Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.2% → 96.8%

Time: 14.6s

Alternatives: 5

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.2% 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: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp((abs((n - m)) - (l + (((0.5d0 * (n + m)) - m_1) ** 2.0d0)))) * cos(m_1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in K around 0 95.7%

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

5. Simplified 95.7%

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

6. Final simplification 95.7%

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

Alternative 2: 36.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ \mathbf{if}\;K \leq -5.9 \cdot 10^{-242}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;K \leq 3 \cdot 10^{-152}:\\ \;\;\;\;-0.5 \cdot \left(K \cdot \left(m \cdot \left(t\_0 \cdot \sin \left(0.5 \cdot \left(n \cdot K\right) - M\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))))
   (if (<= K -5.9e-242)
     (/ (cos M) (exp l))
     (if (<= K 3e-152)
       (* -0.5 (* K (* m (* t_0 (sin (- (* 0.5 (* n K)) M))))))
       (* (cos M) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double tmp;
	if (K <= -5.9e-242) {
		tmp = cos(M) / exp(l);
	} else if (K <= 3e-152) {
		tmp = -0.5 * (K * (m * (t_0 * sin(((0.5 * (n * K)) - M)))));
	} else {
		tmp = cos(M) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-l)
    if (k <= (-5.9d-242)) then
        tmp = cos(m_1) / exp(l)
    else if (k <= 3d-152) then
        tmp = (-0.5d0) * (k * (m * (t_0 * sin(((0.5d0 * (n * k)) - m_1)))))
    else
        tmp = cos(m_1) * t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(-l);
	double tmp;
	if (K <= -5.9e-242) {
		tmp = Math.cos(M) / Math.exp(l);
	} else if (K <= 3e-152) {
		tmp = -0.5 * (K * (m * (t_0 * Math.sin(((0.5 * (n * K)) - M)))));
	} else {
		tmp = Math.cos(M) * t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-l)
	tmp = 0
	if K <= -5.9e-242:
		tmp = math.cos(M) / math.exp(l)
	elif K <= 3e-152:
		tmp = -0.5 * (K * (m * (t_0 * math.sin(((0.5 * (n * K)) - M)))))
	else:
		tmp = math.cos(M) * t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	tmp = 0.0
	if (K <= -5.9e-242)
		tmp = Float64(cos(M) / exp(l));
	elseif (K <= 3e-152)
		tmp = Float64(-0.5 * Float64(K * Float64(m * Float64(t_0 * sin(Float64(Float64(0.5 * Float64(n * K)) - M))))));
	else
		tmp = Float64(cos(M) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-l);
	tmp = 0.0;
	if (K <= -5.9e-242)
		tmp = cos(M) / exp(l);
	elseif (K <= 3e-152)
		tmp = -0.5 * (K * (m * (t_0 * sin(((0.5 * (n * K)) - M)))));
	else
		tmp = cos(M) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[K, -5.9e-242], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[K, 3e-152], N[(-0.5 * N[(K * N[(m * N[(t$95$0 * N[Sin[N[(N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if K < -5.89999999999999999e-242

   1. Initial program 72.2%

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

   3. Taylor expanded in l around inf 26.2%

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

      1. mul-1-neg 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in K around 0 33.3%

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

      1. cos-neg 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in l around -inf 33.3%

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

       1. neg-mul-1 33.3%

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

       2. exp-neg 33.3%

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

       3. associate-*r/ 33.3%

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

       4. *-rgt-identity 33.3%

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

   11. Simplified 33.3%

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

   if -5.89999999999999999e-242 < K < 3e-152

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 18.2%

      \[\leadsto \left(\cos \left(0.5 \cdot \left(n \cdot K\right) - M\right) + -0.5 \cdot \left(\left(m \cdot K\right) \cdot \sin \left(0.5 \cdot \left(n \cdot K\right) - M\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg 18.2%

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

   8. Simplified 18.2%

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

   9. Taylor expanded in K around inf 42.0%

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

   if 3e-152 < K

   1. Initial program 61.1%

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

   3. Taylor expanded in l around inf 35.3%

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

      1. mul-1-neg 35.3%

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

   5. Simplified 35.3%

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

   6. Taylor expanded in K around 0 41.9%

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

      1. cos-neg 41.9%

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

   8. Simplified 41.9%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq -5.9 \cdot 10^{-242}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;K \leq 3 \cdot 10^{-152}:\\ \;\;\;\;-0.5 \cdot \left(K \cdot \left(m \cdot \left(e^{-\ell} \cdot \sin \left(0.5 \cdot \left(n \cdot K\right) - M\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. sub-neg 72.5%

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

   2. distribute-neg-out 72.5%

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

   3. div-inv 72.5%

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

   4. fmm-def 72.5%

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

   5. metadata-eval 72.5%

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

   6. add-sqr-sqrt 32.2%

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

   7. fabs-sqr 32.2%

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

   8. add-sqr-sqrt 72.5%

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

4. Applied egg-rr 72.5%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left(m - n\right)\right)\right)}} \]
5. Step-by-step derivation

   1. +-commutative 72.5%

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

   2. distribute-neg-in 72.5%

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

   3. sub-neg 72.5%

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

   4. sub-neg 72.5%

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

   5. distribute-neg-in 72.5%

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

   6. sub-neg 72.5%

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

   7. mul-1-neg 72.5%

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

   8. distribute-neg-in 72.5%

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

   9. mul-1-neg 72.5%

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

   10. mul-1-neg 72.5%

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

   11. remove-double-neg 72.5%

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

   12. distribute-neg-in 72.5%

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

   13. mul-1-neg 72.5%

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

   14. remove-double-neg 72.5%

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

   15. sub-neg 72.5%

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

   16. fmm-undef 72.5%

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

   17. *-commutative 72.5%

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

6. Simplified 72.5%

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

7. Taylor expanded in K around 0 95.7%

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

8. Taylor expanded in M around 0 95.7%

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

9. Final simplification 95.7%

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

Alternative 4: 35.0% 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 72.5%

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

3. Taylor expanded in l around inf 28.4%

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

   1. mul-1-neg 28.4%

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

5. Simplified 28.4%

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

6. Taylor expanded in K around 0 34.1%

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

   1. cos-neg 34.1%

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

8. Simplified 34.1%

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

Alternative 5: 34.7% 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 72.5%

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

3. Taylor expanded in l around inf 28.4%

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

   1. mul-1-neg 28.4%

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

5. Simplified 28.4%

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

6. Taylor expanded in K around 0 34.1%

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

   1. cos-neg 34.1%

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

8. Simplified 34.1%

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

9. Taylor expanded in M around 0 34.1%

   \[\leadsto \color{blue}{e^{-\ell}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024184 
(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)))))))