Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.3% → 96.9%

Time: 23.9s

Alternatives: 7

Speedup: 1.3×

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: 76.3% 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: 96.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = ((m + n) * 0.5) - M;
	return cos(M) * exp(((fabs((m - n)) - l) - (t_0 * t_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
    real(8) :: t_0
    t_0 = ((m + n) * 0.5d0) - m_1
    code = cos(m_1) * exp(((abs((m - n)) - l) - (t_0 * t_0)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = ((m + n) * 0.5) - M;
	return Math.cos(M) * Math.exp(((Math.abs((m - n)) - l) - (t_0 * t_0)));
}

Python

def code(K, m, n, M, l):
	t_0 = ((m + n) * 0.5) - M
	return math.cos(M) * math.exp(((math.fabs((m - n)) - l) - (t_0 * t_0)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

3. Taylor expanded in K around 0 95.9%

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

   1. cos-neg 95.9%

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

5. Simplified 95.9%

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

   1. unpow2 95.9%

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

   2. div-inv 95.9%

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

   3. metadata-eval 95.9%

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

   4. div-inv 95.9%

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

   5. metadata-eval 95.9%

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

7. Applied egg-rr 95.9%

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

8. Final simplification 95.9%

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

Alternative 2: 87.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -185:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + \left(\left|m - n\right| - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -185.0)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (*
    (cos M)
    (exp (+ (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) (- (fabs (- m n)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -185.0) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (fabs((m - n)) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-185.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) + (abs((m - n)) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -185.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (Math.abs((m - n)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -185.0:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (math.fabs((m - n)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -185.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) + Float64(abs(Float64(m - n)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -185.0)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (abs((m - n)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -185.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -185

   1. Initial program 64.4%

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

   3. Taylor expanded in K around 0 100.0%

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

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

   if -185 < m

   1. Initial program 78.6%

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

   3. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   5. Simplified 94.7%

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

   6. Taylor expanded in m around 0 79.6%

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

      1. +-commutative 79.6%

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

      2. unpow2 79.6%

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

      3. distribute-rgt-out 83.2%

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

      4. *-commutative 83.2%

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

      5. *-commutative 83.2%

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

   8. Simplified 83.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -185:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + \left(\left|m - n\right| - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -2.1 \cdot 10^{+24} \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{m - \left(n + \ell\right)} \cdot \cos \left(\left(\left(m + n\right) \cdot 0.5\right) \cdot K\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -2.1e+24) (not (<= M 27.0)))
   (* (cos M) (exp (- (pow M 2.0))))
   (* (exp (- m (+ n l))) (cos (* (* (+ m n) 0.5) K)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2.1e+24) || !(M <= 27.0)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = exp((m - (n + l))) * cos((((m + n) * 0.5) * K));
	}
	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 <= (-2.1d+24)) .or. (.not. (m_1 <= 27.0d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = exp((m - (n + l))) * cos((((m + n) * 0.5d0) * k))
    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 <= -2.1e+24) || !(M <= 27.0)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp((m - (n + l))) * Math.cos((((m + n) * 0.5) * K));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -2.1e+24) or not (M <= 27.0):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp((m - (n + l))) * math.cos((((m + n) * 0.5) * K))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -2.1e+24) || !(M <= 27.0))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(exp(Float64(m - Float64(n + l))) * cos(Float64(Float64(Float64(m + n) * 0.5) * K)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -2.1e+24) || ~((M <= 27.0)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = exp((m - (n + l))) * cos((((m + n) * 0.5) * K));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -2.1e+24], N[Not[LessEqual[M, 27.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(m - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -2.1000000000000001e24 or 27 < M

   1. Initial program 79.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 98.3%

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

      1. mul-1-neg 98.3%

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

   8. Simplified 98.3%

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

   if -2.1000000000000001e24 < M < 27

   1. Initial program 71.4%

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

      1. *-un-lft-identity 71.4%

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

      2. *-commutative 71.4%

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

   4. Applied egg-rr 29.2%

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

   5. Taylor expanded in M around inf 51.4%

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

   6. Taylor expanded in M around 0 51.4%

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

      1. *-commutative 51.4%

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

      2. +-commutative 51.4%

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

      3. distribute-lft-in 51.4%

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

      4. distribute-lft-out 51.4%

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

      5. *-commutative 51.4%

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

      6. associate-*l* 51.4%

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

      7. *-commutative 51.4%

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

      8. *-commutative 51.4%

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

      9. associate-*r* 51.4%

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

      10. distribute-lft-in 51.4%

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

      11. *-commutative 51.4%

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

      12. distribute-lft-in 51.4%

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

      13. +-commutative 51.4%

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

   8. Simplified 51.4%

      \[\leadsto \color{blue}{e^{m - \left(n + \ell\right)} \cdot \cos \left(K \cdot \left(0.5 \cdot \left(n + m\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.1 \cdot 10^{+24} \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{m - \left(n + \ell\right)} \cdot \cos \left(\left(\left(m + n\right) \cdot 0.5\right) \cdot K\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;e^{m - \left(n + \ell\right)} \cdot \cos \left(\left(\left(m + n\right) \cdot 0.5\right) \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 720.0)
   (* (exp (- m (+ n l))) (cos (* (* (+ m n) 0.5) K)))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = exp((m - (n + l))) * cos((((m + n) * 0.5) * K));
	} else {
		tmp = 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 <= 720.0d0) then
        tmp = exp((m - (n + l))) * cos((((m + n) * 0.5d0) * k))
    else
        tmp = 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 <= 720.0) {
		tmp = Math.exp((m - (n + l))) * Math.cos((((m + n) * 0.5) * K));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 720.0:
		tmp = math.exp((m - (n + l))) * math.cos((((m + n) * 0.5) * K))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 720.0)
		tmp = Float64(exp(Float64(m - Float64(n + l))) * cos(Float64(Float64(Float64(m + n) * 0.5) * K)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 720.0)
		tmp = exp((m - (n + l))) * cos((((m + n) * 0.5) * K));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 720.0], N[(N[Exp[N[(m - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 720

   1. Initial program 76.2%

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

      1. *-un-lft-identity 76.2%

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

      2. *-commutative 76.2%

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

   4. Applied egg-rr 14.1%

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

   5. Taylor expanded in M around inf 28.6%

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

   6. Taylor expanded in M around 0 36.7%

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

      1. *-commutative 36.7%

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

      2. +-commutative 36.7%

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

      3. distribute-lft-in 36.7%

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

      4. distribute-lft-out 36.7%

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

      5. *-commutative 36.7%

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

      6. associate-*l* 36.7%

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

      7. *-commutative 36.7%

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

      8. *-commutative 36.7%

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

      9. associate-*r* 36.7%

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

      10. distribute-lft-in 36.7%

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

      11. *-commutative 36.7%

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

      12. distribute-lft-in 36.7%

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

      13. +-commutative 36.7%

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

   8. Simplified 36.7%

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

   if 720 < l

   1. Initial program 72.4%

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

      1. add-exp-log 39.7%

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

   4. Applied egg-rr 39.7%

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

   5. Taylor expanded in l around inf 39.7%

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

      1. mul-1-neg 39.7%

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

   7. Simplified 39.7%

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

   8. Taylor expanded in m around inf 82.8%

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

      1. *-commutative 82.8%

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

   10. Simplified 82.8%

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

   11. Taylor expanded in m around 0 100.0%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;e^{m - \left(n + \ell\right)} \cdot \cos \left(\left(\left(m + n\right) \cdot 0.5\right) \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{m}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -6.5e-9)
   (* (cos (- (/ (* (+ m n) K) 2.0) M)) (exp m))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -6.5e-9) {
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(m);
	} 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 (m <= (-6.5d-9)) then
        tmp = cos(((((m + n) * k) / 2.0d0) - m_1)) * exp(m)
    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 (m <= -6.5e-9) {
		tmp = Math.cos(((((m + n) * K) / 2.0) - M)) * Math.exp(m);
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -6.5e-9:
		tmp = math.cos(((((m + n) * K) / 2.0) - M)) * math.exp(m)
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -6.5e-9)
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) * K) / 2.0) - M)) * exp(m));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -6.5e-9)
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(m);
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -6.5e-9], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[m], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -6.5000000000000003e-9

   1. Initial program 65.0%

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

      1. *-un-lft-identity 65.0%

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

      2. *-commutative 65.0%

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

   4. Applied egg-rr 1.0%

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

   5. Taylor expanded in M around inf 33.8%

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

   6. Taylor expanded in m around inf 63.4%

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

   if -6.5000000000000003e-9 < m

   1. Initial program 78.5%

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

   3. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   5. Simplified 94.7%

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

   6. Taylor expanded in l around inf 39.5%

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

      1. mul-1-neg 39.5%

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

   8. Simplified 39.5%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{m}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

3. Taylor expanded in K around 0 95.9%

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

   1. cos-neg 95.9%

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

5. Simplified 95.9%

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

6. Taylor expanded in l around inf 35.1%

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

   1. mul-1-neg 35.1%

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

8. Simplified 35.1%

   \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]
9. Add Preprocessing

Alternative 7: 35.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. add-exp-log 40.5%

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

4. Applied egg-rr 40.5%

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

5. Taylor expanded in l around inf 17.4%

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

   1. mul-1-neg 17.4%

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

7. Simplified 17.4%

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

8. Taylor expanded in m around inf 30.1%

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

   1. *-commutative 30.1%

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

10. Simplified 30.1%

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

11. Taylor expanded in m around 0 34.3%

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

12. Final simplification 34.3%

    \[\leadsto e^{-\ell} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(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)))))))