Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.7% → 96.6%

Time: 15.5s

Alternatives: 9

Speedup: 2.0×

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: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) - l) - ((((m + n) / 2.0d0) - m_1) ** 2.0d0)))
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)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

3. Taylor expanded in K around 0 97.1%

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

   1. cos-neg 97.1%

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

5. Simplified 97.1%

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

6. Final simplification 97.1%

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

Alternative 2: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -3.5 \cdot 10^{+15} \lor \neg \left(M \leq 27\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -3.5e+15) (not (<= M 27.0)))
   (exp (- (pow M 2.0)))
   (exp (- (- (fabs (- m n)) l) (* 0.25 (pow (+ m n) 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -3.5e+15) || !(M <= 27.0)) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = exp(((fabs((m - n)) - 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) :: tmp
    if ((m_1 <= (-3.5d+15)) .or. (.not. (m_1 <= 27.0d0))) then
        tmp = exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((abs((m - n)) - 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 tmp;
	if ((M <= -3.5e+15) || !(M <= 27.0)) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(((Math.abs((m - n)) - l) - (0.25 * Math.pow((m + n), 2.0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -3.5e+15) or not (M <= 27.0):
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(((math.fabs((m - n)) - l) - (0.25 * math.pow((m + n), 2.0))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -3.5e+15) || !(M <= 27.0))
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64(0.25 * (Float64(m + n) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -3.5e+15) || ~((M <= 27.0)))
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp(((abs((m - n)) - l) - (0.25 * ((m + n) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -3.5e+15], N[Not[LessEqual[M, 27.0]], $MachinePrecision]], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -3.5e15 or 27 < M

   1. Initial program 80.8%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around inf 99.2%

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

      1. mul-1-neg 99.2%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   8. Simplified 99.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   9. Taylor expanded in M around 0 99.2%

      \[\leadsto \color{blue}{1} \cdot e^{-{M}^{2}} \]

   if -3.5e15 < M < 27

   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 94.6%

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

      1. cos-neg 94.6%

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

   5. Simplified 94.6%

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

   6. Taylor expanded in M around 0 94.6%

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

      1. associate--r+ 94.6%

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

      2. fabs-sub 94.6%

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

   8. Simplified 94.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3.5 \cdot 10^{+15} \lor \neg \left(M \leq 27\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 57:\\ \;\;\;\;e^{\left(n - m\right) - \left(\ell + 0.25 \cdot {m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 57.0)
   (exp (- (- n m) (+ l (* 0.25 (pow m 2.0)))))
   (exp (* (* n n) -0.25))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 57.0) {
		tmp = exp(((n - m) - (l + (0.25 * pow(m, 2.0)))));
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 57.0d0) then
        tmp = exp(((n - m) - (l + (0.25d0 * (m ** 2.0d0)))))
    else
        tmp = exp(((n * n) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 57.0) {
		tmp = Math.exp(((n - m) - (l + (0.25 * Math.pow(m, 2.0)))));
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 57.0:
		tmp = math.exp(((n - m) - (l + (0.25 * math.pow(m, 2.0)))))
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 57.0)
		tmp = exp(Float64(Float64(n - m) - Float64(l + Float64(0.25 * (m ^ 2.0)))));
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 57.0)
		tmp = exp(((n - m) - (l + (0.25 * (m ^ 2.0)))));
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 57.0], N[Exp[N[(N[(n - m), $MachinePrecision] - N[(l + N[(0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 57

   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 96.1%

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

      1. cos-neg 96.1%

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

   5. Simplified 96.1%

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

   6. Taylor expanded in M around 0 86.9%

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

      1. associate--r+ 86.9%

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

      2. fabs-sub 86.9%

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

   8. Simplified 86.9%

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

   9. Taylor expanded in n around 0 68.8%

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

       1. rem-square-sqrt 28.8%

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

       2. fabs-sqr 28.8%

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

       3. rem-square-sqrt 85.4%

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

   11. Simplified 85.4%

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

   if 57 < n

   1. Initial program 69.7%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 97.0%

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

      1. associate--r+ 97.0%

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

      2. fabs-sub 97.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in n around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

       1. unpow2 100.0%

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

   13. Applied egg-rr 100.0%

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

Alternative 4: 62.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.3 \cdot 10^{-175}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-11}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 2.3e-175)
   (exp (* (pow m 2.0) -0.25))
   (if (<= n 1.95e-11) (* (cos M) (exp (- l))) (exp (* (* n n) -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2.3e-175) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else if (n <= 1.95e-11) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 2.3d-175) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else if (n <= 1.95d-11) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = exp(((n * n) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2.3e-175) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else if (n <= 1.95e-11) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 2.3e-175:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	elif n <= 1.95e-11:
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 2.3e-175)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	elseif (n <= 1.95e-11)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 2.3e-175)
		tmp = exp(((m ^ 2.0) * -0.25));
	elseif (n <= 1.95e-11)
		tmp = cos(M) * exp(-l);
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 2.3e-175], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.95e-11], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 2.3e-175

   1. Initial program 77.5%

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

   3. Taylor expanded in K around 0 97.6%

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

      1. cos-neg 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in M around 0 88.6%

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

      1. associate--r+ 88.6%

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

      2. fabs-sub 88.6%

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

   8. Simplified 88.6%

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

   9. Taylor expanded in m around inf 53.0%

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

       1. *-commutative 53.0%

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

   11. Simplified 53.0%

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

   if 2.3e-175 < n < 1.95000000000000005e-11

   1. Initial program 77.3%

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

   3. Taylor expanded in l around inf 47.8%

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

      1. mul-1-neg 47.8%

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

   5. Simplified 47.8%

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

   6. Taylor expanded in K around 0 60.3%

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

      1. cos-neg 60.3%

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

   8. Simplified 60.3%

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

   if 1.95000000000000005e-11 < n

   1. Initial program 70.6%

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

   3. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in M around 0 95.6%

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

      1. associate--r+ 95.6%

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

      2. fabs-sub 95.6%

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

   8. Simplified 95.6%

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

   9. Taylor expanded in n around inf 97.1%

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

       1. *-commutative 97.1%

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

   11. Simplified 97.1%

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

       1. unpow2 97.1%

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

   13. Applied egg-rr 97.1%

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

Alternative 5: 65.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.4 \cdot 10^{-244}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.4e-244)
   (exp (* (pow m 2.0) -0.25))
   (if (<= n 55.0) (exp (- (pow M 2.0))) (exp (* (* n n) -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.4e-244) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else if (n <= 55.0) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 1.4d-244) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else if (n <= 55.0d0) then
        tmp = exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((n * n) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.4e-244) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else if (n <= 55.0) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.4e-244:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	elif n <= 55.0:
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.4e-244)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	elseif (n <= 55.0)
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1.4e-244)
		tmp = exp(((m ^ 2.0) * -0.25));
	elseif (n <= 55.0)
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.4e-244], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 55.0], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.40000000000000007e-244

   1. Initial program 76.2%

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

   3. Taylor expanded in K around 0 97.4%

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

      1. cos-neg 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in M around 0 91.4%

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

      1. associate--r+ 91.4%

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

      2. fabs-sub 91.4%

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

   8. Simplified 91.4%

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

   9. Taylor expanded in m around inf 54.4%

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

       1. *-commutative 54.4%

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

   11. Simplified 54.4%

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

   if 1.40000000000000007e-244 < n < 55

   1. Initial program 82.3%

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

   3. Taylor expanded in K around 0 92.0%

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

      1. cos-neg 92.0%

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

   5. Simplified 92.0%

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

   6. Taylor expanded in M around inf 58.3%

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

      1. mul-1-neg 58.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   8. Simplified 58.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   9. Taylor expanded in M around 0 58.3%

      \[\leadsto \color{blue}{1} \cdot e^{-{M}^{2}} \]

   if 55 < n

   1. Initial program 69.7%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 97.0%

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

      1. associate--r+ 97.0%

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

      2. fabs-sub 97.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in n around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

       1. unpow2 100.0%

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

   13. Applied egg-rr 100.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.4 \cdot 10^{-244}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.0072:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -0.0072) (exp (* (pow m 2.0) -0.25)) (exp (* (* n n) -0.25))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.0072) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-0.0072d0)) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = exp(((n * n) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.0072) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -0.0072:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -0.0072)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -0.0072)
		tmp = exp(((m ^ 2.0) * -0.25));
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -0.0072], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.0071999999999999998

   1. Initial program 72.6%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 100.0%

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

      1. associate--r+ 100.0%

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

      2. fabs-sub 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in m around inf 95.3%

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

       1. *-commutative 95.3%

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

   11. Simplified 95.3%

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

   if -0.0071999999999999998 < m

   1. Initial program 76.6%

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

   3. Taylor expanded in K around 0 96.2%

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

      1. cos-neg 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in M around 0 86.2%

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

      1. associate--r+ 86.2%

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

      2. fabs-sub 86.2%

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

   8. Simplified 86.2%

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

   9. Taylor expanded in n around inf 56.7%

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

       1. *-commutative 56.7%

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

   11. Simplified 56.7%

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

       1. unpow2 56.7%

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

   13. Applied egg-rr 56.7%

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

Alternative 7: 54.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(n \cdot n\right) \cdot -0.25} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (exp (* (* n n) -0.25)))

C

double code(double K, double m, double n, double M, double l) {
	return exp(((n * n) * -0.25));
}

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(((n * n) * (-0.25d0)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(((n * n) * -0.25));
}

Python

def code(K, m, n, M, l):
	return math.exp(((n * n) * -0.25))

Julia

function code(K, m, n, M, l)
	return exp(Float64(Float64(n * n) * -0.25))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((n * n) * -0.25));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 75.6%

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

3. Taylor expanded in K around 0 97.1%

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

   1. cos-neg 97.1%

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

5. Simplified 97.1%

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

6. Taylor expanded in M around 0 89.5%

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

   1. associate--r+ 89.5%

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

   2. fabs-sub 89.5%

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

8. Simplified 89.5%

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

9. Taylor expanded in n around inf 56.2%

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

    1. *-commutative 56.2%

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

11. Simplified 56.2%

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

    1. unpow2 56.2%

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

13. Applied egg-rr 56.2%

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

Alternative 8: 7.0% 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 75.6%

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

3. Taylor expanded in l around inf 31.3%

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

   1. mul-1-neg 31.3%

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

5. Simplified 31.3%

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

6. Taylor expanded in l around 0 6.0%

   \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right) - M\right)} \]
7. Step-by-step derivation

   1. *-commutative 6.0%

      \[\leadsto \cos \left(0.5 \cdot \color{blue}{\left(\left(m + n\right) \cdot K\right)} - M\right) \]

   2. *-commutative 6.0%

      \[\leadsto \cos \left(\color{blue}{\left(\left(m + n\right) \cdot K\right) \cdot 0.5} - M\right) \]

   3. associate-*l* 6.0%

      \[\leadsto \cos \left(\color{blue}{\left(m + n\right) \cdot \left(K \cdot 0.5\right)} - M\right) \]

   4. *-commutative 6.0%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \color{blue}{\left(0.5 \cdot K\right)} - M\right) \]

8. Simplified 6.0%

   \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \left(0.5 \cdot K\right) - M\right)} \]

9. Taylor expanded in K around 0 6.7%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \]
10. Step-by-step derivation

    1. cos-neg 6.7%

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

11. Simplified 6.7%

    \[\leadsto \color{blue}{\cos M} \]
12. Add Preprocessing

Alternative 9: 7.0% accurate, 425.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

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
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

Python

def code(K, m, n, M, l):
	return 1.0

Julia

function code(K, m, n, M, l)
	return 1.0
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 75.6%

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

3. Taylor expanded in K around 0 97.1%

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

   1. cos-neg 97.1%

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

5. Simplified 97.1%

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

6. Taylor expanded in M around 0 89.5%

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

   1. associate--r+ 89.5%

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

   2. fabs-sub 89.5%

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

8. Simplified 89.5%

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

9. Taylor expanded in n around inf 56.2%

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

    1. *-commutative 56.2%

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

11. Simplified 56.2%

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

12. Taylor expanded in n around 0 6.7%

    \[\leadsto \color{blue}{1} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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)))))))