Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.6% → 96.7%

Time: 18.3s

Alternatives: 11

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: 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, 0.7× speedup

Math

\[\begin{array}{l} \\ e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (exp (- (fabs (- n m)) (+ l (pow (- (* 0.5 (+ n m)) M) 2.0))))
  (log1p (expm1 (cos M)))))

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - (l + pow(((0.5 * (n + m)) - M), 2.0)))) * log1p(expm1(cos(M)));
}

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((Math.abs((n - m)) - (l + Math.pow(((0.5 * (n + m)) - M), 2.0)))) * Math.log1p(Math.expm1(Math.cos(M)));
}

Python

def code(K, m, n, M, l):
	return math.exp((math.fabs((n - m)) - (l + math.pow(((0.5 * (n + m)) - M), 2.0)))) * math.log1p(math.expm1(math.cos(M)))

Julia

function code(K, m, n, M, l)
	return Float64(exp(Float64(abs(Float64(n - m)) - Float64(l + (Float64(Float64(0.5 * Float64(n + m)) - M) ^ 2.0)))) * log1p(expm1(cos(M))))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[Power[N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[1 + N[(Exp[N[Cos[M], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in K around 0 96.6%

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

5. Simplified 96.6%

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]
6. Step-by-step derivation

   1. log1p-expm1-u 96.6%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

7. Applied egg-rr 96.6%

   \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

8. Final simplification 96.6%

   \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right) \]
9. Add Preprocessing

Alternative 2: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - (l + pow(((0.5 * (n + m)) - M), 2.0)))) * 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 = exp((abs((n - m)) - (l + (((0.5d0 * (n + m)) - m_1) ** 2.0d0)))) * cos(m_1)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((Math.abs((n - m)) - (l + Math.pow(((0.5 * (n + m)) - M), 2.0)))) * Math.cos(M);
}

Python

def code(K, m, n, M, l):
	return math.exp((math.fabs((n - m)) - (l + math.pow(((0.5 * (n + m)) - M), 2.0)))) * math.cos(M)

Julia

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

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp((abs((n - m)) - (l + (((0.5 * (n + m)) - M) ^ 2.0)))) * cos(M);
end

Wolfram

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

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in K around 0 96.6%

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

5. Simplified 96.6%

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

6. Final simplification 96.6%

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

Alternative 3: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{-30} \lor \neg \left(m \leq 7 \cdot 10^{-77}\right):\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - 0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot e^{\left(-\ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -5e-30) (not (<= m 7e-77)))
   (exp (- (- (fabs (- n m)) l) (* 0.25 (* (+ n m) (+ n m)))))
   (*
    (cos (- (/ (* (+ n m) K) 2.0) M))
    (exp (- (- l) (pow (- (/ (+ n m) 2.0) M) 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -5e-30) || !(m <= 7e-77)) {
		tmp = exp(((fabs((n - m)) - l) - (0.25 * ((n + m) * (n + m)))));
	} else {
		tmp = cos(((((n + m) * K) / 2.0) - M)) * exp((-l - pow((((n + m) / 2.0) - M), 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 <= (-5d-30)) .or. (.not. (m <= 7d-77))) then
        tmp = exp(((abs((n - m)) - l) - (0.25d0 * ((n + m) * (n + m)))))
    else
        tmp = cos(((((n + m) * k) / 2.0d0) - m_1)) * exp((-l - ((((n + m) / 2.0d0) - m_1) ** 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 <= -5e-30) || !(m <= 7e-77)) {
		tmp = Math.exp(((Math.abs((n - m)) - l) - (0.25 * ((n + m) * (n + m)))));
	} else {
		tmp = Math.cos(((((n + m) * K) / 2.0) - M)) * Math.exp((-l - Math.pow((((n + m) / 2.0) - M), 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -5e-30) or not (m <= 7e-77):
		tmp = math.exp(((math.fabs((n - m)) - l) - (0.25 * ((n + m) * (n + m)))))
	else:
		tmp = math.cos(((((n + m) * K) / 2.0) - M)) * math.exp((-l - math.pow((((n + m) / 2.0) - M), 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -5e-30) || !(m <= 7e-77))
		tmp = exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(0.25 * Float64(Float64(n + m) * Float64(n + m)))));
	else
		tmp = Float64(cos(Float64(Float64(Float64(Float64(n + m) * K) / 2.0) - M)) * exp(Float64(Float64(-l) - (Float64(Float64(Float64(n + m) / 2.0) - M) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -5e-30) || ~((m <= 7e-77)))
		tmp = exp(((abs((n - m)) - l) - (0.25 * ((n + m) * (n + m)))));
	else
		tmp = cos(((((n + m) * K) / 2.0) - M)) * exp((-l - ((((n + m) / 2.0) - M) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -5e-30], N[Not[LessEqual[m, 7e-77]], $MachinePrecision]], N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(0.25 * N[(N[(n + m), $MachinePrecision] * N[(n + m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(N[(N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-l) - N[Power[N[(N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -4.99999999999999972e-30 or 7.00000000000000026e-77 < m

   1. Initial program 67.9%

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

   3. Taylor expanded in K around 0 97.0%

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

   5. Simplified 97.0%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 97.0%

         \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   7. Applied egg-rr 97.0%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   8. Taylor expanded in M around 0 93.8%

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

      1. associate--r+ 93.8%

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

      2. +-commutative 93.8%

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

   10. Simplified 93.8%

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

       1. unpow2 93.8%

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

   12. Applied egg-rr 93.8%

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

   if -4.99999999999999972e-30 < m < 7.00000000000000026e-77

   1. Initial program 84.5%

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

   3. Taylor expanded in l around inf 82.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{-30} \lor \neg \left(m \leq 7 \cdot 10^{-77}\right):\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - 0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot e^{\left(-\ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 8.5 \cdot 10^{+164}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - 0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot e^{M \cdot \left(m + \left(n - M\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M 8.5e+164)
   (exp (- (- (fabs (- n m)) l) (* 0.25 (* (+ n m) (+ n m)))))
   (* (cos (- (/ (* (+ n m) K) 2.0) M)) (exp (* M (+ m (- n M)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= 8.5e+164) {
		tmp = exp(((fabs((n - m)) - l) - (0.25 * ((n + m) * (n + m)))));
	} else {
		tmp = cos(((((n + m) * K) / 2.0) - M)) * exp((M * (m + (n - M))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m_1 <= 8.5d+164) then
        tmp = exp(((abs((n - m)) - l) - (0.25d0 * ((n + m) * (n + m)))))
    else
        tmp = cos(((((n + m) * k) / 2.0d0) - m_1)) * exp((m_1 * (m + (n - m_1))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= 8.5e+164) {
		tmp = Math.exp(((Math.abs((n - m)) - l) - (0.25 * ((n + m) * (n + m)))));
	} else {
		tmp = Math.cos(((((n + m) * K) / 2.0) - M)) * Math.exp((M * (m + (n - M))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= 8.5e+164:
		tmp = math.exp(((math.fabs((n - m)) - l) - (0.25 * ((n + m) * (n + m)))))
	else:
		tmp = math.cos(((((n + m) * K) / 2.0) - M)) * math.exp((M * (m + (n - M))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= 8.5e+164)
		tmp = exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(0.25 * Float64(Float64(n + m) * Float64(n + m)))));
	else
		tmp = Float64(cos(Float64(Float64(Float64(Float64(n + m) * K) / 2.0) - M)) * exp(Float64(M * Float64(m + Float64(n - M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= 8.5e+164)
		tmp = exp(((abs((n - m)) - l) - (0.25 * ((n + m) * (n + m)))));
	else
		tmp = cos(((((n + m) * K) / 2.0) - M)) * exp((M * (m + (n - M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, 8.5e+164], N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(0.25 * N[(N[(n + m), $MachinePrecision] * N[(n + m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(N[(N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(M * N[(m + N[(n - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.50000000000000027e164

   1. Initial program 72.2%

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

   3. Taylor expanded in K around 0 96.2%

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

   5. Simplified 96.2%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 96.2%

         \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   7. Applied egg-rr 96.2%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   8. Taylor expanded in M around 0 85.9%

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

      1. associate--r+ 85.9%

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

      2. +-commutative 85.9%

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

   10. Simplified 85.9%

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

       1. unpow2 85.9%

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

   12. Applied egg-rr 85.9%

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

   if 8.50000000000000027e164 < M

   1. Initial program 89.3%

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

   3. Taylor expanded in M around inf 89.3%

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

   4. Taylor expanded in M around 0 89.3%

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

      1. neg-mul-1 89.3%

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

   6. Simplified 89.3%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{M \cdot \left(m + \left(n + \left(-M\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 8.5 \cdot 10^{+164}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - 0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot e^{M \cdot \left(m + \left(n - M\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in K around 0 96.6%

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

5. Simplified 96.6%

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]
6. Step-by-step derivation

   1. log1p-expm1-u 96.6%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

7. Applied egg-rr 96.6%

   \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

8. Taylor expanded in M around 0 83.6%

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

   1. associate--r+ 83.6%

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

   2. +-commutative 83.6%

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

10. Simplified 83.6%

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

    1. unpow2 83.6%

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

12. Applied egg-rr 83.6%

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

Alternative 6: 65.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -8.7 \cdot 10^{-33}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -55.0)
   (exp (* (pow m 2.0) -0.25))
   (if (<= m -8.7e-33) (* (cos M) (exp (- l))) (exp (* -0.25 (pow n 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -55.0) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else if (m <= -8.7e-33) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = exp((-0.25 * pow(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 <= (-55.0d0)) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else if (m <= (-8.7d-33)) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = exp(((-0.25d0) * (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 <= -55.0) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else if (m <= -8.7e-33) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -55.0:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	elif m <= -8.7e-33:
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -55.0)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	elseif (m <= -8.7e-33)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -55.0)
		tmp = exp(((m ^ 2.0) * -0.25));
	elseif (m <= -8.7e-33)
		tmp = cos(M) * exp(-l);
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -55.0], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -8.7e-33], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -55

   1. Initial program 67.6%

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

   3. Taylor expanded in K around 0 98.6%

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

   5. Simplified 98.6%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 98.6%

         \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   7. Applied egg-rr 98.6%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   8. Taylor expanded in M around 0 98.6%

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

      1. associate--r+ 98.6%

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

      2. +-commutative 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in m around inf 97.3%

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

       1. *-commutative 97.3%

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

   13. Simplified 97.3%

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

   if -55 < m < -8.7000000000000004e-33

   1. Initial program 42.9%

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

   3. Taylor expanded in l around inf 6.6%

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

      1. mul-1-neg 6.6%

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

   5. Simplified 6.6%

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

   6. Taylor expanded in K around 0 49.9%

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

      1. cos-neg 49.9%

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

   8. Simplified 49.9%

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

   if -8.7000000000000004e-33 < m

   1. Initial program 78.0%

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

   3. Taylor expanded in K around 0 95.7%

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

   5. Simplified 95.7%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 95.7%

         \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   7. Applied egg-rr 95.7%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   8. Taylor expanded in M around 0 77.2%

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

      1. associate--r+ 77.2%

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

      2. +-commutative 77.2%

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

   10. Simplified 77.2%

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

   11. Taylor expanded in n around inf 50.9%

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

       1. *-commutative 50.9%

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

   13. Simplified 50.9%

       \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -8.7 \cdot 10^{-33}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -950000000000:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -950000000000.0)
   (exp (* (pow m 2.0) -0.25))
   (exp (* -0.25 (pow n 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -950000000000.0) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = exp((-0.25 * pow(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 <= (-950000000000.0d0)) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = exp(((-0.25d0) * (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 <= -950000000000.0) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -950000000000.0:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -950000000000.0)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -950000000000.0)
		tmp = exp(((m ^ 2.0) * -0.25));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -950000000000.0], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -9.5e11

   1. Initial program 66.7%

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

   3. Taylor expanded in K around 0 98.6%

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

   5. Simplified 98.6%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 98.6%

         \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   7. Applied egg-rr 98.6%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   8. Taylor expanded in M around 0 98.6%

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

      1. associate--r+ 98.6%

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

      2. +-commutative 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in m around inf 98.6%

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

       1. *-commutative 98.6%

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

   13. Simplified 98.6%

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

   if -9.5e11 < m

   1. Initial program 76.9%

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

   3. Taylor expanded in K around 0 95.9%

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

   5. Simplified 95.9%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 95.9%

         \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   7. Applied egg-rr 95.9%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

   8. Taylor expanded in M around 0 77.8%

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

      1. associate--r+ 77.8%

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

      2. +-commutative 77.8%

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

   10. Simplified 77.8%

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

   11. Taylor expanded in n around inf 50.9%

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

       1. *-commutative 50.9%

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

   13. Simplified 50.9%

       \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -950000000000:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{{m}^{2} \cdot -0.25} \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return exp((pow(m, 2.0) * -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(((m ** 2.0d0) * (-0.25d0)))
end function

Java

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

Python

def code(K, m, n, M, l):
	return math.exp((math.pow(m, 2.0) * -0.25))

Julia

function code(K, m, n, M, l)
	return exp(Float64((m ^ 2.0) * -0.25))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((m ^ 2.0) * -0.25));
end

Wolfram

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

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in K around 0 96.6%

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

5. Simplified 96.6%

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos M} \]
6. Step-by-step derivation

   1. log1p-expm1-u 96.6%

      \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

7. Applied egg-rr 96.6%

   \[\leadsto e^{\left|n - m\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos M\right)\right)} \]

8. Taylor expanded in M around 0 83.6%

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

   1. associate--r+ 83.6%

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

   2. +-commutative 83.6%

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

10. Simplified 83.6%

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

11. Taylor expanded in m around inf 56.0%

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

    1. *-commutative 56.0%

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

13. Simplified 56.0%

    \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \]
14. Add Preprocessing

Alternative 9: 9.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{m \cdot K}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* m K) 2.0) M))
  (+ 1.0 (* l (+ (* l (+ 0.5 (* l -0.16666666666666666))) -1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((m * K) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -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 = cos((((m * k) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * (0.5d0 + (l * (-0.16666666666666666d0)))) + (-1.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((m * K) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
}

Python

def code(K, m, n, M, l):
	return math.cos((((m * K) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(m * K) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * Float64(0.5 + Float64(l * -0.16666666666666666))) + -1.0))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((m * K) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(m * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * N[(0.5 + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in l around inf 24.0%

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

   1. mul-1-neg 24.0%

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

5. Simplified 24.0%

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

6. Taylor expanded in m around inf 27.0%

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

   1. *-commutative 27.0%

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

8. Simplified 27.0%

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

9. Taylor expanded in l around 0 6.7%

   \[\leadsto \cos \left(\frac{m \cdot K}{2} - M\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + -0.16666666666666666 \cdot \ell\right) - 1\right)\right)} \]

10. Final simplification 6.7%

    \[\leadsto \cos \left(\frac{m \cdot K}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right) \]
11. Add Preprocessing

Alternative 10: 9.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{m \cdot K}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos (- (/ (* m K) 2.0) M)) (+ 1.0 (* l (+ (* l 0.5) -1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((m * K) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -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 = cos((((m * k) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * 0.5d0) + (-1.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((m * K) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
}

Python

def code(K, m, n, M, l):
	return math.cos((((m * K) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(m * K) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * 0.5) + -1.0))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((m * K) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(m * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in l around inf 24.0%

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

   1. mul-1-neg 24.0%

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

5. Simplified 24.0%

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

6. Taylor expanded in m around inf 27.0%

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

   1. *-commutative 27.0%

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

8. Simplified 27.0%

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

9. Taylor expanded in l around 0 6.4%

   \[\leadsto \cos \left(\frac{m \cdot K}{2} - M\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(0.5 \cdot \ell - 1\right)\right)} \]

10. Final simplification 6.4%

    \[\leadsto \cos \left(\frac{m \cdot K}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right) \]
11. Add Preprocessing

Alternative 11: 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 74.1%

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

3. Taylor expanded in M around inf 41.0%

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

4. Taylor expanded in M around 0 4.6%

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

   1. *-commutative 4.6%

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

6. Simplified 4.6%

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

7. Taylor expanded in n around 0 4.9%

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

   1. *-commutative 4.9%

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

   2. *-commutative 4.9%

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

   3. associate-*l* 4.9%

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

   4. *-commutative 4.9%

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

9. Simplified 4.9%

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

10. Taylor expanded in m around 0 5.3%

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

Reproduce

herbie shell --seed 2024136 
(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)))))))