n\right)}} \]

   11. Applied rewrites 50.7%

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

       1. lift--.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift--.f64 N/A

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

       6. lift-exp.f64 N/A

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

       7. *-lft-identity 50.7

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

       8. lift--.f64 N/A

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

       9. sub-neg N/A

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

       10. +-commutative N/A

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

       11. lift-*.f64 N/A

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

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

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

       13. metadata-eval N/A

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

       14. lift-*.f64 N/A

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

       17. lower-fma.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 50.7

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

   13. Applied rewrites 50.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{-210}:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(n, n \cdot -0.25, \left|n - m\right|\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{-210}:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -55.0)
   (exp (* -0.25 (* m m)))
   (if (<= m 1.02e-210) (exp (- (* M M))) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -55.0) {
		tmp = exp((-0.25 * (m * m)));
	} else if (m <= 1.02e-210) {
		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 (m <= (-55.0d0)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (m <= 1.02d-210) 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 (m <= -55.0) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (m <= 1.02e-210) {
		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 m <= -55.0:
		tmp = math.exp((-0.25 * (m * m)))
	elif m <= 1.02e-210:
		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 (m <= -55.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (m <= 1.02e-210)
		tmp = exp(Float64(-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 <= -55.0)
		tmp = exp((-0.25 * (m * m)));
	elseif (m <= 1.02e-210)
		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[m, -55.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 1.02e-210], 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 m < -55

   1. Initial program 75.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in m around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 96.8

          \[\leadsto 1 \cdot e^{-0.25 \cdot \color{blue}{\left(m \cdot m\right)}} \]

   11. Applied rewrites 96.8%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(m \cdot m\right)}} \]

   if -55 < m < 1.02000000000000002e-210

   1. Initial program 80.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 92.7%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 47.0

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

   11. Applied rewrites 47.0%

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

   if 1.02000000000000002e-210 < m

   1. Initial program 75.9%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 95.5%

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

   9. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 57.5

          \[\leadsto 1 \cdot e^{-0.25 \cdot \color{blue}{\left(n \cdot n\right)}} \]

   11. Applied rewrites 57.5%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(n \cdot n\right)}} \]

   12. Taylor expanded in n around inf

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

       1. lower-exp.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 57.5

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

   14. Applied rewrites 57.5%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{-210}:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{if}\;n \leq -54:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (* n n)))))
   (if (<= n -54.0) t_0 (if (<= n 55.0) (exp (- (* M M))) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * (n * n)));
	double tmp;
	if (n <= -54.0) {
		tmp = t_0;
	} else if (n <= 55.0) {
		tmp = exp(-(M * M));
	} 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(((-0.25d0) * (n * n)))
    if (n <= (-54.0d0)) then
        tmp = t_0
    else if (n <= 55.0d0) then
        tmp = exp(-(m_1 * m_1))
    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((-0.25 * (n * n)));
	double tmp;
	if (n <= -54.0) {
		tmp = t_0;
	} else if (n <= 55.0) {
		tmp = Math.exp(-(M * M));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * (n * n)))
	tmp = 0
	if n <= -54.0:
		tmp = t_0
	elif n <= 55.0:
		tmp = math.exp(-(M * M))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * Float64(n * n)))
	tmp = 0.0
	if (n <= -54.0)
		tmp = t_0;
	elseif (n <= 55.0)
		tmp = exp(Float64(-Float64(M * M)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (n * n)));
	tmp = 0.0;
	if (n <= -54.0)
		tmp = t_0;
	elseif (n <= 55.0)
		tmp = exp(-(M * M));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -54.0], t$95$0, If[LessEqual[n, 55.0], N[Exp[(-N[(M * M), $MachinePrecision])], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -54 or 55 < n

   1. Initial program 69.7%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 95.8

          \[\leadsto 1 \cdot e^{-0.25 \cdot \color{blue}{\left(n \cdot n\right)}} \]

   11. Applied rewrites 95.8%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(n \cdot n\right)}} \]

   12. Taylor expanded in n around inf

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

       1. lower-exp.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 95.8

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

   14. Applied rewrites 95.8%

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

   if -54 < n < 55

   1. Initial program 87.4%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 91.9%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 54.2

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

   11. Applied rewrites 54.2%

       \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -54:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;e^{-M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{if}\;n \leq -53:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 52:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (* n n)))))
   (if (<= n -53.0) t_0 (if (<= n 52.0) (exp (- l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * (n * n)));
	double tmp;
	if (n <= -53.0) {
		tmp = t_0;
	} else if (n <= 52.0) {
		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(((-0.25d0) * (n * n)))
    if (n <= (-53.0d0)) then
        tmp = t_0
    else if (n <= 52.0d0) 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((-0.25 * (n * n)));
	double tmp;
	if (n <= -53.0) {
		tmp = t_0;
	} else if (n <= 52.0) {
		tmp = Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * (n * n)))
	tmp = 0
	if n <= -53.0:
		tmp = t_0
	elif n <= 52.0:
		tmp = math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * Float64(n * n)))
	tmp = 0.0
	if (n <= -53.0)
		tmp = t_0;
	elseif (n <= 52.0)
		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((-0.25 * (n * n)));
	tmp = 0.0;
	if (n <= -53.0)
		tmp = t_0;
	elseif (n <= 52.0)
		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[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -53.0], t$95$0, If[LessEqual[n, 52.0], N[Exp[(-l)], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -53 or 52 < n

   1. Initial program 69.7%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 95.8

          \[\leadsto 1 \cdot e^{-0.25 \cdot \color{blue}{\left(n \cdot n\right)}} \]

   11. Applied rewrites 95.8%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(n \cdot n\right)}} \]

   12. Taylor expanded in n around inf

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

       1. lower-exp.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 95.8

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

   14. Applied rewrites 95.8%

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

   if -53 < n < 52

   1. Initial program 87.4%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 36.7

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)} \cdot e^{\mathsf{neg}\left(\ell\right)} \]

      2. *-commutative N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot \color{blue}{\left(n \cdot K\right)}\right) \cdot e^{\mathsf{neg}\left(\ell\right)} \]

      3. lower-*.f64 37.2

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

   8. Applied rewrites 37.2%

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

   9. Taylor expanded in n around 0

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

       1. neg-mul-1 N/A

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

       2. lower-exp.f64 N/A

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

       3. neg-mul-1 N/A

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

       4. lower-neg.f64 35.3

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

   11. Applied rewrites 35.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;n \leq 52:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in l around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 27.6

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

5. Applied rewrites 27.6%

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

6. Taylor expanded in n around inf

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

   1. lower-*.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)} \cdot e^{\mathsf{neg}\left(\ell\right)} \]

   2. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{2} \cdot \color{blue}{\left(n \cdot K\right)}\right) \cdot e^{\mathsf{neg}\left(\ell\right)} \]

   3. lower-*.f64 29.1

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

8. Applied rewrites 29.1%

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

9. Taylor expanded in n around 0

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

    1. neg-mul-1 N/A

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

    2. lower-exp.f64 N/A

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

    3. neg-mul-1 N/A

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

    4. lower-neg.f64 30.5

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

11. Applied rewrites 30.5%

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

Alternative 9: 9.3% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(K, K \cdot \left(\left(n \cdot n\right) \cdot -0.125\right), 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (fma K (* K (* (* n n) -0.125)) 1.0)
  (fma l (fma l (fma l -0.16666666666666666 0.5) -1.0) 1.0)))

C

double code(double K, double m, double n, double M, double l) {
	return fma(K, (K * ((n * n) * -0.125)), 1.0) * fma(l, fma(l, fma(l, -0.16666666666666666, 0.5), -1.0), 1.0);
}

Julia

function code(K, m, n, M, l)
	return Float64(fma(K, Float64(K * Float64(Float64(n * n) * -0.125)), 1.0) * fma(l, fma(l, fma(l, -0.16666666666666666, 0.5), -1.0), 1.0))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[(K * N[(K * N[(N[(n * n), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(l * N[(l * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in l around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 27.6

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

5. Applied rewrites 27.6%

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

6. Taylor expanded in n around inf

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

   1. lower-*.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)} \cdot e^{\mathsf{neg}\left(\ell\right)} \]

   2. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{2} \cdot \color{blue}{\left(n \cdot K\right)}\right) \cdot e^{\mathsf{neg}\left(\ell\right)} \]

   3. lower-*.f64 29.1

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

8. Applied rewrites 29.1%

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

9. Taylor expanded in l around 0

   \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right) + 1\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1, 1\right)} \]

    3. sub-neg N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

    4. metadata-eval N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + \color{blue}{-1}, 1\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\mathsf{fma}\left(\ell, \frac{1}{2} + \frac{-1}{6} \cdot \ell, -1\right)}, 1\right) \]

    6. +-commutative N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \color{blue}{\frac{-1}{6} \cdot \ell + \frac{1}{2}}, -1\right), 1\right) \]

    7. *-commutative N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

    8. lower-fma.f64 9.6

       \[\leadsto \cos \left(0.5 \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \color{blue}{\mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

11. Applied rewrites 9.6%

    \[\leadsto \cos \left(0.5 \cdot \left(n \cdot K\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

12. Taylor expanded in n around 0

    \[\leadsto \color{blue}{\left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {n}^{2}\right)\right)} \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]
13. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \left(1 + \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot {n}^{2}}\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    2. +-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot {n}^{2} + 1\right)} \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    3. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left({K}^{2} \cdot \frac{-1}{8}\right)} \cdot {n}^{2} + 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    4. associate-*r* N/A

       \[\leadsto \left(\color{blue}{{K}^{2} \cdot \left(\frac{-1}{8} \cdot {n}^{2}\right)} + 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    5. unpow2 N/A

       \[\leadsto \left(\color{blue}{\left(K \cdot K\right)} \cdot \left(\frac{-1}{8} \cdot {n}^{2}\right) + 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    6. associate-*l* N/A

       \[\leadsto \left(\color{blue}{K \cdot \left(K \cdot \left(\frac{-1}{8} \cdot {n}^{2}\right)\right)} + 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    7. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(K, K \cdot \left(\frac{-1}{8} \cdot {n}^{2}\right), 1\right)} \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(K, \color{blue}{K \cdot \left(\frac{-1}{8} \cdot {n}^{2}\right)}, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    9. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(K, K \cdot \color{blue}{\left(\frac{-1}{8} \cdot {n}^{2}\right)}, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(K, K \cdot \left(\frac{-1}{8} \cdot \color{blue}{\left(n \cdot n\right)}\right), 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    11. lower-*.f64 8.3

        \[\leadsto \mathsf{fma}\left(K, K \cdot \left(-0.125 \cdot \color{blue}{\left(n \cdot n\right)}\right), 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right) \]

14. Applied rewrites 8.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(K, K \cdot \left(-0.125 \cdot \left(n \cdot n\right)\right), 1\right)} \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right) \]

15. Final simplification 8.3%

    \[\leadsto \mathsf{fma}\left(K, K \cdot \left(\left(n \cdot n\right) \cdot -0.125\right), 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right) \]
16. Add Preprocessing

Alternative 10: 9.8% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (fma l (fma l (fma l -0.16666666666666666 0.5) -1.0) 1.0))

C

double code(double K, double m, double n, double M, double l) {
	return fma(l, fma(l, fma(l, -0.16666666666666666, 0.5), -1.0), 1.0);
}

Julia

function code(K, m, n, M, l)
	return fma(l, fma(l, fma(l, -0.16666666666666666, 0.5), -1.0), 1.0)
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(l * N[(l * N[(l * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in l around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 27.6

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

5. Applied rewrites 27.6%

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

6. Taylor expanded in n around inf

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

   1. lower-*.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)} \cdot e^{\mathsf{neg}\left(\ell\right)} \]

   2. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{2} \cdot \color{blue}{\left(n \cdot K\right)}\right) \cdot e^{\mathsf{neg}\left(\ell\right)} \]

   3. lower-*.f64 29.1

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

8. Applied rewrites 29.1%

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

9. Taylor expanded in l around 0

   \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right) + 1\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1, 1\right)} \]

    3. sub-neg N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

    4. metadata-eval N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + \color{blue}{-1}, 1\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\mathsf{fma}\left(\ell, \frac{1}{2} + \frac{-1}{6} \cdot \ell, -1\right)}, 1\right) \]

    6. +-commutative N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \color{blue}{\frac{-1}{6} \cdot \ell + \frac{1}{2}}, -1\right), 1\right) \]

    7. *-commutative N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

    8. lower-fma.f64 9.6

       \[\leadsto \cos \left(0.5 \cdot \left(n \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \color{blue}{\mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

11. Applied rewrites 9.6%

    \[\leadsto \cos \left(0.5 \cdot \left(n \cdot K\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

12. Taylor expanded in n around 0

    \[\leadsto \color{blue}{1 + \ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right)} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right) + 1} \]

    2. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1, 1\right)} \]

    3. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + \color{blue}{-1}, 1\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\mathsf{fma}\left(\ell, \frac{1}{2} + \frac{-1}{6} \cdot \ell, -1\right)}, 1\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \color{blue}{\frac{-1}{6} \cdot \ell + \frac{1}{2}}, -1\right), 1\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

    8. lower-fma.f64 8.7

       \[\leadsto \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \color{blue}{\mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

14. Applied rewrites 8.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
15. Add Preprocessing

Alternative 11: 6.7% accurate, 359.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

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 = 1.0d0
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

Python

def code(K, m, n, M, l):
	return 1.0

Julia

function code(K, m, n, M, l)
	return 1.0
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in m around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 37.7

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

5. Applied rewrites 37.7%

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

6. Taylor expanded in m around 0

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

   1. lower-cos.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 5.5

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

8. Applied rewrites 5.5%

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

9. Taylor expanded in n around 0

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

    1. cos-neg N/A

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

    2. lower-cos.f64 5.8

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

11. Applied rewrites 5.8%

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

12. Taylor expanded in M around 0

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

14. Applied rewrites 5.8%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))