Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.7% → 96.8%

Time: 35.8s

Alternatives: 9

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 76.7% 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: 96.8% 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 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. *-commutative 72.5%

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

   2. associate-*r/ 72.5%

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

   3. associate--r- 72.5%

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

   4. +-commutative 72.5%

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

   5. associate-+r- 72.5%

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

   6. unsub-neg 72.5%

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

   7. associate--r+ 72.5%

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

   8. +-commutative 72.5%

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

   9. associate--r+ 72.5%

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

3. Simplified 72.5%

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

4. Taylor expanded in K around 0 96.8%

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

   1. cos-neg 96.8%

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

6. Simplified 96.8%

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

7. Final simplification 96.8%

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

Alternative 2: 63.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-29}:\\ \;\;\;\;\cos \left(\left(K \cdot 0.5\right) \cdot \left(m + n\right) - M\right) \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- (- (fabs (- m n)) l) (* M M))))))
   (if (<= l -3.8e+159)
     t_0
     (if (<= l -1.5e-29)
       (* (cos (- (* (* K 0.5) (+ m n)) M)) (exp l))
       (if (<= l 5.3e-17) t_0 (exp (- l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp(((fabs((m - n)) - l) - (M * M)));
	double tmp;
	if (l <= -3.8e+159) {
		tmp = t_0;
	} else if (l <= -1.5e-29) {
		tmp = cos((((K * 0.5) * (m + n)) - M)) * exp(l);
	} else if (l <= 5.3e-17) {
		tmp = t_0;
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(m_1) * exp(((abs((m - n)) - l) - (m_1 * m_1)))
    if (l <= (-3.8d+159)) then
        tmp = t_0
    else if (l <= (-1.5d-29)) then
        tmp = cos((((k * 0.5d0) * (m + n)) - m_1)) * exp(l)
    else if (l <= 5.3d-17) then
        tmp = t_0
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(M) * Math.exp(((Math.abs((m - n)) - l) - (M * M)));
	double tmp;
	if (l <= -3.8e+159) {
		tmp = t_0;
	} else if (l <= -1.5e-29) {
		tmp = Math.cos((((K * 0.5) * (m + n)) - M)) * Math.exp(l);
	} else if (l <= 5.3e-17) {
		tmp = t_0;
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp(((math.fabs((m - n)) - l) - (M * M)))
	tmp = 0
	if l <= -3.8e+159:
		tmp = t_0
	elif l <= -1.5e-29:
		tmp = math.cos((((K * 0.5) * (m + n)) - M)) * math.exp(l)
	elif l <= 5.3e-17:
		tmp = t_0
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64(M * M))))
	tmp = 0.0
	if (l <= -3.8e+159)
		tmp = t_0;
	elseif (l <= -1.5e-29)
		tmp = Float64(cos(Float64(Float64(Float64(K * 0.5) * Float64(m + n)) - M)) * exp(l));
	elseif (l <= 5.3e-17)
		tmp = t_0;
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp(((abs((m - n)) - l) - (M * M)));
	tmp = 0.0;
	if (l <= -3.8e+159)
		tmp = t_0;
	elseif (l <= -1.5e-29)
		tmp = cos((((K * 0.5) * (m + n)) - M)) * exp(l);
	elseif (l <= 5.3e-17)
		tmp = t_0;
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.8e+159], t$95$0, If[LessEqual[l, -1.5e-29], N[(N[Cos[N[(N[(N[(K * 0.5), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.3e-17], t$95$0, N[Exp[(-l)], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.79999999999999965e159 or -1.5000000000000001e-29 < l < 5.2999999999999998e-17

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

      2. associate-*r/ 66.2%

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

      3. associate--r- 66.2%

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

      4. +-commutative 66.2%

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

      5. associate-+r- 66.2%

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

      6. unsub-neg 66.2%

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

      7. associate--r+ 66.2%

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

      8. +-commutative 66.2%

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

      9. associate--r+ 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in K around 0 94.5%

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

      1. cos-neg 94.5%

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

   6. Simplified 94.5%

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

   7. Taylor expanded in M around inf 52.9%

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

      1. unpow2 52.9%

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

   9. Simplified 52.9%

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

   if -3.79999999999999965e159 < l < -1.5000000000000001e-29

   1. Initial program 81.6%

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

      1. sub-neg 81.6%

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

      2. associate--r+ 81.6%

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

      3. exp-diff 13.2%

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

      4. associate-*r/ 13.2%

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

      5. associate-/l* 13.2%

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

      6. associate-*r/ 13.2%

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

      7. exp-diff 5.3%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in l around inf 9.2%

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

      1. div-inv 9.2%

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

      2. div-inv 9.2%

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

      3. metadata-eval 9.2%

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

      4. *-commutative 9.2%

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

      5. +-commutative 9.2%

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

      6. associate-*r* 9.2%

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

      7. +-commutative 9.2%

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

      8. rec-exp 9.2%

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

   6. Applied egg-rr 9.2%

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

      1. expm1-log1p-u 6.1%

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

      2. expm1-udef 6.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(\left(K \cdot 0.5\right) \cdot \left(m + n\right) - M\right) \cdot e^{-\ell}\right)} - 1} \]

      3. associate-*l* 6.1%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(\color{blue}{K \cdot \left(0.5 \cdot \left(m + n\right)\right)} - M\right) \cdot e^{-\ell}\right)} - 1 \]

      4. +-commutative 6.1%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(0.5 \cdot \color{blue}{\left(n + m\right)}\right) - M\right) \cdot e^{-\ell}\right)} - 1 \]

      5. *-commutative 6.1%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \color{blue}{\left(\left(n + m\right) \cdot 0.5\right)} - M\right) \cdot e^{-\ell}\right)} - 1 \]

      6. add-sqr-sqrt 6.1%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

      7. sqrt-unprod 6.1%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

      8. sqr-neg 6.1%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

      9. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

      10. add-sqr-sqrt 63.7%

          \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   8. Applied egg-rr 63.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\ell}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 63.7%

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

      2. expm1-log1p 63.7%

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

      3. associate-*r* 63.7%

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

      4. *-commutative 63.7%

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

      5. associate-*r* 63.7%

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

      6. *-commutative 63.7%

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

   10. Simplified 63.7%

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

   if 5.2999999999999998e-17 < l

   1. Initial program 81.2%

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

      1. sub-neg 81.2%

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

      2. associate--r+ 81.2%

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

      3. exp-diff 27.5%

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

      4. associate-*r/ 27.5%

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

      5. associate-/l* 27.5%

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

      6. associate-*r/ 27.5%

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

      7. exp-diff 27.5%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in l around inf 79.7%

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

   5. Taylor expanded in n around inf 87.0%

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

      1. *-commutative 87.0%

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

      2. associate-*r* 87.0%

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

      3. *-commutative 87.0%

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

      4. associate-*l* 87.0%

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

   7. Simplified 87.0%

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

   8. Taylor expanded in K around 0 97.2%

      \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
   9. Step-by-step derivation

      1. rec-exp 97.2%

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

   10. Simplified 97.2%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+159}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-29}:\\ \;\;\;\;\cos \left(\left(K \cdot 0.5\right) \cdot \left(m + n\right) - M\right) \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{-17}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 3: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right| - \ell\\ \mathbf{if}\;M \leq -2.1 \cdot 10^{+125} \lor \neg \left(M \leq 3.1 \cdot 10^{+61}\right):\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- m n)) l)))
   (if (or (<= M -2.1e+125) (not (<= M 3.1e+61)))
     (* (cos M) (exp (- t_0 (* M M))))
     (* (cos M) (exp (- t_0 (* 0.25 (* n n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n)) - l;
	double tmp;
	if ((M <= -2.1e+125) || !(M <= 3.1e+61)) {
		tmp = cos(M) * exp((t_0 - (M * M)));
	} else {
		tmp = cos(M) * exp((t_0 - (0.25 * (n * n))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((m - n)) - l
    if ((m_1 <= (-2.1d+125)) .or. (.not. (m_1 <= 3.1d+61))) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * m_1)))
    else
        tmp = cos(m_1) * exp((t_0 - (0.25d0 * (n * n))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((m - n)) - l;
	double tmp;
	if ((M <= -2.1e+125) || !(M <= 3.1e+61)) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * M)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - (0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n)) - l
	tmp = 0
	if (M <= -2.1e+125) or not (M <= 3.1e+61):
		tmp = math.cos(M) * math.exp((t_0 - (M * M)))
	else:
		tmp = math.cos(M) * math.exp((t_0 - (0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(m - n)) - l)
	tmp = 0.0
	if ((M <= -2.1e+125) || !(M <= 3.1e+61))
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n)) - l;
	tmp = 0.0;
	if ((M <= -2.1e+125) || ~((M <= 3.1e+61)))
		tmp = cos(M) * exp((t_0 - (M * M)));
	else
		tmp = cos(M) * exp((t_0 - (0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, If[Or[LessEqual[M, -2.1e+125], N[Not[LessEqual[M, 3.1e+61]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -2.1000000000000001e125 or 3.0999999999999999e61 < M

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

      2. associate-*r/ 73.9%

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

      3. associate--r- 73.9%

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

      4. +-commutative 73.9%

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

      5. associate-+r- 73.9%

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

      6. unsub-neg 73.9%

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

      7. associate--r+ 73.9%

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

      8. +-commutative 73.9%

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

      9. associate--r+ 73.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around inf 95.7%

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

      1. unpow2 95.7%

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

   9. Simplified 95.7%

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

   if -2.1000000000000001e125 < M < 3.0999999999999999e61

   1. Initial program 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*r/ 71.7%

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

      3. associate--r- 71.7%

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

      4. +-commutative 71.7%

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

      5. associate-+r- 71.7%

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

      6. unsub-neg 71.7%

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

      7. associate--r+ 71.7%

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

      8. +-commutative 71.7%

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

      9. associate--r+ 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in K around 0 95.0%

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

      1. cos-neg 95.0%

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

   6. Simplified 95.0%

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

   7. Taylor expanded in n around inf 72.2%

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

      1. *-commutative 72.2%

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

      2. unpow2 72.2%

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

   9. Simplified 72.2%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.1 \cdot 10^{+125} \lor \neg \left(M \leq 3.1 \cdot 10^{+61}\right):\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 4: 71.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;n \leq 8 \cdot 10^{+60}:\\ \;\;\;\;\frac{\cos M}{e^{\ell + \left(\left(m \cdot m\right) \cdot 0.25 - t_0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(t_0 - \ell\right) - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (<= n 8e+60)
     (/ (cos M) (exp (+ l (- (* (* m m) 0.25) t_0))))
     (* (cos M) (exp (- (- t_0 l) (* 0.25 (* n n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if (n <= 8e+60) {
		tmp = cos(M) / exp((l + (((m * m) * 0.25) - t_0)));
	} else {
		tmp = cos(M) * exp(((t_0 - l) - (0.25 * (n * n))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((m - n))
    if (n <= 8d+60) then
        tmp = cos(m_1) / exp((l + (((m * m) * 0.25d0) - t_0)))
    else
        tmp = cos(m_1) * exp(((t_0 - l) - (0.25d0 * (n * n))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((m - n));
	double tmp;
	if (n <= 8e+60) {
		tmp = Math.cos(M) / Math.exp((l + (((m * m) * 0.25) - t_0)));
	} else {
		tmp = Math.cos(M) * Math.exp(((t_0 - l) - (0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	tmp = 0
	if n <= 8e+60:
		tmp = math.cos(M) / math.exp((l + (((m * m) * 0.25) - t_0)))
	else:
		tmp = math.cos(M) * math.exp(((t_0 - l) - (0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if (n <= 8e+60)
		tmp = Float64(cos(M) / exp(Float64(l + Float64(Float64(Float64(m * m) * 0.25) - t_0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(t_0 - l) - Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	tmp = 0.0;
	if (n <= 8e+60)
		tmp = cos(M) / exp((l + (((m * m) * 0.25) - t_0)));
	else
		tmp = cos(M) * exp(((t_0 - l) - (0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 8e+60], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(l + N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] - N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 7.9999999999999996e60

   1. Initial program 77.3%

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

      1. sub-neg 77.3%

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

      2. associate--r+ 77.3%

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

      3. exp-diff 27.8%

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

      4. associate-*r/ 27.8%

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

      5. associate-/l* 27.8%

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

      6. associate-*r/ 27.8%

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

      7. exp-diff 21.7%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around inf 57.3%

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

      1. *-commutative 57.3%

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

      2. unpow2 57.3%

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

   6. Simplified 57.3%

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

   7. Taylor expanded in K around 0 72.3%

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

      1. cos-neg 72.3%

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

      2. exp-diff 21.2%

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

      3. fabs-sub 21.2%

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

      4. sub-neg 21.2%

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

      5. mul-1-neg 21.2%

         \[\leadsto \frac{\cos M}{\frac{e^{\ell + 0.25 \cdot {m}^{2}}}{e^{\left|m + \color{blue}{-1 \cdot n}\right|}}} \]

      6. +-commutative 21.2%

         \[\leadsto \frac{\cos M}{\frac{e^{\ell + 0.25 \cdot {m}^{2}}}{e^{\left|\color{blue}{-1 \cdot n + m}\right|}}} \]

      7. fabs-neg 21.2%

         \[\leadsto \frac{\cos M}{\frac{e^{\ell + 0.25 \cdot {m}^{2}}}{e^{\color{blue}{\left|-\left(-1 \cdot n + m\right)\right|}}}} \]

      8. exp-diff 72.3%

         \[\leadsto \frac{\cos M}{\color{blue}{e^{\left(\ell + 0.25 \cdot {m}^{2}\right) - \left|-\left(-1 \cdot n + m\right)\right|}}} \]

      9. associate--l+ 72.3%

         \[\leadsto \frac{\cos M}{e^{\color{blue}{\ell + \left(0.25 \cdot {m}^{2} - \left|-\left(-1 \cdot n + m\right)\right|\right)}}} \]

      10. *-commutative 72.3%

          \[\leadsto \frac{\cos M}{e^{\ell + \left(\color{blue}{{m}^{2} \cdot 0.25} - \left|-\left(-1 \cdot n + m\right)\right|\right)}} \]

      11. unpow2 72.3%

          \[\leadsto \frac{\cos M}{e^{\ell + \left(\color{blue}{\left(m \cdot m\right)} \cdot 0.25 - \left|-\left(-1 \cdot n + m\right)\right|\right)}} \]

      12. fabs-neg 72.3%

          \[\leadsto \frac{\cos M}{e^{\ell + \left(\left(m \cdot m\right) \cdot 0.25 - \color{blue}{\left|-1 \cdot n + m\right|}\right)}} \]

      13. +-commutative 72.3%

          \[\leadsto \frac{\cos M}{e^{\ell + \left(\left(m \cdot m\right) \cdot 0.25 - \left|\color{blue}{m + -1 \cdot n}\right|\right)}} \]

      14. mul-1-neg 72.3%

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

      15. sub-neg 72.3%

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

      16. fabs-sub 72.3%

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

   9. Simplified 72.3%

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

   if 7.9999999999999996e60 < n

   1. Initial program 56.7%

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

      1. *-commutative 56.7%

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

      2. associate-*r/ 56.7%

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

      3. associate--r- 56.7%

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

      4. +-commutative 56.7%

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

      5. associate-+r- 56.7%

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

      6. unsub-neg 56.7%

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

      7. associate--r+ 56.7%

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

      8. +-commutative 56.7%

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

      9. associate--r+ 56.7%

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

   3. Simplified 56.7%

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

   4. Taylor expanded in K around 0 98.3%

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

      1. cos-neg 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in n around inf 95.1%

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

      1. *-commutative 95.1%

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

      2. unpow2 95.1%

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

   9. Simplified 95.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 8 \cdot 10^{+60}:\\ \;\;\;\;\frac{\cos M}{e^{\ell + \left(\left(m \cdot m\right) \cdot 0.25 - \left|m - n\right|\right)}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 5: 44.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -13:\\ \;\;\;\;\cos \left(\left(K \cdot 0.5\right) \cdot \left(m + n\right) - M\right) \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -13.0)
   (* (cos (- (* (* K 0.5) (+ m n)) M)) (exp l))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -13.0) {
		tmp = cos((((K * 0.5) * (m + n)) - M)) * exp(l);
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-13.0d0)) then
        tmp = cos((((k * 0.5d0) * (m + n)) - m_1)) * exp(l)
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -13.0) {
		tmp = Math.cos((((K * 0.5) * (m + n)) - M)) * Math.exp(l);
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -13.0:
		tmp = math.cos((((K * 0.5) * (m + n)) - M)) * math.exp(l)
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -13.0)
		tmp = Float64(cos(Float64(Float64(Float64(K * 0.5) * Float64(m + n)) - M)) * exp(l));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -13.0)
		tmp = cos((((K * 0.5) * (m + n)) - M)) * exp(l);
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -13.0], N[(N[Cos[N[(N[(N[(K * 0.5), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -13

   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. sub-neg 73.3%

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

      2. associate--r+ 73.3%

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

      3. exp-diff 24.0%

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

      4. associate-*r/ 24.0%

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

      5. associate-/l* 24.0%

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

      6. associate-*r/ 24.0%

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

      7. exp-diff 6.7%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in l around inf 19.5%

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

      1. div-inv 19.5%

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

      2. div-inv 19.5%

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

      3. metadata-eval 19.5%

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

      4. *-commutative 19.5%

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

      5. +-commutative 19.5%

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

      6. associate-*r* 19.5%

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

      7. +-commutative 19.5%

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

      8. rec-exp 19.5%

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

   6. Applied egg-rr 19.5%

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

      1. expm1-log1p-u 13.7%

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

      2. expm1-udef 13.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(\left(K \cdot 0.5\right) \cdot \left(m + n\right) - M\right) \cdot e^{-\ell}\right)} - 1} \]

      3. associate-*l* 13.7%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(\color{blue}{K \cdot \left(0.5 \cdot \left(m + n\right)\right)} - M\right) \cdot e^{-\ell}\right)} - 1 \]

      4. +-commutative 13.7%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(0.5 \cdot \color{blue}{\left(n + m\right)}\right) - M\right) \cdot e^{-\ell}\right)} - 1 \]

      5. *-commutative 13.7%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \color{blue}{\left(\left(n + m\right) \cdot 0.5\right)} - M\right) \cdot e^{-\ell}\right)} - 1 \]

      6. add-sqr-sqrt 13.7%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

      7. sqrt-unprod 13.7%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

      8. sqr-neg 13.7%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

      9. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

      10. add-sqr-sqrt 53.7%

          \[\leadsto e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   8. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(K \cdot \left(\left(n + m\right) \cdot 0.5\right) - M\right) \cdot e^{\ell}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 53.7%

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

      2. expm1-log1p 53.7%

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

      3. associate-*r* 53.7%

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

      4. *-commutative 53.7%

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

      5. associate-*r* 53.7%

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

      6. *-commutative 53.7%

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

   10. Simplified 53.7%

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

   if -13 < l

   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. sub-neg 72.2%

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

      2. associate--r+ 72.2%

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

      3. exp-diff 20.8%

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

      4. associate-*r/ 20.8%

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

      5. associate-/l* 20.8%

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

      6. associate-*r/ 20.8%

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

      7. exp-diff 20.7%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in l around inf 35.6%

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

      1. div-inv 35.6%

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

      2. div-inv 35.6%

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

      3. metadata-eval 35.6%

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

      4. *-commutative 35.6%

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

      5. +-commutative 35.6%

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

      6. associate-*r* 35.6%

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

      7. +-commutative 35.6%

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

      8. rec-exp 35.6%

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

   6. Applied egg-rr 35.6%

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

   7. Taylor expanded in K around 0 43.0%

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

      1. *-commutative 43.0%

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

      2. cos-neg 43.0%

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

   9. Simplified 43.0%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -13:\\ \;\;\;\;\cos \left(\left(K \cdot 0.5\right) \cdot \left(m + n\right) - M\right) \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 6: 35.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. sub-neg 72.5%

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

   2. associate--r+ 72.5%

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

   3. exp-diff 21.7%

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

   4. associate-*r/ 21.7%

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

   5. associate-/l* 21.7%

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

   6. associate-*r/ 21.7%

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

   7. exp-diff 16.6%

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

3. Simplified 72.5%

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

4. Taylor expanded in l around inf 30.9%

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

   1. div-inv 30.9%

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

   2. div-inv 30.9%

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

   3. metadata-eval 30.9%

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

   4. *-commutative 30.9%

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

   5. +-commutative 30.9%

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

   6. associate-*r* 30.9%

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

   7. +-commutative 30.9%

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

   8. rec-exp 30.9%

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

6. Applied egg-rr 30.9%

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

7. Taylor expanded in K around 0 36.2%

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

   1. *-commutative 36.2%

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

   2. cos-neg 36.2%

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

9. Simplified 36.2%

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

10. Final simplification 36.2%

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

Alternative 7: 35.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. sub-neg 72.5%

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

   2. associate--r+ 72.5%

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

   3. exp-diff 21.7%

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

   4. associate-*r/ 21.7%

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

   5. associate-/l* 21.7%

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

   6. associate-*r/ 21.7%

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

   7. exp-diff 16.6%

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

3. Simplified 72.5%

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

4. Taylor expanded in l around inf 30.9%

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

5. Taylor expanded in K around 0 36.2%

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

   1. cos-neg 36.2%

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

7. Simplified 36.2%

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

8. Final simplification 36.2%

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

Alternative 8: 35.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. sub-neg 72.5%

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

   2. associate--r+ 72.5%

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

   3. exp-diff 21.7%

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

   4. associate-*r/ 21.7%

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

   5. associate-/l* 21.7%

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

   6. associate-*r/ 21.7%

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

   7. exp-diff 16.6%

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

3. Simplified 72.5%

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

4. Taylor expanded in l around inf 30.9%

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

5. Taylor expanded in n around inf 33.6%

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

   1. *-commutative 33.6%

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

   2. associate-*r* 33.6%

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

   3. *-commutative 33.6%

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

   4. associate-*l* 33.6%

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

7. Simplified 33.6%

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

8. Taylor expanded in K around 0 35.4%

   \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
9. Step-by-step derivation

   1. rec-exp 35.4%

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

10. Simplified 35.4%

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

11. Final simplification 35.4%

    \[\leadsto e^{-\ell} \]

Alternative 9: 7.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \cos M \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(K, m, n, M, l)
	return cos(M)
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[Cos[M], $MachinePrecision]

Derivation

1. Initial program 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. sub-neg 72.5%

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

   2. associate--r+ 72.5%

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

   3. exp-diff 21.7%

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

   4. associate-*r/ 21.7%

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

   5. associate-/l* 21.7%

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

   6. associate-*r/ 21.7%

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

   7. exp-diff 16.6%

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

3. Simplified 72.5%

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

4. Taylor expanded in l around inf 30.9%

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

5. Taylor expanded in l around 0 5.4%

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

   1. associate-*r* 5.4%

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

   2. *-commutative 5.4%

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

   3. +-commutative 5.4%

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

   4. fma-neg 5.4%

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

   5. +-commutative 5.4%

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

   6. *-lft-identity 5.4%

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

   7. metadata-eval 5.4%

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

   8. cancel-sign-sub-inv 5.4%

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

   9. fma-neg 5.4%

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

   10. sub-neg 5.4%

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

   11. mul-1-neg 5.4%

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

   12. remove-double-neg 5.4%

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

   13. +-commutative 5.4%

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

   14. associate-*l* 5.4%

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

   15. +-commutative 5.4%

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

7. Simplified 5.4%

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

8. Taylor expanded in K around 0 6.2%

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

   1. cos-neg 6.2%

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

10. Simplified 6.2%

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

11. Final simplification 6.2%

    \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023187 
(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)))))))