Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.0% → 96.7%

Time: 8.7s

Alternatives: 9

Speedup: 2.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.0%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 97.9%

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

7. Applied rewrites 97.9%

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

8. Final simplification 97.9%

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

Alternative 2: 96.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.0%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 97.9%

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

7. Applied rewrites 97.9%

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

8. Taylor expanded in M around 0

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

10. Applied rewrites 97.5%

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

11. Final simplification 97.5%

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

Alternative 3: 86.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{+23}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(\mathsf{fma}\left(0.5, n, -M\right), \mathsf{fma}\left(-0.5, n, M\right), \left|m - n\right|\right) - \ell} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1.4e+23)
   (* (exp (* (* m m) -0.25)) 1.0)
   (* (exp (- (fma (fma 0.5 n (- M)) (fma -0.5 n M) (fabs (- m n))) l)) 1.0)))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.4e+23) {
		tmp = exp(((m * m) * -0.25)) * 1.0;
	} else {
		tmp = exp((fma(fma(0.5, n, -M), fma(-0.5, n, M), fabs((m - n))) - l)) * 1.0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1.4e+23)
		tmp = Float64(exp(Float64(Float64(m * m) * -0.25)) * 1.0);
	else
		tmp = Float64(exp(Float64(fma(fma(0.5, n, Float64(-M)), fma(-0.5, n, M), abs(Float64(m - n))) - l)) * 1.0);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1.4e+23], N[(N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(N[(0.5 * n + (-M)), $MachinePrecision] * N[(-0.5 * n + M), $MachinePrecision] + N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.4e23

   1. Initial program 65.0%

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

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 100.0%

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

   if -1.4e23 < m

   1. Initial program 79.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 97.3%

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 96.8%

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

   11. Taylor expanded in m around 0

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

   13. Applied rewrites 87.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{+23}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(\mathsf{fma}\left(0.5, n, -M\right), \mathsf{fma}\left(-0.5, n, M\right), \left|m - n\right|\right) - \ell} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -520000000000:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot 1\\ \mathbf{elif}\;m \leq -3 \cdot 10^{-156}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, n \cdot n, \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -520000000000.0)
   (* (exp (* (* m m) -0.25)) 1.0)
   (if (<= m -3e-156)
     (* (exp (* (- M) M)) 1.0)
     (exp (- (fabs (- m n)) (fma 0.25 (* n n) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -520000000000.0) {
		tmp = exp(((m * m) * -0.25)) * 1.0;
	} else if (m <= -3e-156) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = exp((fabs((m - n)) - fma(0.25, (n * n), l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -520000000000.0)
		tmp = Float64(exp(Float64(Float64(m * m) * -0.25)) * 1.0);
	elseif (m <= -3e-156)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = exp(Float64(abs(Float64(m - n)) - fma(0.25, Float64(n * n), l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -520000000000.0], N[(N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[m, -3e-156], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(n * n), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.2e11

   1. Initial program 66.1%

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

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 100.0%

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

   if -5.2e11 < m < -3e-156

   1. Initial program 84.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 96.1%

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 96.1%

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

   11. Taylor expanded in M around inf

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

   13. Applied rewrites 69.9%

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

   if -3e-156 < m

   1. Initial program 78.4%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 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 86.5%

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

   9. Taylor expanded in m around 0

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

   11. Applied rewrites 63.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -520000000000:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot 1\\ \mathbf{elif}\;m \leq -3 \cdot 10^{-156}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, n \cdot n, \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -54 \lor \neg \left(n \leq 54\right):\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -54.0) (not (<= n 54.0)))
   (* (exp (* (* n n) -0.25)) 1.0)
   (* (exp (* (- M) M)) 1.0)))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -54.0) || !(n <= 54.0)) {
		tmp = exp(((n * n) * -0.25)) * 1.0;
	} else {
		tmp = exp((-M * M)) * 1.0;
	}
	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) :: tmp
    if ((n <= (-54.0d0)) .or. (.not. (n <= 54.0d0))) then
        tmp = exp(((n * n) * (-0.25d0))) * 1.0d0
    else
        tmp = exp((-m_1 * m_1)) * 1.0d0
    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 <= -54.0) || !(n <= 54.0)) {
		tmp = Math.exp(((n * n) * -0.25)) * 1.0;
	} else {
		tmp = Math.exp((-M * M)) * 1.0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -54.0) or not (n <= 54.0):
		tmp = math.exp(((n * n) * -0.25)) * 1.0
	else:
		tmp = math.exp((-M * M)) * 1.0
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -54.0) || !(n <= 54.0))
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -54.0) || ~((n <= 54.0)))
		tmp = exp(((n * n) * -0.25)) * 1.0;
	else
		tmp = exp((-M * M)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -54.0], N[Not[LessEqual[n, 54.0]], $MachinePrecision]], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -54 or 54 < n

   1. Initial program 71.0%

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

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 99.2%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 97.7%

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

   if -54 < n < 54

   1. Initial program 81.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 96.6%

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

   7. Applied rewrites 96.6%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 95.8%

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

   11. Taylor expanded in M around inf

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

   13. Applied rewrites 65.9%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -54 \lor \neg \left(n \leq 54\right):\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -520000000000:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot 1\\ \mathbf{elif}\;m \leq -1 \cdot 10^{-287}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -520000000000.0)
   (* (exp (* (* m m) -0.25)) 1.0)
   (if (<= m -1e-287)
     (* (exp (* (- M) M)) 1.0)
     (* (exp (* (* n n) -0.25)) 1.0))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -520000000000.0) {
		tmp = exp(((m * m) * -0.25)) * 1.0;
	} else if (m <= -1e-287) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = exp(((n * n) * -0.25)) * 1.0;
	}
	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) :: tmp
    if (m <= (-520000000000.0d0)) then
        tmp = exp(((m * m) * (-0.25d0))) * 1.0d0
    else if (m <= (-1d-287)) then
        tmp = exp((-m_1 * m_1)) * 1.0d0
    else
        tmp = exp(((n * n) * (-0.25d0))) * 1.0d0
    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 <= -520000000000.0) {
		tmp = Math.exp(((m * m) * -0.25)) * 1.0;
	} else if (m <= -1e-287) {
		tmp = Math.exp((-M * M)) * 1.0;
	} else {
		tmp = Math.exp(((n * n) * -0.25)) * 1.0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -520000000000.0:
		tmp = math.exp(((m * m) * -0.25)) * 1.0
	elif m <= -1e-287:
		tmp = math.exp((-M * M)) * 1.0
	else:
		tmp = math.exp(((n * n) * -0.25)) * 1.0
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -520000000000.0)
		tmp = Float64(exp(Float64(Float64(m * m) * -0.25)) * 1.0);
	elseif (m <= -1e-287)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -520000000000.0)
		tmp = exp(((m * m) * -0.25)) * 1.0;
	elseif (m <= -1e-287)
		tmp = exp((-M * M)) * 1.0;
	else
		tmp = exp(((n * n) * -0.25)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -520000000000.0], N[(N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[m, -1e-287], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.2e11

   1. Initial program 66.1%

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

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 100.0%

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

   if -5.2e11 < m < -1.00000000000000002e-287

   1. Initial program 88.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 96.2%

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

   7. Applied rewrites 96.2%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 96.2%

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

   11. Taylor expanded in M around inf

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

   13. Applied rewrites 68.7%

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

   if -1.00000000000000002e-287 < m

   1. Initial program 75.7%

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

   3. Taylor expanded in K around 0

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

      1. *-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 97.7%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 97.0%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 59.5%

       \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -520000000000:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot 1\\ \mathbf{elif}\;m \leq -1 \cdot 10^{-287}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 45:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, m \cdot m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 45.0)
   (exp (- (fabs (- m n)) (fma 0.25 (* m m) l)))
   (* (exp (* (* n n) -0.25)) 1.0)))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 45.0) {
		tmp = exp((fabs((m - n)) - fma(0.25, (m * m), l)));
	} else {
		tmp = exp(((n * n) * -0.25)) * 1.0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 45.0)
		tmp = exp(Float64(abs(Float64(m - n)) - fma(0.25, Float64(m * m), l)));
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * 1.0);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 45.0], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(m * m), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 45

   1. Initial program 77.7%

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

   3. Taylor expanded in K around 0

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

      1. *-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 97.7%

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

   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 85.1%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 69.9%

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

   if 45 < n

   1. Initial program 71.4%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 98.6%

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 98.6%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 98.6%

       \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 45:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, m \cdot m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -22500000000000 \lor \neg \left(M \leq 4.4 \cdot 10^{-13}\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -22500000000000.0) (not (<= M 4.4e-13)))
   (* (exp (* (- M) M)) 1.0)
   (* 1.0 (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -22500000000000.0) || !(M <= 4.4e-13)) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = 1.0 * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-22500000000000.0d0)) .or. (.not. (m_1 <= 4.4d-13))) then
        tmp = exp((-m_1 * m_1)) * 1.0d0
    else
        tmp = 1.0d0 * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -22500000000000.0) || !(M <= 4.4e-13)) {
		tmp = Math.exp((-M * M)) * 1.0;
	} else {
		tmp = 1.0 * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -22500000000000.0) or not (M <= 4.4e-13):
		tmp = math.exp((-M * M)) * 1.0
	else:
		tmp = 1.0 * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -22500000000000.0) || !(M <= 4.4e-13))
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(1.0 * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -22500000000000.0) || ~((M <= 4.4e-13)))
		tmp = exp((-M * M)) * 1.0;
	else
		tmp = 1.0 * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -22500000000000.0], N[Not[LessEqual[M, 4.4e-13]], $MachinePrecision]], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -2.25e13 or 4.39999999999999993e-13 < M

   1. Initial program 76.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 99.3%

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

   11. Taylor expanded in M around inf

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

   13. Applied rewrites 95.2%

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

   if -2.25e13 < M < 4.39999999999999993e-13

   1. Initial program 74.7%

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

   3. Taylor expanded in 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 45.0

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

   5. Applied rewrites 45.0%

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

   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 50.9

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

   8. Applied rewrites 50.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 50.9%

       \[\leadsto 1 \cdot e^{-\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -22500000000000 \lor \neg \left(M \leq 4.4 \cdot 10^{-13}\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot e^{-\ell} \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 * 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 = 1.0d0 * exp(-l)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 * Math.exp(-l);
}

Python

def code(K, m, n, M, l):
	return 1.0 * math.exp(-l)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 31.7

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

5. Applied rewrites 31.7%

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

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 36.5

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

8. Applied rewrites 36.5%

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

9. Taylor expanded in M around 0

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

11. Applied rewrites 36.1%

    \[\leadsto 1 \cdot e^{-\ell} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(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)))))))