Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.3% → 96.7%

Time: 13.0s

Alternatives: 7

Speedup: 2.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma 0.5 (+ m n) (- 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, (m + n), -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(m + n), 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[(m + n), $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 74.4%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 97.8%

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

6. Final simplification 97.8%

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

Alternative 2: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.4%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 97.8%

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

6. Taylor expanded in M around 0

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

8. Applied rewrites 97.3%

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

9. Final simplification 97.3%

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

Alternative 3: 92.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ t_1 := 1 \cdot e^{t\_0 - M \cdot M}\\ \mathbf{if}\;M \leq -1.5 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq 3.9 \cdot 10^{+119}:\\ \;\;\;\;e^{t\_0 - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))) (t_1 (* 1.0 (exp (- t_0 (* M M))))))
   (if (<= M -1.5e+107)
     t_1
     (if (<= M 3.9e+119) (exp (- t_0 (fma 0.25 (* (+ m n) (+ m n)) l))) t_1))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double t_1 = 1.0 * exp((t_0 - (M * M)));
	double tmp;
	if (M <= -1.5e+107) {
		tmp = t_1;
	} else if (M <= 3.9e+119) {
		tmp = exp((t_0 - fma(0.25, ((m + n) * (m + n)), l)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	t_1 = Float64(1.0 * exp(Float64(t_0 - Float64(M * M))))
	tmp = 0.0
	if (M <= -1.5e+107)
		tmp = t_1;
	elseif (M <= 3.9e+119)
		tmp = exp(Float64(t_0 - fma(0.25, Float64(Float64(m + n) * Float64(m + n)), l)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 * N[Exp[N[(t$95$0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -1.5e+107], t$95$1, If[LessEqual[M, 3.9e+119], N[Exp[N[(t$95$0 - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if M < -1.50000000000000012e107 or 3.8999999999999998e119 < M

   1. Initial program 77.2%

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

   3. Taylor expanded in K around 0

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

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 98.9%

       \[\leadsto 1 \cdot e^{\left|n - m\right| - M \cdot M} \]

   if -1.50000000000000012e107 < M < 3.8999999999999998e119

   1. Initial program 72.8%

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

   3. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. fabs-sub N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. lower-fabs.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 91.3%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.5 \cdot 10^{+107}:\\ \;\;\;\;1 \cdot e^{\left|n - m\right| - M \cdot M}\\ \mathbf{elif}\;M \leq 3.9 \cdot 10^{+119}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{\left|n - m\right| - M \cdot M}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.0032:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq 1.95 \cdot 10^{-273}:\\ \;\;\;\;1 \cdot e^{\left|n - m\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -0.0032)
   (exp (* (* m m) -0.25))
   (if (<= m 1.95e-273)
     (* 1.0 (exp (- (fabs (- n m)) (* M M))))
     (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.0032) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= 1.95e-273) {
		tmp = 1.0 * exp((fabs((n - m)) - (M * M)));
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-0.0032d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= 1.95d-273) then
        tmp = 1.0d0 * exp((abs((n - m)) - (m_1 * m_1)))
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.0032) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= 1.95e-273) {
		tmp = 1.0 * Math.exp((Math.abs((n - m)) - (M * M)));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -0.0032:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= 1.95e-273:
		tmp = 1.0 * math.exp((math.fabs((n - m)) - (M * M)))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -0.0032)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= 1.95e-273)
		tmp = Float64(1.0 * exp(Float64(abs(Float64(n - m)) - Float64(M * M))));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -0.0032)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= 1.95e-273)
		tmp = 1.0 * exp((abs((n - m)) - (M * M)));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -0.0032], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 1.95e-273], N[(1.0 * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.00320000000000000015

   1. Initial program 62.4%

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

   3. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. fabs-sub N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. lower-fabs.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 93.0%

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

   if -0.00320000000000000015 < m < 1.9500000000000002e-273

   1. Initial program 82.3%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 96.6%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 95.3%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 53.2%

       \[\leadsto 1 \cdot e^{\left|n - m\right| - M \cdot M} \]

   if 1.9500000000000002e-273 < m

   1. Initial program 76.1%

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

   3. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. fabs-sub N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. lower-fabs.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 90.1%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 61.8%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0032:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq 1.95 \cdot 10^{-273}:\\ \;\;\;\;1 \cdot e^{\left|n - m\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -1.05 \cdot 10^{-75}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -53.0)
   (exp (* (* m m) -0.25))
   (if (<= m -1.05e-75) (exp (- l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= -1.05e-75) {
		tmp = exp(-l);
	} 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 <= (-53.0d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= (-1.05d-75)) then
        tmp = exp(-l)
    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 <= -53.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= -1.05e-75) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -53.0:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= -1.05e-75:
		tmp = math.exp(-l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -53.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= -1.05e-75)
		tmp = exp(Float64(-l));
	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 <= -53.0)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= -1.05e-75)
		tmp = exp(-l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -53.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -1.05e-75], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -53

   1. Initial program 63.1%

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

   3. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. fabs-sub N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. lower-fabs.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

   5. Applied rewrites 61.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 96.9%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 97.0%

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

   if -53 < m < -1.0500000000000001e-75

   1. Initial program 74.9%

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

   3. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. fabs-sub N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. lower-fabs.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 53.2%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 44.4%

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

   if -1.0500000000000001e-75 < m

   1. Initial program 78.7%

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

   3. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. fabs-sub N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. lower-fabs.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 61.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -1.05 \cdot 10^{-75}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 720.0) (exp (* (* m m) -0.25)) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = exp(((m * m) * -0.25));
	} else {
		tmp = exp(-l);
	}
	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 (l <= 720.0d0) then
        tmp = exp(((m * m) * (-0.25d0)))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 720.0:
		tmp = math.exp(((m * m) * -0.25))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 720.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 720.0)
		tmp = exp(((m * m) * -0.25));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 720.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 720

   1. Initial program 76.3%

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

   3. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. fabs-sub N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. lower-fabs.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 80.9%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 54.7%

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

   if 720 < l

   1. Initial program 68.8%

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

   3. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. fabs-sub N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. lower-fabs.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 100.0%

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

Alternative 7: 35.2% 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 74.4%

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

3. Taylor expanded in M around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower--.f64 N/A

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

   10. fabs-sub N/A

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

   11. sub-neg N/A

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

   12. mul-1-neg N/A

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

   13. lower-fabs.f64 N/A

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

   14. mul-1-neg N/A

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

   15. sub-neg N/A

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

   16. lower--.f64 N/A

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

   17. +-commutative N/A

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

5. Applied rewrites 62.2%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 85.7%

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

9. Taylor expanded in l around inf

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

11. Applied rewrites 32.4%

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

Reproduce

herbie shell --seed 2024221 
(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)))))))