Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.7% → 96.4%

Time: 23.6s

Alternatives: 7

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% 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 78.0%

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

3. Taylor expanded in K around 0 96.6%

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

   1. cos-neg 96.6%

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

5. Simplified 96.6%

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

6. Final simplification 96.6%

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

Alternative 2: 86.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M - n \cdot 0.5\\ t_1 := \left|m - n\right| - \ell\\ \mathbf{if}\;m \leq -49:\\ \;\;\;\;e^{t\_1 - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_1 - t\_0 \cdot \left(t\_0 - m\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* n 0.5))) (t_1 (- (fabs (- m n)) l)))
   (if (<= m -49.0)
     (exp (- t_1 (* 0.25 (pow (+ m n) 2.0))))
     (exp (- t_1 (* t_0 (- t_0 m)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (n * 0.5);
	double t_1 = fabs((m - n)) - l;
	double tmp;
	if (m <= -49.0) {
		tmp = exp((t_1 - (0.25 * pow((m + n), 2.0))));
	} else {
		tmp = exp((t_1 - (t_0 * (t_0 - m))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = m_1 - (n * 0.5d0)
    t_1 = abs((m - n)) - l
    if (m <= (-49.0d0)) then
        tmp = exp((t_1 - (0.25d0 * ((m + n) ** 2.0d0))))
    else
        tmp = exp((t_1 - (t_0 * (t_0 - m))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (n * 0.5);
	double t_1 = Math.abs((m - n)) - l;
	double tmp;
	if (m <= -49.0) {
		tmp = Math.exp((t_1 - (0.25 * Math.pow((m + n), 2.0))));
	} else {
		tmp = Math.exp((t_1 - (t_0 * (t_0 - m))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M - (n * 0.5)
	t_1 = math.fabs((m - n)) - l
	tmp = 0
	if m <= -49.0:
		tmp = math.exp((t_1 - (0.25 * math.pow((m + n), 2.0))))
	else:
		tmp = math.exp((t_1 - (t_0 * (t_0 - m))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M - Float64(n * 0.5))
	t_1 = Float64(abs(Float64(m - n)) - l)
	tmp = 0.0
	if (m <= -49.0)
		tmp = exp(Float64(t_1 - Float64(0.25 * (Float64(m + n) ^ 2.0))));
	else
		tmp = exp(Float64(t_1 - Float64(t_0 * Float64(t_0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = M - (n * 0.5);
	t_1 = abs((m - n)) - l;
	tmp = 0.0;
	if (m <= -49.0)
		tmp = exp((t_1 - (0.25 * ((m + n) ^ 2.0))));
	else
		tmp = exp((t_1 - (t_0 * (t_0 - m))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, If[LessEqual[m, -49.0], N[Exp[N[(t$95$1 - N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$1 - N[(t$95$0 * N[(t$95$0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if m < -49

   1. Initial program 81.3%

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

   3. Taylor expanded in K around 0 83.3%

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

      1. cos-neg 83.3%

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

      2. *-commutative 83.3%

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

      3. associate-*l* 83.3%

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

      4. metadata-eval 83.3%

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

      5. distribute-rgt-neg-in 83.3%

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

      6. *-commutative 83.3%

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

      7. associate-*r* 83.3%

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

      8. distribute-lft-neg-in 83.3%

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

      9. sin-neg 83.3%

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

      10. distribute-rgt-neg-out 83.3%

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

      11. remove-double-neg 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in M around 0 100.0%

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

      1. associate--r+ 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -49 < m

   1. Initial program 77.2%

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

   3. Taylor expanded in M around 0 77.2%

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

      1. *-commutative 77.2%

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

      2. *-commutative 77.2%

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

      3. associate-*l* 77.2%

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

   5. Simplified 77.2%

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

   6. Taylor expanded in K around 0 94.9%

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

   7. Taylor expanded in m around 0 78.8%

      \[\leadsto 1 \cdot e^{\left(-\color{blue}{\left(m \cdot \left(0.5 \cdot n - M\right) + {\left(0.5 \cdot n - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   8. Step-by-step derivation

      1. +-commutative 78.8%

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

      2. unpow2 78.8%

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

      3. distribute-rgt-out 83.7%

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

      4. *-commutative 83.7%

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

      5. *-commutative 83.7%

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

   9. Simplified 83.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -49:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - \left(M - n \cdot 0.5\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (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 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 = 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.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.exp(((math.fabs((m - n)) - l) - math.pow((((m + n) / 2.0) - M), 2.0)))

Julia

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in M around 0 78.0%

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

   1. *-commutative 78.0%

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

   2. *-commutative 78.0%

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

   3. associate-*l* 78.0%

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

5. Simplified 78.0%

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

6. Taylor expanded in K around 0 95.9%

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

7. Final simplification 95.9%

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

Alternative 4: 82.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M - n \cdot 0.5\\ \mathbf{if}\;m \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\cos M \cdot e^{m - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - t\_0 \cdot \left(t\_0 - m\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* n 0.5))))
   (if (<= m -5e+51)
     (* (cos M) (exp (- m (+ n l))))
     (exp (- (- (fabs (- m n)) l) (* t_0 (- t_0 m)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (n * 0.5);
	double tmp;
	if (m <= -5e+51) {
		tmp = cos(M) * exp((m - (n + l)));
	} else {
		tmp = exp(((fabs((m - n)) - l) - (t_0 * (t_0 - m))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_1 - (n * 0.5d0)
    if (m <= (-5d+51)) then
        tmp = cos(m_1) * exp((m - (n + l)))
    else
        tmp = exp(((abs((m - n)) - l) - (t_0 * (t_0 - m))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (n * 0.5);
	double tmp;
	if (m <= -5e+51) {
		tmp = Math.cos(M) * Math.exp((m - (n + l)));
	} else {
		tmp = Math.exp(((Math.abs((m - n)) - l) - (t_0 * (t_0 - m))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M - (n * 0.5)
	tmp = 0
	if m <= -5e+51:
		tmp = math.cos(M) * math.exp((m - (n + l)))
	else:
		tmp = math.exp(((math.fabs((m - n)) - l) - (t_0 * (t_0 - m))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M - Float64(n * 0.5))
	tmp = 0.0
	if (m <= -5e+51)
		tmp = Float64(cos(M) * exp(Float64(m - Float64(n + l))));
	else
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64(t_0 * Float64(t_0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = M - (n * 0.5);
	tmp = 0.0;
	if (m <= -5e+51)
		tmp = cos(M) * exp((m - (n + l)));
	else
		tmp = exp(((abs((m - n)) - l) - (t_0 * (t_0 - m))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -5e+51], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(t$95$0 * N[(t$95$0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -5e51

   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. Step-by-step derivation

      1. *-un-lft-identity 78.6%

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

      2. *-commutative 78.6%

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

   4. Applied egg-rr 8.3%

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

   5. Taylor expanded in l around inf 71.5%

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

      1. neg-mul-1 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in K around 0 88.2%

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

      1. cos-neg 88.2%

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

   10. Simplified 88.2%

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

   if -5e51 < m

   1. Initial program 77.8%

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

   3. Taylor expanded in M around 0 77.8%

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

      1. *-commutative 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-*l* 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in K around 0 95.0%

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

   7. Taylor expanded in m around 0 79.4%

      \[\leadsto 1 \cdot e^{\left(-\color{blue}{\left(m \cdot \left(0.5 \cdot n - M\right) + {\left(0.5 \cdot n - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   8. Step-by-step derivation

      1. +-commutative 79.4%

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

      2. unpow2 79.4%

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

      3. distribute-rgt-out 84.1%

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

      4. *-commutative 84.1%

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

      5. *-commutative 84.1%

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

   9. Simplified 84.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\cos M \cdot e^{m - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - \left(M - n \cdot 0.5\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.95:\\ \;\;\;\;e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{m - \left(n + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.95)
   (exp (+ (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) (- (fabs (- m n)) l)))
   (* (cos M) (exp (- m (+ n l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.95) {
		tmp = exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (fabs((m - n)) - l)));
	} else {
		tmp = cos(M) * exp((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 (n <= 1.95d0) then
        tmp = exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) + (abs((m - n)) - l)))
    else
        tmp = cos(m_1) * exp((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 (n <= 1.95) {
		tmp = Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (Math.abs((m - n)) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((m - (n + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.95:
		tmp = math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (math.fabs((m - n)) - l)))
	else:
		tmp = math.cos(M) * math.exp((m - (n + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.95)
		tmp = exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) + Float64(abs(Float64(m - n)) - l)));
	else
		tmp = Float64(cos(M) * exp(Float64(m - Float64(n + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1.95)
		tmp = exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (abs((m - n)) - l)));
	else
		tmp = cos(M) * exp((m - (n + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.95], N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.94999999999999996

   1. Initial program 82.4%

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

   3. Taylor expanded in M around 0 82.4%

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

      1. *-commutative 82.4%

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

      2. *-commutative 82.4%

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

      3. associate-*l* 82.4%

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

   5. Simplified 82.4%

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

   6. Taylor expanded in K around 0 95.5%

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

   7. Taylor expanded in n around 0 81.4%

      \[\leadsto 1 \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   8. Step-by-step derivation

      1. +-commutative 81.4%

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

      2. unpow2 81.4%

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

      3. distribute-rgt-out 83.6%

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

      4. *-commutative 83.6%

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

      5. *-commutative 83.6%

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

   9. Simplified 83.6%

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

   if 1.94999999999999996 < n

   1. Initial program 65.2%

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

      1. *-un-lft-identity 65.2%

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

      2. *-commutative 65.2%

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

   4. Applied egg-rr 5.5%

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

   5. Taylor expanded in l around inf 54.7%

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

      1. neg-mul-1 54.7%

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

   7. Simplified 54.7%

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

   8. Taylor expanded in K around 0 82.1%

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

      1. cos-neg 82.1%

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

   10. Simplified 82.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.95:\\ \;\;\;\;e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{m - \left(n + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. *-un-lft-identity 78.0%

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

   2. *-commutative 78.0%

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

4. Applied egg-rr 19.2%

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

5. Taylor expanded in l around inf 44.3%

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

   1. neg-mul-1 44.3%

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

7. Simplified 44.3%

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

8. Taylor expanded in K around 0 53.4%

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

   1. cos-neg 53.4%

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

10. Simplified 53.4%

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

11. Final simplification 53.4%

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

Alternative 7: 47.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. *-un-lft-identity 78.0%

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

   2. *-commutative 78.0%

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

4. Applied egg-rr 19.2%

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

5. Taylor expanded in l around inf 44.3%

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

   1. neg-mul-1 44.3%

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

7. Simplified 44.3%

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

8. Taylor expanded in m around 0 46.3%

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

   1. *-commutative 46.3%

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

   2. associate-*l* 46.3%

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

10. Simplified 46.3%

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

11. Taylor expanded in n around 0 45.2%

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

    1. cos-neg 45.2%

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

13. Simplified 45.2%

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

Reproduce

herbie shell --seed 2024101 
(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)))))))