Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.3% → 96.1%

Time: 6.3s

Alternatives: 9

Speedup: 1.6×

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: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{if}\;M \leq -1 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) (cos M))))
   (if (<= M -1e+159)
     t_0
     (if (<= M 5e+51)
       (*
        (fma (* M M) -0.5 1.0)
        (exp (- (fabs (- n m)) (+ (pow (fma 0.5 (+ n m) (- M)) 2.0) l))))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * cos(M);
	double tmp;
	if (M <= -1e+159) {
		tmp = t_0;
	} else if (M <= 5e+51) {
		tmp = fma((M * M), -0.5, 1.0) * exp((fabs((n - m)) - (pow(fma(0.5, (n + m), -M), 2.0) + l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * cos(M))
	tmp = 0.0
	if (M <= -1e+159)
		tmp = t_0;
	elseif (M <= 5e+51)
		tmp = Float64(fma(Float64(M * M), -0.5, 1.0) * exp(Float64(abs(Float64(n - m)) - Float64((fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0) + l))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -1e+159], t$95$0, If[LessEqual[M, 5e+51], N[(N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[Power[N[(0.5 * N[(n + m), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -9.9999999999999993e158 or 5e51 < M

   1. Initial program 81.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 98.0%

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

   if -9.9999999999999993e158 < M < 5e51

   1. Initial program 75.1%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 96.0%

      \[\leadsto e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+159}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{elif}\;M \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (fabs (- n m)) (+ (pow (fma 0.5 (+ n m) (- M)) 2.0) l)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp((fabs((n - m)) - (pow(fma(0.5, (n + m), -M), 2.0) + l)));
}

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(abs(Float64(n - m)) - Float64((fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0) + l))))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[Power[N[(0.5 * N[(n + m), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.8%

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

6. Final simplification 96.8%

   \[\leadsto \cos M \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \]
7. Add Preprocessing

Alternative 3: 95.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{if}\;M \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 2.95 \cdot 10^{+106}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) (cos M))))
   (if (<= M -1e+20)
     t_0
     (if (<= M 2.95e+106)
       (exp (- (fabs (- n m)) (fma 0.25 (pow (+ n m) 2.0) l)))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * cos(M);
	double tmp;
	if (M <= -1e+20) {
		tmp = t_0;
	} else if (M <= 2.95e+106) {
		tmp = exp((fabs((n - m)) - fma(0.25, pow((n + m), 2.0), l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * cos(M))
	tmp = 0.0
	if (M <= -1e+20)
		tmp = t_0;
	elseif (M <= 2.95e+106)
		tmp = exp(Float64(abs(Float64(n - m)) - fma(0.25, (Float64(n + m) ^ 2.0), l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -1e+20], t$95$0, If[LessEqual[M, 2.95e+106], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[Power[N[(n + m), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -1e20 or 2.95000000000000014e106 < M

   1. Initial program 80.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 99.1%

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

   if -1e20 < M < 2.95000000000000014e106

   1. Initial program 75.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 94.4%

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

Alternative 4: 65.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -4 \cdot 10^{-285}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-0.25 \cdot n\right) \cdot n} \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -2.85e-5)
   (exp (* -0.25 (* m m)))
   (if (<= m -4e-285)
     (* (exp (* (- M) M)) (cos M))
     (* (exp (* (* -0.25 n) n)) (cos M)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.85e-5) {
		tmp = exp((-0.25 * (m * m)));
	} else if (m <= -4e-285) {
		tmp = exp((-M * M)) * cos(M);
	} else {
		tmp = exp(((-0.25 * n) * n)) * cos(M);
	}
	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 <= (-2.85d-5)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (m <= (-4d-285)) then
        tmp = exp((-m_1 * m_1)) * cos(m_1)
    else
        tmp = exp((((-0.25d0) * n) * n)) * cos(m_1)
    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 <= -2.85e-5) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (m <= -4e-285) {
		tmp = Math.exp((-M * M)) * Math.cos(M);
	} else {
		tmp = Math.exp(((-0.25 * n) * n)) * Math.cos(M);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -2.85e-5:
		tmp = math.exp((-0.25 * (m * m)))
	elif m <= -4e-285:
		tmp = math.exp((-M * M)) * math.cos(M)
	else:
		tmp = math.exp(((-0.25 * n) * n)) * math.cos(M)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -2.85e-5)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (m <= -4e-285)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * cos(M));
	else
		tmp = Float64(exp(Float64(Float64(-0.25 * n) * n)) * cos(M));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -2.85e-5)
		tmp = exp((-0.25 * (m * m)));
	elseif (m <= -4e-285)
		tmp = exp((-M * M)) * cos(M);
	else
		tmp = exp(((-0.25 * n) * n)) * cos(M);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -2.85e-5], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -4e-285], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(-0.25 * n), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.8500000000000002e-5

   1. Initial program 73.4%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 96.9%

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

   if -2.8500000000000002e-5 < m < -4.0000000000000003e-285

   1. Initial program 85.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 67.9%

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

   if -4.0000000000000003e-285 < 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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 52.2%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -4 \cdot 10^{-285}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-0.25 \cdot n\right) \cdot n} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -4 \cdot 10^{-285}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -2.85e-5)
   (exp (* -0.25 (* m m)))
   (if (<= m -4e-285)
     (* (exp (* (- M) M)) (cos M))
     (exp (- (fabs (- n m)) (* (* n n) 0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.85e-5) {
		tmp = exp((-0.25 * (m * m)));
	} else if (m <= -4e-285) {
		tmp = exp((-M * M)) * cos(M);
	} else {
		tmp = exp((fabs((n - m)) - ((n * n) * 0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-2.85d-5)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (m <= (-4d-285)) then
        tmp = exp((-m_1 * m_1)) * cos(m_1)
    else
        tmp = exp((abs((n - m)) - ((n * n) * 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.85e-5) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (m <= -4e-285) {
		tmp = Math.exp((-M * M)) * Math.cos(M);
	} else {
		tmp = Math.exp((Math.abs((n - m)) - ((n * n) * 0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -2.85e-5:
		tmp = math.exp((-0.25 * (m * m)))
	elif m <= -4e-285:
		tmp = math.exp((-M * M)) * math.cos(M)
	else:
		tmp = math.exp((math.fabs((n - m)) - ((n * n) * 0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -2.85e-5)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (m <= -4e-285)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * cos(M));
	else
		tmp = exp(Float64(abs(Float64(n - m)) - Float64(Float64(n * n) * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -2.85e-5)
		tmp = exp((-0.25 * (m * m)));
	elseif (m <= -4e-285)
		tmp = exp((-M * M)) * cos(M);
	else
		tmp = exp((abs((n - m)) - ((n * n) * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -2.85e-5], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -4e-285], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.8500000000000002e-5

   1. Initial program 73.4%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 96.9%

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

   if -2.8500000000000002e-5 < m < -4.0000000000000003e-285

   1. Initial program 85.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 67.9%

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

   if -4.0000000000000003e-285 < 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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 85.2%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 49.7%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -4 \cdot 10^{-285}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -4.2 \cdot 10^{-282}:\\ \;\;\;\;e^{-\ell} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -2.85e-5)
   (exp (* -0.25 (* m m)))
   (if (<= m -4.2e-282)
     (* (exp (- l)) (cos M))
     (exp (- (fabs (- n m)) (* (* n n) 0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.85e-5) {
		tmp = exp((-0.25 * (m * m)));
	} else if (m <= -4.2e-282) {
		tmp = exp(-l) * cos(M);
	} else {
		tmp = exp((fabs((n - m)) - ((n * n) * 0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-2.85d-5)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (m <= (-4.2d-282)) then
        tmp = exp(-l) * cos(m_1)
    else
        tmp = exp((abs((n - m)) - ((n * n) * 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.85e-5) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (m <= -4.2e-282) {
		tmp = Math.exp(-l) * Math.cos(M);
	} else {
		tmp = Math.exp((Math.abs((n - m)) - ((n * n) * 0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -2.85e-5:
		tmp = math.exp((-0.25 * (m * m)))
	elif m <= -4.2e-282:
		tmp = math.exp(-l) * math.cos(M)
	else:
		tmp = math.exp((math.fabs((n - m)) - ((n * n) * 0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -2.85e-5)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (m <= -4.2e-282)
		tmp = Float64(exp(Float64(-l)) * cos(M));
	else
		tmp = exp(Float64(abs(Float64(n - m)) - Float64(Float64(n * n) * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -2.85e-5)
		tmp = exp((-0.25 * (m * m)));
	elseif (m <= -4.2e-282)
		tmp = exp(-l) * cos(M);
	else
		tmp = exp((abs((n - m)) - ((n * n) * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -2.85e-5], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -4.2e-282], N[(N[Exp[(-l)], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.8500000000000002e-5

   1. Initial program 73.4%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 96.9%

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

   if -2.8500000000000002e-5 < m < -4.20000000000000023e-282

   1. Initial program 85.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 40.5

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

   8. Applied rewrites 40.5%

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

   if -4.20000000000000023e-282 < 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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 85.2%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 49.7%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -4.2 \cdot 10^{-282}:\\ \;\;\;\;e^{-\ell} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.75 \cdot 10^{-98}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-0.25 \cdot n\right) \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.75e-98)
   (exp (* -0.25 (* m m)))
   (if (<= n 54.0) (exp (- l)) (exp (* (* -0.25 n) n)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.75e-98) {
		tmp = exp((-0.25 * (m * m)));
	} else if (n <= 54.0) {
		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 (n <= 1.75d-98) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (n <= 54.0d0) 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 (n <= 1.75e-98) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (n <= 54.0) {
		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 n <= 1.75e-98:
		tmp = math.exp((-0.25 * (m * m)))
	elif n <= 54.0:
		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 (n <= 1.75e-98)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (n <= 54.0)
		tmp = exp(Float64(-l));
	else
		tmp = exp(Float64(Float64(-0.25 * n) * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1.75e-98)
		tmp = exp((-0.25 * (m * m)));
	elseif (n <= 54.0)
		tmp = exp(-l);
	else
		tmp = exp(((-0.25 * n) * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.75e-98], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 54.0], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(N[(-0.25 * n), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.7500000000000001e-98

   1. Initial program 82.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 85.0%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 54.4%

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

   if 1.7500000000000001e-98 < n < 54

   1. Initial program 82.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 77.8%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 49.5%

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

   if 54 < n

   1. Initial program 64.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 97.4%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.75 \cdot 10^{-98}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-0.25 \cdot n\right) \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.7% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* (* -0.25 n) n))))
   (if (<= n -54.0) t_0 (if (<= n 54.0) (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 <= -54.0) {
		tmp = t_0;
	} else if (n <= 54.0) {
		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 <= (-54.0d0)) then
        tmp = t_0
    else if (n <= 54.0d0) 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 <= -54.0) {
		tmp = t_0;
	} else if (n <= 54.0) {
		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 <= -54.0:
		tmp = t_0
	elif n <= 54.0:
		tmp = math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(-0.25 * n) * n))
	tmp = 0.0
	if (n <= -54.0)
		tmp = t_0;
	elseif (n <= 54.0)
		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 <= -54.0)
		tmp = t_0;
	elseif (n <= 54.0)
		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[(N[(-0.25 * n), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -54.0], t$95$0, If[LessEqual[n, 54.0], N[Exp[(-l)], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -54 or 54 < n

   1. Initial program 65.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 96.0%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 97.6%

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

   if -54 < n < 54

   1. Initial program 88.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 78.2%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 45.4%

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

Alternative 9: 35.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.8%

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

6. Taylor expanded in M around 0

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

8. Applied rewrites 86.9%

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

9. Taylor expanded in l around inf

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

11. Applied rewrites 37.1%

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

Reproduce

herbie shell --seed 2024308 
(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)))))))