Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.5% → 96.7%

Time: 35.0s

Alternatives: 5

Speedup: 1.0×

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: 75.5% 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: 96.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (fabs (- m n)) (+ l (pow (- (* (+ m n) 0.5) 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) * 0.5) - 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) * 0.5d0) - 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) * 0.5) - 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) * 0.5) - M), 2.0))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Taylor expanded in K around 0 97.6%

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

   1. cos-neg 97.6%

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

   2. sub-neg 97.6%

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

   3. sub-neg 97.6%

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

   4. +-commutative 97.6%

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

   5. neg-mul-1 97.6%

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

   6. sub-neg 97.6%

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

5. Simplified 97.6%

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

6. Final simplification 97.6%

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

Alternative 2: 72.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -3.1 \cdot 10^{+75} \lor \neg \left(M \leq 6.6 \cdot 10^{+72}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left(n + \left(\ell - m\right)\right) - {M}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -3.1e+75) (not (<= M 6.6e+72)))
   (* (cos M) (exp (- (pow M 2.0))))
   (/ (cos M) (exp (- (+ n (- l m)) (pow M 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -3.1e+75) || !(M <= 6.6e+72)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) / exp(((n + (l - m)) - pow(M, 2.0)));
	}
	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) :: tmp
    if ((m_1 <= (-3.1d+75)) .or. (.not. (m_1 <= 6.6d+72))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) / exp(((n + (l - m)) - (m_1 ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -3.1e+75) || !(M <= 6.6e+72)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) / Math.exp(((n + (l - m)) - Math.pow(M, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -3.1e+75) or not (M <= 6.6e+72):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) / math.exp(((n + (l - m)) - math.pow(M, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -3.1e+75) || !(M <= 6.6e+72))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) / exp(Float64(Float64(n + Float64(l - m)) - (M ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -3.1e+75) || ~((M <= 6.6e+72)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) / exp(((n + (l - m)) - (M ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -3.1e+75], N[Not[LessEqual[M, 6.6e+72]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(n + N[(l - m), $MachinePrecision]), $MachinePrecision] - N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -3.1000000000000001e75 or 6.6e72 < M

   1. Initial program 79.8%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. neg-mul-1 100.0%

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

      6. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around inf 95.5%

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

   7. Taylor expanded in M around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   if -3.1000000000000001e75 < M < 6.6e72

   1. Initial program 67.4%

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

      1. exp-diff 23.8%

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

      2. associate-*r/ 23.8%

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

   4. Applied egg-rr 23.4%

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

      1. associate-/l* 23.4%

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

      2. div-exp 27.2%

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

      3. associate--r- 27.2%

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

      4. +-commutative 27.2%

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

   6. Simplified 27.2%

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

   7. Taylor expanded in K around 0 30.8%

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

      1. cos-neg 30.8%

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

   9. Simplified 30.8%

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

   10. Taylor expanded in M around inf 68.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3.1 \cdot 10^{+75} \lor \neg \left(M \leq 6.6 \cdot 10^{+72}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left(n + \left(\ell - m\right)\right) - {M}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.1 \cdot 10^{+67} \lor \neg \left(M \leq 820\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1.1e+67) (not (<= M 820.0)))
   (* (cos M) (exp (- (pow M 2.0))))
   (/ (cos M) (exp (- l m)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.1e+67) || !(M <= 820.0)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) / exp((l - m));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-1.1d+67)) .or. (.not. (m_1 <= 820.0d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) / exp((l - m))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.1e+67) || !(M <= 820.0)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) / Math.exp((l - m));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1.1e+67) or not (M <= 820.0):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) / math.exp((l - m))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1.1e+67) || !(M <= 820.0))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) / exp(Float64(l - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1.1e+67) || ~((M <= 820.0)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) / exp((l - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1.1e+67], N[Not[LessEqual[M, 820.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(l - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1.1e67 or 820 < M

   1. Initial program 78.8%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. neg-mul-1 100.0%

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

      6. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around inf 90.8%

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

   7. Taylor expanded in M around inf 98.3%

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

      1. mul-1-neg 98.3%

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

   9. Simplified 98.3%

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

   if -1.1e67 < M < 820

   1. Initial program 67.4%

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

      1. exp-diff 24.7%

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

      2. associate-*r/ 24.7%

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

   4. Applied egg-rr 24.2%

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

      1. associate-/l* 24.2%

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

      2. div-exp 28.2%

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

      3. associate--r- 28.2%

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

      4. +-commutative 28.2%

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

   6. Simplified 28.2%

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

   7. Taylor expanded in n around inf 38.2%

      \[\leadsto \frac{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right)}{e^{\left(n + \left(\ell - m\right)\right) - \color{blue}{0.25 \cdot {n}^{2}}}} \]

   8. Taylor expanded in n around 0 52.8%

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

   9. Taylor expanded in K around 0 62.8%

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

       1. cos-neg 62.8%

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

   11. Simplified 62.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.1 \cdot 10^{+67} \lor \neg \left(M \leq 820\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell - m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. exp-diff 24.2%

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

   2. associate-*r/ 24.2%

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

4. Applied egg-rr 13.8%

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

   1. associate-/l* 13.8%

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

   2. div-exp 16.5%

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

   3. associate--r- 16.5%

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

   4. +-commutative 16.5%

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

6. Simplified 16.5%

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

7. Taylor expanded in n around inf 30.9%

   \[\leadsto \frac{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right)}{e^{\left(n + \left(\ell - m\right)\right) - \color{blue}{0.25 \cdot {n}^{2}}}} \]

8. Taylor expanded in n around 0 42.0%

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

9. Taylor expanded in K around 0 51.0%

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

    1. cos-neg 51.0%

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

11. Simplified 51.0%

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

12. Final simplification 51.0%

    \[\leadsto \frac{\cos M}{e^{\ell - m}} \]
13. Add Preprocessing

Alternative 5: 35.7% 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 72.7%

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

   1. exp-diff 24.2%

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

   2. associate-*r/ 24.2%

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

4. Applied egg-rr 13.8%

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

   1. associate-/l* 13.8%

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

   2. div-exp 16.5%

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

   3. associate--r- 16.5%

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

   4. +-commutative 16.5%

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

6. Simplified 16.5%

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

7. Taylor expanded in n around inf 30.9%

   \[\leadsto \frac{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right)}{e^{\left(n + \left(\ell - m\right)\right) - \color{blue}{0.25 \cdot {n}^{2}}}} \]

8. Taylor expanded in n around 0 42.0%

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

9. Taylor expanded in m around 0 33.3%

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

    1. cos-neg 33.3%

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

11. Simplified 33.3%

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

12. Final simplification 33.3%

    \[\leadsto \frac{\cos M}{e^{\ell}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024035 
(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)))))))