Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.7% → 96.7%

Time: 9.2s

Alternatives: 8

Speedup: 2.7×

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: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[K_, m_, n_, M_, l_] := N[(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] * N[Cos[M], $MachinePrecision]), $MachinePrecision]

Derivation

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 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 96.1%

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

6. Final simplification 96.1%

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

Alternative 2: 81.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M}\\ \mathbf{if}\;M \leq -7.6 \cdot 10^{-16}:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{elif}\;M \leq 7.8 \cdot 10^{-8}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(n \cdot n, 0.25, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* (- M) M))))
   (if (<= M -7.6e-16)
     (* t_0 1.0)
     (if (<= M 7.8e-8)
       (exp (- (fabs (- n m)) (fma (* n n) 0.25 l)))
       (* t_0 (cos M))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M));
	double tmp;
	if (M <= -7.6e-16) {
		tmp = t_0 * 1.0;
	} else if (M <= 7.8e-8) {
		tmp = exp((fabs((n - m)) - fma((n * n), 0.25, l)));
	} else {
		tmp = t_0 * cos(M);
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(-M) * M))
	tmp = 0.0
	if (M <= -7.6e-16)
		tmp = Float64(t_0 * 1.0);
	elseif (M <= 7.8e-8)
		tmp = exp(Float64(abs(Float64(n - m)) - fma(Float64(n * n), 0.25, l)));
	else
		tmp = Float64(t_0 * cos(M));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -7.6e-16], N[(t$95$0 * 1.0), $MachinePrecision], If[LessEqual[M, 7.8e-8], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[(n * n), $MachinePrecision] * 0.25 + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$0 * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < -7.60000000000000024e-16

   1. Initial program 75.9%

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

   3. Taylor expanded in K around 0

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

      1. *-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 98.3%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 89.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 89.9%

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

   if -7.60000000000000024e-16 < M < 7.7999999999999997e-8

   1. Initial program 75.2%

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

   3. Taylor expanded in M around 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(\left(\left(n + m\right) \cdot K\right) \cdot 0.5\right)} \]

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 59.9%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 68.3%

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

   if 7.7999999999999997e-8 < M

   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 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 95.4%

      \[\leadsto e^{-M \cdot M} \cdot \cos M \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7.6 \cdot 10^{-16}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq 7.8 \cdot 10^{-8}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(n \cdot n, 0.25, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6800000000:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(n \cdot n, 0.25, \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -6800000000.0)
   (* (cos M) (exp (* (* m m) -0.25)))
   (exp (- (fabs (- n m)) (fma (* n n) 0.25 l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -6800000000.0) {
		tmp = cos(M) * exp(((m * m) * -0.25));
	} else {
		tmp = exp((fabs((n - m)) - fma((n * n), 0.25, l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -6800000000.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(m * m) * -0.25)));
	else
		tmp = exp(Float64(abs(Float64(n - m)) - fma(Float64(n * n), 0.25, l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -6800000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[(n * n), $MachinePrecision] * 0.25 + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -6.8e9

   1. Initial program 69.3%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 94.8

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

   8. Applied rewrites 94.8%

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

   if -6.8e9 < 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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 63.7%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(\left(\left(n + m\right) \cdot K\right) \cdot 0.5\right)} \]

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 68.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6800000000:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(n \cdot n, 0.25, \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -7.6 \cdot 10^{-16} \lor \neg \left(M \leq 7.8 \cdot 10^{-8}\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(n \cdot n, 0.25, \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -7.6e-16) (not (<= M 7.8e-8)))
   (* (exp (* (- M) M)) 1.0)
   (exp (- (fabs (- n m)) (fma (* n n) 0.25 l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -7.6e-16) || !(M <= 7.8e-8)) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = exp((fabs((n - m)) - fma((n * n), 0.25, l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -7.6e-16) || !(M <= 7.8e-8))
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = exp(Float64(abs(Float64(n - m)) - fma(Float64(n * n), 0.25, l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -7.6e-16], N[Not[LessEqual[M, 7.8e-8]], $MachinePrecision]], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[(n * n), $MachinePrecision] * 0.25 + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -7.60000000000000024e-16 or 7.7999999999999997e-8 < M

   1. Initial program 74.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 99.2%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 92.7%

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

   if -7.60000000000000024e-16 < M < 7.7999999999999997e-8

   1. Initial program 75.2%

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

   3. Taylor expanded in M around 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(\left(\left(n + m\right) \cdot K\right) \cdot 0.5\right)} \]

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 59.9%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 68.3%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7.6 \cdot 10^{-16} \lor \neg \left(M \leq 7.8 \cdot 10^{-8}\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(n \cdot n, 0.25, \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -27 \lor \neg \left(M \leq 2.35 \cdot 10^{-39}\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -27.0) (not (<= M 2.35e-39)))
   (* (exp (* (- M) M)) 1.0)
   (* (fma (* M M) -0.5 1.0) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -27.0) || !(M <= 2.35e-39)) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = fma((M * M), -0.5, 1.0) * exp(-l);
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -27.0) || !(M <= 2.35e-39))
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(fma(Float64(M * M), -0.5, 1.0) * exp(Float64(-l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -27.0], N[Not[LessEqual[M, 2.35e-39]], $MachinePrecision]], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -27 or 2.3500000000000001e-39 < M

   1. Initial program 74.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 99.2%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 90.7%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 90.6%

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

   if -27 < M < 2.3500000000000001e-39

   1. Initial program 75.2%

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

   3. Taylor expanded in 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 39.2

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

   5. Applied rewrites 39.2%

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

   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 44.4

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

   8. Applied rewrites 44.4%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 44.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -27 \lor \neg \left(M \leq 2.35 \cdot 10^{-39}\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -7.6 \cdot 10^{-16} \lor \neg \left(M \leq 2.35 \cdot 10^{-39}\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -7.6e-16) (not (<= M 2.35e-39)))
   (* (exp (* (- M) M)) 1.0)
   (exp (- (fabs (- n m)) l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -7.6e-16) || !(M <= 2.35e-39)) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = exp((fabs((n - m)) - 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 ((m_1 <= (-7.6d-16)) .or. (.not. (m_1 <= 2.35d-39))) then
        tmp = exp((-m_1 * m_1)) * 1.0d0
    else
        tmp = exp((abs((n - m)) - 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 ((M <= -7.6e-16) || !(M <= 2.35e-39)) {
		tmp = Math.exp((-M * M)) * 1.0;
	} else {
		tmp = Math.exp((Math.abs((n - m)) - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -7.6e-16) or not (M <= 2.35e-39):
		tmp = math.exp((-M * M)) * 1.0
	else:
		tmp = math.exp((math.fabs((n - m)) - l))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -7.6e-16) || !(M <= 2.35e-39))
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = exp(Float64(abs(Float64(n - m)) - l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -7.6e-16) || ~((M <= 2.35e-39)))
		tmp = exp((-M * M)) * 1.0;
	else
		tmp = exp((abs((n - m)) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -7.6e-16], N[Not[LessEqual[M, 2.35e-39]], $MachinePrecision]], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -7.60000000000000024e-16 or 2.3500000000000001e-39 < M

   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. *-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 99.2%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 88.7%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 88.6%

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

   if -7.60000000000000024e-16 < M < 2.3500000000000001e-39

   1. Initial program 75.4%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(\left(\left(n + m\right) \cdot K\right) \cdot 0.5\right)} \]

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 37.1%

       \[\leadsto e^{\left|m - n\right| - \ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7.6 \cdot 10^{-16} \lor \neg \left(M \leq 2.35 \cdot 10^{-39}\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 24.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ e^{\left|n - m\right| - \ell} \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - 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((abs((n - m)) - l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 65.3%

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(\left(\left(n + m\right) \cdot K\right) \cdot 0.5\right)} \]

6. Taylor expanded in m around 0

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

8. Applied rewrites 50.1%

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

9. Taylor expanded in n around 0

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

11. Applied rewrites 27.3%

    \[\leadsto e^{\left|m - n\right| - \ell} \]

12. Final simplification 27.3%

    \[\leadsto e^{\left|n - m\right| - \ell} \]
13. Add Preprocessing

Alternative 8: 7.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ e^{\left|n - m\right|} \end{array} \]

FPCore

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

C

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

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(abs((n - m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 65.3%

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(\left(\left(n + m\right) \cdot K\right) \cdot 0.5\right)} \]

6. Taylor expanded in m around 0

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

8. Applied rewrites 50.1%

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

9. Taylor expanded in n around 0

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

11. Applied rewrites 27.3%

    \[\leadsto e^{\left|m - n\right| - \ell} \]

12. Taylor expanded in l around 0

    \[\leadsto e^{\left|m - n\right|} \]
13. Step-by-step derivation

14. Applied rewrites 9.0%

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

15. Final simplification 9.0%

    \[\leadsto e^{\left|n - m\right|} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(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)))))))