Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.4% → 96.7%

Time: 14.4s

Alternatives: 9

Speedup: 3.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.0× speedup

Math

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

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 78.1%

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

3. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

5. Simplified 98.1%

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

6. Final simplification 98.1%

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

Alternative 2: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -340000000000 \lor \neg \left(M \leq 4.8 \cdot 10^{+67}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -340000000000.0) (not (<= M 4.8e+67)))
   (* (cos M) (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 <= -340000000000.0) || !(M <= 4.8e+67)) {
		tmp = cos(M) * 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 <= (-340000000000.0d0)) .or. (.not. (m_1 <= 4.8d+67))) then
        tmp = cos(m_1) * 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 <= -340000000000.0) || !(M <= 4.8e+67)) {
		tmp = Math.cos(M) * 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 <= -340000000000.0) or not (M <= 4.8e+67):
		tmp = math.cos(M) * 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 <= -340000000000.0) || !(M <= 4.8e+67))
		tmp = Float64(cos(M) * 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 <= -340000000000.0) || ~((M <= 4.8e+67)))
		tmp = cos(M) * 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, -340000000000.0], N[Not[LessEqual[M, 4.8e+67]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $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 < -3.4e11 or 4.80000000000000004e67 < M

   1. Initial program 84.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 98.2%

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

      1. mul-1-neg 98.2%

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

   8. Simplified 98.2%

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

   if -3.4e11 < M < 4.80000000000000004e67

   1. Initial program 72.8%

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

   3. Taylor expanded in K around 0 96.7%

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

      1. cos-neg 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in M around 0 94.7%

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

      1. fabs-sub 94.7%

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

      2. associate--r+ 94.7%

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

   8. Simplified 94.7%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -340000000000 \lor \neg \left(M \leq 4.8 \cdot 10^{+67}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -27.0) || !(M <= 27.0)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = exp(((n - m) - (0.25 * ((m + n) * (m + n)))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-27.0d0)) .or. (.not. (m_1 <= 27.0d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((n - m) - (0.25d0 * ((m + n) * (m + n)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -27.0) || !(M <= 27.0)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(((n - m) - (0.25 * ((m + n) * (m + n)))));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if M < -27 or 27 < M

   1. Initial program 81.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 96.3%

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

      1. mul-1-neg 96.3%

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

   8. Simplified 96.3%

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

   if -27 < M < 27

   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 96.1%

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

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

      1. fabs-sub 96.1%

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

      2. associate--r+ 96.1%

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

   8. Simplified 96.1%

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

   9. Taylor expanded in l around 0 82.0%

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

       1. rem-square-sqrt 39.1%

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

       2. fabs-sqr 39.1%

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

       3. rem-square-sqrt 81.5%

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

   11. Simplified 81.5%

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

       1. unpow2 81.5%

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

       2. +-commutative 81.5%

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

       3. +-commutative 81.5%

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

   13. Applied egg-rr 81.5%

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

4. Final simplification 89.1%

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

Alternative 4: 83.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 700:\\ \;\;\;\;e^{\left(n - m\right) - 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 700.0)
   (exp (- (- n m) (* 0.25 (* (+ m n) (+ m n)))))
   (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 700.0) {
		tmp = exp(((n - m) - (0.25 * ((m + n) * (m + n)))));
	} else {
		tmp = cos(M) / exp(l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 700.0d0) then
        tmp = exp(((n - m) - (0.25d0 * ((m + n) * (m + n)))))
    else
        tmp = cos(m_1) / exp(l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 700.0) {
		tmp = Math.exp(((n - m) - (0.25 * ((m + n) * (m + n)))));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 700.0:
		tmp = math.exp(((n - m) - (0.25 * ((m + n) * (m + n)))))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 700.0)
		tmp = exp(Float64(Float64(n - m) - Float64(0.25 * Float64(Float64(m + n) * Float64(m + n)))));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 700.0)
		tmp = exp(((n - m) - (0.25 * ((m + n) * (m + n)))));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 700.0], N[Exp[N[(N[(n - m), $MachinePrecision] - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 700

   1. Initial program 78.4%

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

   3. Taylor expanded in K around 0 97.8%

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

      1. cos-neg 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in M around 0 81.7%

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

      1. fabs-sub 81.7%

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

      2. associate--r+ 81.7%

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

   8. Simplified 81.7%

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

   9. Taylor expanded in l around 0 75.9%

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

       1. rem-square-sqrt 37.5%

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

       2. fabs-sqr 37.5%

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

       3. rem-square-sqrt 75.6%

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

   11. Simplified 75.6%

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

       1. unpow2 75.6%

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

       2. +-commutative 75.6%

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

       3. +-commutative 75.6%

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

   13. Applied egg-rr 75.6%

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

   if 700 < l

   1. Initial program 76.7%

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

   3. Taylor expanded in l around inf 76.7%

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

      1. mul-1-neg 76.7%

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

   5. Simplified 76.7%

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

      1. clear-num 76.7%

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

      2. inv-pow 76.7%

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

      3. *-commutative 76.7%

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

   7. Applied egg-rr 76.7%

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

      1. unpow-1 76.7%

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

   9. Simplified 76.7%

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

       1. exp-neg 76.7%

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

   11. Applied egg-rr 76.7%

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

   12. Taylor expanded in K around 0 100.0%

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

       1. cos-neg 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 700:\\ \;\;\;\;e^{\left(n - m\right) - 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-113} \lor \neg \left(n \leq 300000\right):\\ \;\;\;\;e^{\left(n - m\right) - 0.25 \cdot \left(n \cdot \left(n + m \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(-M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -8e-113) (not (<= n 300000.0)))
   (exp (- (- n m) (* 0.25 (* n (+ n (* m 2.0))))))
   (* (cos (- M)) (+ 1.0 (* l (+ (* l 0.5) -1.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -8e-113) || !(n <= 300000.0)) {
		tmp = exp(((n - m) - (0.25 * (n * (n + (m * 2.0))))));
	} else {
		tmp = cos(-M) * (1.0 + (l * ((l * 0.5) + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((n <= (-8d-113)) .or. (.not. (n <= 300000.0d0))) then
        tmp = exp(((n - m) - (0.25d0 * (n * (n + (m * 2.0d0))))))
    else
        tmp = cos(-m_1) * (1.0d0 + (l * ((l * 0.5d0) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -8e-113) || !(n <= 300000.0)) {
		tmp = Math.exp(((n - m) - (0.25 * (n * (n + (m * 2.0))))));
	} else {
		tmp = Math.cos(-M) * (1.0 + (l * ((l * 0.5) + -1.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -8e-113) or not (n <= 300000.0):
		tmp = math.exp(((n - m) - (0.25 * (n * (n + (m * 2.0))))))
	else:
		tmp = math.cos(-M) * (1.0 + (l * ((l * 0.5) + -1.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -8e-113) || !(n <= 300000.0))
		tmp = exp(Float64(Float64(n - m) - Float64(0.25 * Float64(n * Float64(n + Float64(m * 2.0))))));
	else
		tmp = Float64(cos(Float64(-M)) * Float64(1.0 + Float64(l * Float64(Float64(l * 0.5) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -8e-113) || ~((n <= 300000.0)))
		tmp = exp(((n - m) - (0.25 * (n * (n + (m * 2.0))))));
	else
		tmp = cos(-M) * (1.0 + (l * ((l * 0.5) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -8e-113], N[Not[LessEqual[n, 300000.0]], $MachinePrecision]], N[Exp[N[(N[(n - m), $MachinePrecision] - N[(0.25 * N[(n * N[(n + N[(m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[(-M)], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -7.99999999999999983e-113 or 3e5 < n

   1. Initial program 73.2%

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

   3. Taylor expanded in K around 0 99.3%

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

      1. cos-neg 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in M around 0 94.5%

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

      1. fabs-sub 94.5%

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

      2. associate--r+ 94.5%

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

   8. Simplified 94.5%

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

   9. Taylor expanded in l around 0 90.4%

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

       1. rem-square-sqrt 40.2%

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

       2. fabs-sqr 40.2%

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

       3. rem-square-sqrt 90.0%

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

   11. Simplified 90.0%

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

   12. Taylor expanded in m around 0 76.0%

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

       1. +-commutative 76.0%

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

       2. unpow2 76.0%

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

       3. associate-*r* 76.0%

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

       4. distribute-rgt-in 79.6%

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

       5. *-commutative 79.6%

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

   14. Simplified 79.6%

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

   if -7.99999999999999983e-113 < n < 3e5

   1. Initial program 84.1%

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

   3. Taylor expanded in l around inf 35.5%

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

      1. mul-1-neg 35.5%

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

   5. Simplified 35.5%

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

      1. clear-num 36.4%

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

      2. inv-pow 36.4%

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

      3. *-commutative 36.4%

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

   7. Applied egg-rr 36.4%

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

      1. unpow-1 36.4%

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

   9. Simplified 36.4%

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

   10. Taylor expanded in l around 0 21.0%

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

   11. Taylor expanded in K around 0 20.7%

       \[\leadsto \color{blue}{\cos \left(-M\right) \cdot \left(1 + \ell \cdot \left(0.5 \cdot \ell - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-113} \lor \neg \left(n \leq 300000\right):\\ \;\;\;\;e^{\left(n - m\right) - 0.25 \cdot \left(n \cdot \left(n + m \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(-M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

5. Simplified 98.1%

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

6. Taylor expanded in M around 0 84.8%

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

   1. fabs-sub 84.8%

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

   2. associate--r+ 84.8%

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

8. Simplified 84.8%

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

9. Taylor expanded in l around 0 76.5%

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

    1. rem-square-sqrt 37.5%

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

    2. fabs-sqr 37.5%

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

    3. rem-square-sqrt 76.3%

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

11. Simplified 76.3%

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

    1. unpow2 76.3%

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

    2. +-commutative 76.3%

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

    3. +-commutative 76.3%

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

13. Applied egg-rr 76.3%

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

14. Final simplification 76.3%

    \[\leadsto e^{\left(n - m\right) - 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)} \]
15. Add Preprocessing

Alternative 7: 9.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(-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_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) * (1.0 + (l * ((l * 0.5) + -1.0)));
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Taylor expanded in l around inf 26.9%

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

   1. mul-1-neg 26.9%

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

5. Simplified 26.9%

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

   1. clear-num 27.3%

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

   2. inv-pow 27.3%

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

   3. *-commutative 27.3%

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

7. Applied egg-rr 27.3%

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

   1. unpow-1 27.3%

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

9. Simplified 27.3%

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

10. Taylor expanded in l around 0 13.6%

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

11. Taylor expanded in K around 0 14.2%

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

12. Final simplification 14.2%

    \[\leadsto \cos \left(-M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right) \]
13. 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 78.1%

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

3. Taylor expanded in l around inf 26.9%

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

   1. mul-1-neg 26.9%

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

5. Simplified 26.9%

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

6. Taylor expanded in l around 0 8.6%

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

   1. *-commutative 8.6%

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

   2. *-commutative 8.6%

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

   3. *-commutative 8.6%

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

   4. associate-*r* 8.6%

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

   5. fmm-undef 8.6%

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

   6. fmm-undef 8.6%

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

   7. associate-*r* 8.6%

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

   8. *-commutative 8.6%

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

   9. associate-*l* 8.6%

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

   10. *-commutative 8.6%

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

8. Simplified 8.6%

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

9. Taylor expanded in K around 0 9.0%

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

    1. cos-neg 9.0%

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

11. Simplified 9.0%

    \[\leadsto \color{blue}{\cos M} \]
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 78.1%

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

3. Taylor expanded in l around inf 26.9%

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

   1. mul-1-neg 26.9%

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

5. Simplified 26.9%

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

6. Taylor expanded in l around 0 8.6%

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

   1. *-commutative 8.6%

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

   2. *-commutative 8.6%

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

   3. *-commutative 8.6%

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

   4. associate-*r* 8.6%

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

   5. fmm-undef 8.6%

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

   6. fmm-undef 8.6%

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

   7. associate-*r* 8.6%

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

   8. *-commutative 8.6%

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

   9. associate-*l* 8.6%

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

   10. *-commutative 8.6%

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

8. Simplified 8.6%

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

9. Taylor expanded in K around 0 9.0%

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

    1. cos-neg 9.0%

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

11. Simplified 9.0%

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

12. Taylor expanded in M around 0 9.0%

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

Reproduce

herbie shell --seed 2024170 
(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)))))))