Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.6% → 96.7%

Time: 22.1s

Alternatives: 9

Speedup: 1.4×

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: 76.6% 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, 1.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp((fabs((m - n)) - (pow((((m + n) / 2.0) - M), 2.0) + 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 = cos(m_1) * exp((abs((m - n)) - (((((m + n) / 2.0d0) - m_1) ** 2.0d0) + l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. associate-/l* 71.6%

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

   2. associate--r- 71.6%

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

3. Simplified 71.6%

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

4. Taylor expanded in K around 0 97.8%

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

   1. cos-neg 97.8%

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

6. Simplified 97.8%

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

7. Final simplification 97.8%

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

Alternative 2: 88.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;M \leq -1.12 \cdot 10^{+72}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (<= M -1.12e+72)
     (* (cos M) (exp (- t_0 (* M M))))
     (exp (- t_0 (+ l (* 0.25 (pow (+ m n) 2.0))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if (M <= -1.12e+72) {
		tmp = cos(M) * exp((t_0 - (M * M)));
	} else {
		tmp = exp((t_0 - (l + (0.25 * pow((m + n), 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((m - n))
    if (m_1 <= (-1.12d+72)) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * m_1)))
    else
        tmp = exp((t_0 - (l + (0.25d0 * ((m + n) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((m - n));
	double tmp;
	if (M <= -1.12e+72) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * M)));
	} else {
		tmp = Math.exp((t_0 - (l + (0.25 * Math.pow((m + n), 2.0)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	tmp = 0
	if M <= -1.12e+72:
		tmp = math.cos(M) * math.exp((t_0 - (M * M)))
	else:
		tmp = math.exp((t_0 - (l + (0.25 * math.pow((m + n), 2.0)))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if (M <= -1.12e+72)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	else
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * (Float64(m + n) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	tmp = 0.0;
	if (M <= -1.12e+72)
		tmp = cos(M) * exp((t_0 - (M * M)));
	else
		tmp = exp((t_0 - (l + (0.25 * ((m + n) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -1.12e+72], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -1.12000000000000001e72

   1. Initial program 74.5%

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

      1. associate-/l* 74.5%

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

      2. associate--r- 74.5%

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

   3. Simplified 74.5%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around inf 92.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}} + \left|m - n\right|} \]
   8. Step-by-step derivation

      1. mul-1-neg 92.3%

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

      2. unpow2 92.3%

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

      3. distribute-rgt-neg-in 92.3%

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

   9. Simplified 92.3%

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

   10. Taylor expanded in M around inf 92.3%

       \[\leadsto \color{blue}{e^{\left|m - n\right| - {M}^{2}} \cdot \cos M} \]
   11. Step-by-step derivation

       1. cos-neg 92.3%

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

       2. *-commutative 92.3%

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

       3. sub-neg 92.3%

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

       4. mul-1-neg 92.3%

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

       5. cos-neg 92.3%

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

       6. mul-1-neg 92.3%

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

       7. sub-neg 92.3%

          \[\leadsto \cos M \cdot e^{\color{blue}{\left|m - n\right| - {M}^{2}}} \]

       8. unpow2 92.3%

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

   12. Simplified 92.3%

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

   if -1.12000000000000001e72 < M

   1. Initial program 70.9%

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

      1. associate-/l* 70.9%

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

      2. associate--r- 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in K around 0 97.3%

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

      1. cos-neg 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in M around 0 93.2%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. fabs-sub 93.2%

         \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)} \]

   9. Simplified 93.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.12 \cdot 10^{+72}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \]

Alternative 3: 73.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2000:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -2000.0)
   (exp (* (* m m) -0.25))
   (exp (- (fabs (- m n)) (+ l (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2000.0) {
		tmp = exp(((m * m) * -0.25));
	} else {
		tmp = exp((fabs((m - n)) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-2000.0d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else
        tmp = exp((abs((m - n)) - (l + (0.25d0 * (n * n)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2000.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else {
		tmp = Math.exp((Math.abs((m - n)) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -2000.0:
		tmp = math.exp(((m * m) * -0.25))
	else:
		tmp = math.exp((math.fabs((m - n)) - (l + (0.25 * (n * n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -2000.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -2000.0)
		tmp = exp(((m * m) * -0.25));
	else
		tmp = exp((abs((m - n)) - (l + (0.25 * (n * n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -2000.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2e3

   1. Initial program 57.4%

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

      1. associate-/l* 57.4%

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

      2. associate--r- 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]

   10. Taylor expanded in m around inf 100.0%

       \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

       2. unpow2 100.0%

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

   12. Simplified 100.0%

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

   if -2e3 < m

   1. Initial program 76.1%

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

      1. associate-/l* 76.1%

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

      2. associate--r- 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in K around 0 97.1%

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

      1. cos-neg 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in M around 0 87.3%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. fabs-sub 87.3%

         \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)} \]

   9. Simplified 87.3%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]

   10. Taylor expanded in m around 0 65.1%

       \[\leadsto e^{\left|n - m\right| - \color{blue}{\left(\ell + 0.25 \cdot {n}^{2}\right)}} \]
   11. Step-by-step derivation

       1. *-commutative 65.1%

          \[\leadsto e^{\left|n - m\right| - \left(\ell + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

       2. unpow2 65.1%

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

   12. Simplified 65.1%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2000:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \]

Alternative 4: 63.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -8 \cdot 10^{-216}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -54.0)
   (exp (* (* m m) -0.25))
   (if (<= m -8e-216) (/ (cos M) (exp l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -54.0) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= -8e-216) {
		tmp = cos(M) / exp(l);
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-54.0d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= (-8d-216)) then
        tmp = cos(m_1) / exp(l)
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -54.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= -8e-216) {
		tmp = Math.cos(M) / Math.exp(l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -54.0:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= -8e-216:
		tmp = math.cos(M) / math.exp(l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -54.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= -8e-216)
		tmp = Float64(cos(M) / exp(l));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -54.0)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= -8e-216)
		tmp = cos(M) / exp(l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -54.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -8e-216], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -54

   1. Initial program 57.4%

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

      1. associate-/l* 57.4%

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

      2. associate--r- 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]

   10. Taylor expanded in m around inf 100.0%

       \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

       2. unpow2 100.0%

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

   12. Simplified 100.0%

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

   if -54 < m < -8.0000000000000003e-216

   1. Initial program 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in l around inf 52.1%

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

   5. Taylor expanded in K around 0 56.8%

      \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]
   6. Step-by-step derivation

      1. cos-neg 56.8%

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

   7. Simplified 56.8%

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

   if -8.0000000000000003e-216 < m

   1. Initial program 73.6%

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

      1. associate-/l* 73.8%

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

      2. associate--r- 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in K around 0 97.5%

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

      1. cos-neg 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in M around 0 90.5%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. fabs-sub 90.5%

         \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)} \]

   9. Simplified 90.5%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]

   10. Taylor expanded in n around inf 53.8%

       \[\leadsto e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 53.8%

          \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

       2. unpow2 53.8%

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

   12. Simplified 53.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -8 \cdot 10^{-216}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 5: 68.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -54 \lor \neg \left(m \leq 53\right):\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -54.0) (not (<= m 53.0)))
   (exp (* (* m m) -0.25))
   (/ 1.0 (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -54.0) || !(m <= 53.0)) {
		tmp = exp(((m * m) * -0.25));
	} 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 <= (-54.0d0)) .or. (.not. (m <= 53.0d0))) then
        tmp = exp(((m * m) * (-0.25d0)))
    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 <= -54.0) || !(m <= 53.0)) {
		tmp = Math.exp(((m * m) * -0.25));
	} else {
		tmp = 1.0 / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -54.0) or not (m <= 53.0):
		tmp = math.exp(((m * m) * -0.25))
	else:
		tmp = 1.0 / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -54.0) || !(m <= 53.0))
		tmp = exp(Float64(Float64(m * m) * -0.25));
	else
		tmp = Float64(1.0 / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -54.0) || ~((m <= 53.0)))
		tmp = exp(((m * m) * -0.25));
	else
		tmp = 1.0 / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -54.0], N[Not[LessEqual[m, 53.0]], $MachinePrecision]], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -54 or 53 < m

   1. Initial program 63.7%

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

      1. associate-/l* 63.7%

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

      2. associate--r- 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in K around 0 99.3%

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

      1. cos-neg 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in M around 0 98.5%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. fabs-sub 98.5%

         \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)} \]

   9. Simplified 98.5%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]

   10. Taylor expanded in m around inf 97.8%

       \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 97.8%

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

       2. unpow2 97.8%

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

   12. Simplified 97.8%

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

   if -54 < m < 53

   1. Initial program 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in l around inf 43.3%

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

   5. Taylor expanded in K around 0 48.5%

      \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]
   6. Step-by-step derivation

      1. cos-neg 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in M around 0 48.5%

      \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -54 \lor \neg \left(m \leq 53\right):\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\ell}}\\ \end{array} \]

Alternative 6: 62.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -1.8 \cdot 10^{-215}:\\ \;\;\;\;\frac{1}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -54.0)
   (exp (* (* m m) -0.25))
   (if (<= m -1.8e-215) (/ 1.0 (exp l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -54.0) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= -1.8e-215) {
		tmp = 1.0 / exp(l);
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-54.0d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= (-1.8d-215)) then
        tmp = 1.0d0 / exp(l)
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -54.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= -1.8e-215) {
		tmp = 1.0 / Math.exp(l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -54.0:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= -1.8e-215:
		tmp = 1.0 / math.exp(l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -54.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= -1.8e-215)
		tmp = Float64(1.0 / exp(l));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -54.0)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= -1.8e-215)
		tmp = 1.0 / exp(l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -54.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -1.8e-215], N[(1.0 / N[Exp[l], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -54

   1. Initial program 57.4%

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

      1. associate-/l* 57.4%

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

      2. associate--r- 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]

   10. Taylor expanded in m around inf 100.0%

       \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

       2. unpow2 100.0%

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

   12. Simplified 100.0%

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

   if -54 < m < -1.7999999999999999e-215

   1. Initial program 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in l around inf 52.1%

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

   5. Taylor expanded in K around 0 56.8%

      \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]
   6. Step-by-step derivation

      1. cos-neg 56.8%

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

   7. Simplified 56.8%

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

   8. Taylor expanded in M around 0 56.7%

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

   if -1.7999999999999999e-215 < m

   1. Initial program 73.6%

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

      1. associate-/l* 73.8%

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

      2. associate--r- 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in K around 0 97.5%

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

      1. cos-neg 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in M around 0 90.5%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. fabs-sub 90.5%

         \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)} \]

   9. Simplified 90.5%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]

   10. Taylor expanded in n around inf 53.8%

       \[\leadsto e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 53.8%

          \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

       2. unpow2 53.8%

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

   12. Simplified 53.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -1.8 \cdot 10^{-215}:\\ \;\;\;\;\frac{1}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 7: 35.0% accurate, 4.1× speedup

Math

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

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(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 71.6%

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

3. Simplified 71.6%

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

4. Taylor expanded in l around inf 30.5%

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

5. Taylor expanded in K around 0 39.1%

   \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]
6. Step-by-step derivation

   1. cos-neg 39.1%

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

7. Simplified 39.1%

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

8. Taylor expanded in M around 0 39.1%

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

9. Final simplification 39.1%

   \[\leadsto \frac{1}{e^{\ell}} \]

Alternative 8: 35.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. associate-/l* 71.6%

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

   2. associate--r- 71.6%

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

3. Simplified 71.6%

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

4. Taylor expanded in K around 0 97.8%

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

   1. cos-neg 97.8%

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

6. Simplified 97.8%

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

7. Taylor expanded in M around 0 90.4%

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
8. Step-by-step derivation

   1. fabs-sub 90.4%

      \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)} \]

9. Simplified 90.4%

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]

10. Taylor expanded in l around inf 39.1%

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

    1. mul-1-neg 39.1%

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

12. Simplified 39.1%

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

13. Final simplification 39.1%

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

Alternative 9: 6.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \cos M \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(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 = cos(m_1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Simplified 71.6%

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

4. Taylor expanded in l around inf 30.5%

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

5. Taylor expanded in K around 0 39.1%

   \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]
6. Step-by-step derivation

   1. cos-neg 39.1%

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

7. Simplified 39.1%

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

8. Taylor expanded in l around 0 5.6%

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

9. Final simplification 5.6%

   \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023218 
(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)))))))