Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.4%

Time: 23.5s

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: 76.1% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos \left(\mathsf{expm1}\left(K \cdot \left(-0.125 \cdot \left(K \cdot {\left(m + n\right)}^{2}\right) + 0.5 \cdot \left(m + n\right)\right)\right) - M\right) \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos
   (- (expm1 (* K (+ (* -0.125 (* K (pow (+ m n) 2.0))) (* 0.5 (+ m n))))) M))
  (exp (- (fabs (- m n)) (+ (pow (- (/ (+ m n) 2.0) M) 2.0) l)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((expm1((K * ((-0.125 * (K * pow((m + n), 2.0))) + (0.5 * (m + n))))) - M)) * exp((fabs((m - n)) - (pow((((m + n) / 2.0) - M), 2.0) + l)));
}

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((Math.expm1((K * ((-0.125 * (K * Math.pow((m + n), 2.0))) + (0.5 * (m + n))))) - M)) * Math.exp((Math.abs((m - n)) - (Math.pow((((m + n) / 2.0) - M), 2.0) + l)));
}

Python

def code(K, m, n, M, l):
	return math.cos((math.expm1((K * ((-0.125 * (K * math.pow((m + n), 2.0))) + (0.5 * (m + n))))) - M)) * math.exp((math.fabs((m - n)) - (math.pow((((m + n) / 2.0) - M), 2.0) + l)))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(expm1(Float64(K * Float64(Float64(-0.125 * Float64(K * (Float64(m + n) ^ 2.0))) + Float64(0.5 * Float64(m + n))))) - M)) * exp(Float64(abs(Float64(m - n)) - Float64((Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0) + l))))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(Exp[N[(K * N[(N[(-0.125 * N[(K * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision] + l), $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. Step-by-step derivation

   1. associate-/l* 72.8%

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

   2. +-commutative 72.8%

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

   3. associate-/l* 72.5%

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

   4. associate-/l* 72.8%

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

   5. +-commutative 72.8%

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

   6. exp-diff 22.1%

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

   7. sub-neg 22.1%

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

   8. exp-sum 19.7%

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

   9. associate-/r* 19.7%

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

   10. exp-diff 25.2%

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

3. Simplified 72.8%

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

   1. clear-num 72.5%

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

   2. un-div-inv 72.5%

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

6. Applied egg-rr 72.5%

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

   1. expm1-log1p-u 53.6%

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

   2. div-inv 53.6%

      \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{K \cdot \frac{1}{\frac{2}{m + n}}}\right)\right) - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

   3. clear-num 53.6%

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

   4. div-inv 53.6%

      \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(K \cdot \color{blue}{\left(\left(m + n\right) \cdot \frac{1}{2}\right)}\right)\right) - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

   5. metadata-eval 53.6%

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

   6. +-commutative 53.6%

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

8. Applied egg-rr 53.6%

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

9. Taylor expanded in K around 0 95.3%

   \[\leadsto \cos \left(\mathsf{expm1}\left(\color{blue}{K \cdot \left(-0.125 \cdot \left(K \cdot {\left(m + n\right)}^{2}\right) + 0.5 \cdot \left(m + n\right)\right)}\right) - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

10. Final simplification 95.3%

    \[\leadsto \cos \left(\mathsf{expm1}\left(K \cdot \left(-0.125 \cdot \left(K \cdot {\left(m + n\right)}^{2}\right) + 0.5 \cdot \left(m + n\right)\right)\right) - M\right) \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)} \]
11. Add Preprocessing

Alternative 2: 87.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot 0.5 - M\\ t_1 := \left|m - n\right| - \ell\\ \mathbf{if}\;m \leq -1 \cdot 10^{-27}:\\ \;\;\;\;e^{t\_1 - 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t\_1 - t\_0 \cdot \left(m + t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* n 0.5) M)) (t_1 (- (fabs (- m n)) l)))
   (if (<= m -1e-27)
     (exp (- t_1 (* 0.25 (* (+ m n) (+ m n)))))
     (* (cos M) (exp (- t_1 (* t_0 (+ m t_0))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (n * 0.5) - M;
	double t_1 = fabs((m - n)) - l;
	double tmp;
	if (m <= -1e-27) {
		tmp = exp((t_1 - (0.25 * ((m + n) * (m + n)))));
	} else {
		tmp = cos(M) * exp((t_1 - (t_0 * (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) :: t_1
    real(8) :: tmp
    t_0 = (n * 0.5d0) - m_1
    t_1 = abs((m - n)) - l
    if (m <= (-1d-27)) then
        tmp = exp((t_1 - (0.25d0 * ((m + n) * (m + n)))))
    else
        tmp = cos(m_1) * exp((t_1 - (t_0 * (m + 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 = (n * 0.5) - M;
	double t_1 = Math.abs((m - n)) - l;
	double tmp;
	if (m <= -1e-27) {
		tmp = Math.exp((t_1 - (0.25 * ((m + n) * (m + n)))));
	} else {
		tmp = Math.cos(M) * Math.exp((t_1 - (t_0 * (m + t_0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (n * 0.5) - M
	t_1 = math.fabs((m - n)) - l
	tmp = 0
	if m <= -1e-27:
		tmp = math.exp((t_1 - (0.25 * ((m + n) * (m + n)))))
	else:
		tmp = math.cos(M) * math.exp((t_1 - (t_0 * (m + t_0))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(n * 0.5) - M)
	t_1 = Float64(abs(Float64(m - n)) - l)
	tmp = 0.0
	if (m <= -1e-27)
		tmp = exp(Float64(t_1 - Float64(0.25 * Float64(Float64(m + n) * Float64(m + n)))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_1 - Float64(t_0 * Float64(m + t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (n * 0.5) - M;
	t_1 = abs((m - n)) - l;
	tmp = 0.0;
	if (m <= -1e-27)
		tmp = exp((t_1 - (0.25 * ((m + n) * (m + n)))));
	else
		tmp = cos(M) * exp((t_1 - (t_0 * (m + t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, If[LessEqual[m, -1e-27], N[Exp[N[(t$95$1 - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$1 - N[(t$95$0 * N[(m + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if m < -1e-27

   1. Initial program 66.0%

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

      1. associate-/l* 67.4%

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

      2. +-commutative 67.4%

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

      3. associate-/l* 66.0%

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

      4. associate-/l* 67.4%

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

      5. +-commutative 67.4%

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

      6. exp-diff 4.5%

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

      7. sub-neg 4.5%

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

      8. exp-sum 3.1%

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

      9. associate-/r* 3.1%

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

      10. exp-diff 3.1%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in K around 0 97.4%

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

      1. cos-neg 97.4%

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

      2. sub-neg 97.4%

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

      3. distribute-neg-in 97.4%

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

      4. mul-1-neg 97.4%

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

      5. associate-+l+ 97.4%

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

      6. sub-neg 97.4%

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

      7. mul-1-neg 97.4%

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

      8. sub-neg 97.4%

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

      9. *-commutative 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in M around 0 97.4%

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

      1. associate--r+ 97.4%

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

      2. +-commutative 97.4%

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

   10. Simplified 97.4%

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

       1. unpow2 97.4%

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

   12. Applied egg-rr 97.4%

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

   if -1e-27 < m

   1. Initial program 74.9%

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

      1. associate-/l* 74.9%

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

      2. +-commutative 74.9%

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

      3. associate-/l* 74.9%

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

      4. associate-/l* 74.9%

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

      5. +-commutative 74.9%

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

      6. exp-diff 28.7%

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

      7. sub-neg 28.7%

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

      8. exp-sum 26.0%

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

      9. associate-/r* 26.0%

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

      10. exp-diff 33.5%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in K around 0 94.2%

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

      1. cos-neg 94.2%

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

      2. sub-neg 94.2%

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

      3. distribute-neg-in 94.2%

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

      4. mul-1-neg 94.2%

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

      5. associate-+l+ 94.2%

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

      6. sub-neg 94.2%

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

      7. mul-1-neg 94.2%

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

      8. sub-neg 94.2%

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

      9. *-commutative 94.2%

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

   7. Simplified 94.2%

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

      1. metadata-eval 94.2%

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

      2. div-inv 94.2%

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

      3. unpow2 94.2%

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

      4. div-inv 94.2%

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

      5. metadata-eval 94.2%

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

      6. +-commutative 94.2%

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

      7. div-inv 94.2%

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

      8. metadata-eval 94.2%

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

      9. +-commutative 94.2%

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

   9. Applied egg-rr 94.2%

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

   10. Taylor expanded in m around 0 75.7%

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

       1. +-commutative 75.7%

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

       2. unpow2 75.7%

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

       3. distribute-rgt-out 79.4%

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

       4. *-commutative 79.4%

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

       5. *-commutative 79.4%

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

   12. Simplified 79.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{-27}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot 0.5 - M\\ t_1 := \left|m - n\right|\\ \mathbf{if}\;M \leq 3500000000000:\\ \;\;\;\;e^{\left(t\_1 - \ell\right) - 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t\_1 - t\_0 \cdot \left(m + t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* n 0.5) M)) (t_1 (fabs (- m n))))
   (if (<= M 3500000000000.0)
     (exp (- (- t_1 l) (* 0.25 (* (+ m n) (+ m n)))))
     (* (cos M) (exp (- t_1 (* t_0 (+ m t_0))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (n * 0.5) - M;
	double t_1 = fabs((m - n));
	double tmp;
	if (M <= 3500000000000.0) {
		tmp = exp(((t_1 - l) - (0.25 * ((m + n) * (m + n)))));
	} else {
		tmp = cos(M) * exp((t_1 - (t_0 * (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) :: t_1
    real(8) :: tmp
    t_0 = (n * 0.5d0) - m_1
    t_1 = abs((m - n))
    if (m_1 <= 3500000000000.0d0) then
        tmp = exp(((t_1 - l) - (0.25d0 * ((m + n) * (m + n)))))
    else
        tmp = cos(m_1) * exp((t_1 - (t_0 * (m + 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 = (n * 0.5) - M;
	double t_1 = Math.abs((m - n));
	double tmp;
	if (M <= 3500000000000.0) {
		tmp = Math.exp(((t_1 - l) - (0.25 * ((m + n) * (m + n)))));
	} else {
		tmp = Math.cos(M) * Math.exp((t_1 - (t_0 * (m + t_0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (n * 0.5) - M
	t_1 = math.fabs((m - n))
	tmp = 0
	if M <= 3500000000000.0:
		tmp = math.exp(((t_1 - l) - (0.25 * ((m + n) * (m + n)))))
	else:
		tmp = math.cos(M) * math.exp((t_1 - (t_0 * (m + t_0))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(n * 0.5) - M)
	t_1 = abs(Float64(m - n))
	tmp = 0.0
	if (M <= 3500000000000.0)
		tmp = exp(Float64(Float64(t_1 - l) - Float64(0.25 * Float64(Float64(m + n) * Float64(m + n)))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_1 - Float64(t_0 * Float64(m + t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (n * 0.5) - M;
	t_1 = abs((m - n));
	tmp = 0.0;
	if (M <= 3500000000000.0)
		tmp = exp(((t_1 - l) - (0.25 * ((m + n) * (m + n)))));
	else
		tmp = cos(M) * exp((t_1 - (t_0 * (m + t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, 3500000000000.0], N[Exp[N[(N[(t$95$1 - l), $MachinePrecision] - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$1 - N[(t$95$0 * N[(m + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if M < 3.5e12

   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. Step-by-step derivation

      1. associate-/l* 72.2%

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

      2. +-commutative 72.2%

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

      3. associate-/l* 72.2%

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

      4. associate-/l* 72.2%

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

      5. +-commutative 72.2%

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

      6. exp-diff 21.7%

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

      7. sub-neg 21.7%

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

      8. exp-sum 18.6%

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

      9. associate-/r* 18.6%

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

      10. exp-diff 23.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in K around 0 93.6%

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

      1. cos-neg 93.6%

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

      2. sub-neg 93.6%

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

      3. distribute-neg-in 93.6%

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

      4. mul-1-neg 93.6%

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

      5. associate-+l+ 93.6%

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

      6. sub-neg 93.6%

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

      7. mul-1-neg 93.6%

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

      8. sub-neg 93.6%

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

      9. *-commutative 93.6%

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

   7. Simplified 93.6%

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

   8. Taylor expanded in M around 0 87.7%

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

      1. associate--r+ 87.7%

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

      2. +-commutative 87.7%

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

   10. Simplified 87.7%

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

       1. unpow2 87.7%

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

   12. Applied egg-rr 87.7%

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

   if 3.5e12 < M

   1. Initial program 73.3%

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

      1. associate-/l* 75.0%

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

      2. +-commutative 75.0%

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

      3. associate-/l* 73.3%

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

      4. associate-/l* 75.0%

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

      5. +-commutative 75.0%

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

      6. exp-diff 23.3%

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

      7. sub-neg 23.3%

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

      8. exp-sum 23.3%

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

      9. associate-/r* 23.3%

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

      10. exp-diff 31.7%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. mul-1-neg 100.0%

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

      5. associate-+l+ 100.0%

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

      6. sub-neg 100.0%

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

      7. mul-1-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around 0 100.0%

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

   9. Taylor expanded in m around 0 81.8%

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

       1. +-commutative 81.8%

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

       2. unpow2 81.8%

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

       3. *-commutative 81.8%

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

       4. distribute-lft-out 88.5%

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

       5. *-commutative 88.5%

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

       6. *-commutative 88.5%

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

   11. Simplified 88.5%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3500000000000:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - \left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l* 72.8%

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

   2. +-commutative 72.8%

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

   3. associate-/l* 72.5%

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

   4. associate-/l* 72.8%

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

   5. +-commutative 72.8%

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

   6. exp-diff 22.1%

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

   7. sub-neg 22.1%

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

   8. exp-sum 19.7%

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

   9. associate-/r* 19.7%

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

   10. exp-diff 25.2%

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

3. Simplified 72.8%

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

5. Taylor expanded in K around 0 95.1%

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

   1. cos-neg 95.1%

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

   2. sub-neg 95.1%

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

   3. distribute-neg-in 95.1%

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

   4. mul-1-neg 95.1%

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

   5. associate-+l+ 95.1%

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

   6. sub-neg 95.1%

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

   7. mul-1-neg 95.1%

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

   8. sub-neg 95.1%

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

   9. *-commutative 95.1%

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

7. Simplified 95.1%

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

   1. metadata-eval 95.1%

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

   2. div-inv 95.1%

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

   3. unpow2 95.1%

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

   4. div-inv 95.1%

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

   5. metadata-eval 95.1%

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

   6. +-commutative 95.1%

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

   7. div-inv 95.1%

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

   8. metadata-eval 95.1%

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

   9. +-commutative 95.1%

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

9. Applied egg-rr 95.1%

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

10. Final simplification 95.1%

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

Alternative 5: 86.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l* 72.8%

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

   2. +-commutative 72.8%

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

   3. associate-/l* 72.5%

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

   4. associate-/l* 72.8%

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

   5. +-commutative 72.8%

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

   6. exp-diff 22.1%

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

   7. sub-neg 22.1%

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

   8. exp-sum 19.7%

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

   9. associate-/r* 19.7%

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

   10. exp-diff 25.2%

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

3. Simplified 72.8%

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

5. Taylor expanded in K around 0 95.1%

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

   1. cos-neg 95.1%

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

   2. sub-neg 95.1%

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

   3. distribute-neg-in 95.1%

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

   4. mul-1-neg 95.1%

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

   5. associate-+l+ 95.1%

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

   6. sub-neg 95.1%

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

   7. mul-1-neg 95.1%

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

   8. sub-neg 95.1%

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

   9. *-commutative 95.1%

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

7. Simplified 95.1%

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

8. Taylor expanded in M around 0 86.3%

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

   1. associate--r+ 86.3%

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

   2. +-commutative 86.3%

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

10. Simplified 86.3%

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

    1. unpow2 86.3%

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

12. Applied egg-rr 86.3%

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

13. Final simplification 86.3%

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

Reproduce

herbie shell --seed 2024082 
(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)))))))