Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.6% → 96.6%

Time: 24.5s

Alternatives: 11

Speedup: 1.3×

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: 75.6% 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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp((fabs((m - n)) - (pow((((m + n) / 2.0) - M), 2.0) + 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((abs((m - n)) - (((((m + n) / 2.0d0) - m_1) ** 2.0d0) + l)))
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)) - (Math.pow((((m + n) / 2.0) - M), 2.0) + l)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. associate-/l* 78.8%

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

   2. associate--r- 78.8%

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

3. Simplified 78.8%

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

4. Taylor expanded in K around 0 96.0%

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

   1. cos-neg 96.0%

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

6. Simplified 96.0%

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

7. Final simplification 96.0%

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

Alternative 2: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) / exp((pow(((0.5 * (m + n)) - 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(m_1) / exp(((((0.5d0 * (m + n)) - 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(M) / Math.exp((Math.pow(((0.5 * (m + n)) - M), 2.0) + (l - Math.abs((m - n)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Simplified 79.2%

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

4. Taylor expanded in K around 0 96.0%

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

6. Simplified 96.0%

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

7. Final simplification 96.0%

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

Alternative 3: 92.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;M \leq -6.5 \cdot 10^{+124} \lor \neg \left(M \leq 1.15 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - t_0\right) + M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (or (<= M -6.5e+124) (not (<= M 1.15e+87)))
     (/ (cos M) (exp (+ (- l t_0) (* M M))))
     (exp (- t_0 (+ l (* 0.25 (pow (+ m n) 2.0))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if ((M <= -6.5e+124) || !(M <= 1.15e+87)) {
		tmp = cos(M) / exp(((l - t_0) + (M * M)));
	} else {
		tmp = exp((t_0 - (l + (0.25 * pow((m + n), 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) :: t_0
    real(8) :: tmp
    t_0 = abs((m - n))
    if ((m_1 <= (-6.5d+124)) .or. (.not. (m_1 <= 1.15d+87))) then
        tmp = cos(m_1) / exp(((l - t_0) + (m_1 * m_1)))
    else
        tmp = exp((t_0 - (l + (0.25d0 * ((m + n) ** 2.0d0)))))
    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 ((M <= -6.5e+124) || !(M <= 1.15e+87)) {
		tmp = Math.cos(M) / Math.exp(((l - t_0) + (M * M)));
	} else {
		tmp = Math.exp((t_0 - (l + (0.25 * Math.pow((m + n), 2.0)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	tmp = 0
	if (M <= -6.5e+124) or not (M <= 1.15e+87):
		tmp = math.cos(M) / math.exp(((l - t_0) + (M * M)))
	else:
		tmp = math.exp((t_0 - (l + (0.25 * math.pow((m + n), 2.0)))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if ((M <= -6.5e+124) || !(M <= 1.15e+87))
		tmp = Float64(cos(M) / exp(Float64(Float64(l - t_0) + Float64(M * M))));
	else
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * (Float64(m + n) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	tmp = 0.0;
	if ((M <= -6.5e+124) || ~((M <= 1.15e+87)))
		tmp = cos(M) / exp(((l - t_0) + (M * M)));
	else
		tmp = exp((t_0 - (l + (0.25 * ((m + n) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[M, -6.5e+124], N[Not[LessEqual[M, 1.15e+87]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(l - t$95$0), $MachinePrecision] + N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -6.50000000000000008e124 or 1.1500000000000001e87 < M

   1. Initial program 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in M around inf 83.5%

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

      1. unpow2 83.5%

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

   6. Simplified 83.5%

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

   7. Taylor expanded in K around 0 97.5%

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

      1. cos-neg 97.5%

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

   9. Simplified 97.5%

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

   if -6.50000000000000008e124 < M < 1.1500000000000001e87

   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. associate-/l* 76.6%

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

      2. associate--r- 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in K around 0 94.2%

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

      1. cos-neg 94.2%

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

   6. Simplified 94.2%

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

   7. Taylor expanded in M around 0 88.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6.5 \cdot 10^{+124} \lor \neg \left(M \leq 1.15 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|m - n\right|\right) + M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \]

Alternative 4: 68.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-256}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{+17}:\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|m - n\right|\right) + M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -3.3e-256)
   (exp (* m (* m -0.25)))
   (if (<= n 2.7e+17)
     (/ (cos M) (exp (+ (- l (fabs (- m n))) (* M M))))
     (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -3.3e-256) {
		tmp = exp((m * (m * -0.25)));
	} else if (n <= 2.7e+17) {
		tmp = cos(M) / exp(((l - fabs((m - n))) + (M * M)));
	} else {
		tmp = exp((-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) :: tmp
    if (n <= (-3.3d-256)) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (n <= 2.7d+17) then
        tmp = cos(m_1) / exp(((l - abs((m - n))) + (m_1 * m_1)))
    else
        tmp = exp(((-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 tmp;
	if (n <= -3.3e-256) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (n <= 2.7e+17) {
		tmp = Math.cos(M) / Math.exp(((l - Math.abs((m - n))) + (M * M)));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -3.3e-256:
		tmp = math.exp((m * (m * -0.25)))
	elif n <= 2.7e+17:
		tmp = math.cos(M) / math.exp(((l - math.fabs((m - n))) + (M * M)))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -3.3e-256)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (n <= 2.7e+17)
		tmp = Float64(cos(M) / exp(Float64(Float64(l - abs(Float64(m - n))) + Float64(M * M))));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -3.3e-256)
		tmp = exp((m * (m * -0.25)));
	elseif (n <= 2.7e+17)
		tmp = cos(M) / exp(((l - abs((m - n))) + (M * M)));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -3.3e-256], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.7e+17], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.3e-256

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. associate--r- 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in K around 0 97.1%

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

      1. cos-neg 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in M around 0 83.2%

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

   8. Taylor expanded in m around inf 45.1%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 45.1%

         \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

      2. unpow2 45.1%

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

      3. associate-*l* 45.1%

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

   10. Simplified 45.1%

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

   if -3.3e-256 < n < 2.7e17

   1. Initial program 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in M around inf 67.1%

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

      1. unpow2 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in K around 0 68.2%

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

      1. cos-neg 68.2%

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

   9. Simplified 68.2%

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

   if 2.7e17 < n

   1. Initial program 71.2%

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

      1. associate-/l* 71.2%

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

      2. associate--r- 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 98.3%

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

   8. Taylor expanded in n around inf 95.0%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 95.0%

         \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

      2. unpow2 95.0%

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

   10. Simplified 95.0%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-256}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{+17}:\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|m - n\right|\right) + M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 5: 65.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -116000:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{-178}:\\ \;\;\;\;\frac{\cos \left(0.5 \cdot \left(m \cdot K\right) - M\right)}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -116000.0)
   (exp (* m (* m -0.25)))
   (if (<= m 1.15e-178)
     (/ (cos (- (* 0.5 (* m K)) M)) (exp (* M M)))
     (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -116000.0) {
		tmp = exp((m * (m * -0.25)));
	} else if (m <= 1.15e-178) {
		tmp = cos(((0.5 * (m * K)) - M)) / exp((M * M));
	} else {
		tmp = exp((-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) :: tmp
    if (m <= (-116000.0d0)) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (m <= 1.15d-178) then
        tmp = cos(((0.5d0 * (m * k)) - m_1)) / exp((m_1 * m_1))
    else
        tmp = exp(((-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 tmp;
	if (m <= -116000.0) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (m <= 1.15e-178) {
		tmp = Math.cos(((0.5 * (m * K)) - M)) / Math.exp((M * M));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -116000.0:
		tmp = math.exp((m * (m * -0.25)))
	elif m <= 1.15e-178:
		tmp = math.cos(((0.5 * (m * K)) - M)) / math.exp((M * M))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -116000.0)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (m <= 1.15e-178)
		tmp = Float64(cos(Float64(Float64(0.5 * Float64(m * K)) - M)) / exp(Float64(M * M)));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -116000.0)
		tmp = exp((m * (m * -0.25)));
	elseif (m <= 1.15e-178)
		tmp = cos(((0.5 * (m * K)) - M)) / exp((M * M));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -116000.0], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 1.15e-178], N[(N[Cos[N[(N[(0.5 * N[(m * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -116000

   1. Initial program 71.4%

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

      1. associate-/l* 71.4%

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

      2. associate--r- 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 100.0%

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

   8. Taylor expanded in m around inf 96.9%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 96.9%

         \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

      2. unpow2 96.9%

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

      3. associate-*l* 96.9%

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

   10. Simplified 96.9%

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

   if -116000 < m < 1.14999999999999997e-178

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in M around inf 67.1%

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

      1. unpow2 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in n around 0 69.8%

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

   8. Taylor expanded in M around inf 57.3%

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

      1. unpow2 57.3%

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

   10. Simplified 57.3%

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

   if 1.14999999999999997e-178 < m

   1. Initial program 80.4%

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

      1. associate-/l* 79.4%

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

      2. associate--r- 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in K around 0 95.1%

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

      1. cos-neg 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in M around 0 87.5%

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

   8. Taylor expanded in n around inf 48.6%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 48.6%

         \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

      2. unpow2 48.6%

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

   10. Simplified 48.6%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -116000:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{-178}:\\ \;\;\;\;\frac{\cos \left(0.5 \cdot \left(m \cdot K\right) - M\right)}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 6: 59.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.28 \cdot 10^{+20}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 2.05 \cdot 10^{-227}:\\ \;\;\;\;\frac{\cos \left(K \cdot \left(n \cdot 0.5\right) - M\right)}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1.28e+20)
   (exp (* m (* m -0.25)))
   (if (<= m 2.05e-227)
     (/ (cos (- (* K (* n 0.5)) M)) (exp l))
     (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.28e+20) {
		tmp = exp((m * (m * -0.25)));
	} else if (m <= 2.05e-227) {
		tmp = cos(((K * (n * 0.5)) - M)) / exp(l);
	} else {
		tmp = exp((-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) :: tmp
    if (m <= (-1.28d+20)) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (m <= 2.05d-227) then
        tmp = cos(((k * (n * 0.5d0)) - m_1)) / exp(l)
    else
        tmp = exp(((-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 tmp;
	if (m <= -1.28e+20) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (m <= 2.05e-227) {
		tmp = Math.cos(((K * (n * 0.5)) - M)) / Math.exp(l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1.28e+20:
		tmp = math.exp((m * (m * -0.25)))
	elif m <= 2.05e-227:
		tmp = math.cos(((K * (n * 0.5)) - M)) / math.exp(l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1.28e+20)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (m <= 2.05e-227)
		tmp = Float64(cos(Float64(Float64(K * Float64(n * 0.5)) - M)) / exp(l));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1.28e+20)
		tmp = exp((m * (m * -0.25)));
	elseif (m <= 2.05e-227)
		tmp = cos(((K * (n * 0.5)) - M)) / exp(l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1.28e+20], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 2.05e-227], N[(N[Cos[N[(N[(K * N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.28e20

   1. Initial program 71.0%

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

      1. associate-/l* 71.0%

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

      2. associate--r- 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 100.0%

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

   8. Taylor expanded in m around inf 98.4%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

      2. unpow2 98.4%

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

      3. associate-*l* 98.4%

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

   10. Simplified 98.4%

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

   if -1.28e20 < m < 2.05000000000000005e-227

   1. Initial program 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in M around inf 66.7%

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

      1. unpow2 66.7%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in l around inf 44.2%

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

   8. Taylor expanded in m around 0 43.9%

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

      1. associate-*r* 43.9%

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

      2. *-commutative 43.9%

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

   10. Simplified 43.9%

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

   if 2.05000000000000005e-227 < m

   1. Initial program 80.9%

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

      1. associate-/l* 80.0%

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

      2. associate--r- 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in K around 0 95.5%

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

      1. cos-neg 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in M around 0 84.8%

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

   8. Taylor expanded in n around inf 48.8%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 48.8%

         \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

      2. unpow2 48.8%

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

   10. Simplified 48.8%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.28 \cdot 10^{+20}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 2.05 \cdot 10^{-227}:\\ \;\;\;\;\frac{\cos \left(K \cdot \left(n \cdot 0.5\right) - M\right)}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 7: 60.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.28 \cdot 10^{+20}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 7.6 \cdot 10^{-227}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1.28e+20)
   (exp (* m (* m -0.25)))
   (if (<= m 7.6e-227) (/ (cos M) (exp l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.28e+20) {
		tmp = exp((m * (m * -0.25)));
	} else if (m <= 7.6e-227) {
		tmp = cos(M) / exp(l);
	} else {
		tmp = exp((-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) :: tmp
    if (m <= (-1.28d+20)) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (m <= 7.6d-227) then
        tmp = cos(m_1) / exp(l)
    else
        tmp = exp(((-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 tmp;
	if (m <= -1.28e+20) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (m <= 7.6e-227) {
		tmp = Math.cos(M) / Math.exp(l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1.28e+20:
		tmp = math.exp((m * (m * -0.25)))
	elif m <= 7.6e-227:
		tmp = math.cos(M) / math.exp(l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1.28e+20)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (m <= 7.6e-227)
		tmp = Float64(cos(M) / exp(l));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1.28e+20)
		tmp = exp((m * (m * -0.25)));
	elseif (m <= 7.6e-227)
		tmp = cos(M) / exp(l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1.28e+20], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 7.6e-227], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.28e20

   1. Initial program 71.0%

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

      1. associate-/l* 71.0%

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

      2. associate--r- 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 100.0%

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

   8. Taylor expanded in m around inf 98.4%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

      2. unpow2 98.4%

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

      3. associate-*l* 98.4%

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

   10. Simplified 98.4%

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

   if -1.28e20 < m < 7.60000000000000019e-227

   1. Initial program 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in M around inf 66.7%

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

      1. unpow2 66.7%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in l around inf 44.2%

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

   8. Taylor expanded in K around 0 47.9%

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

      1. cos-neg 47.9%

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

   10. Simplified 47.9%

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

   if 7.60000000000000019e-227 < m

   1. Initial program 80.9%

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

      1. associate-/l* 80.0%

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

      2. associate--r- 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in K around 0 95.5%

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

      1. cos-neg 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in M around 0 84.8%

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

   8. Taylor expanded in n around inf 48.8%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 48.8%

         \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

      2. unpow2 48.8%

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

   10. Simplified 48.8%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.28 \cdot 10^{+20}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 7.6 \cdot 10^{-227}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 8: 68.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.28 \cdot 10^{+20} \lor \neg \left(m \leq 3.2 \cdot 10^{-48}\right):\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -1.28e+20) (not (<= m 3.2e-48)))
   (exp (* m (* m -0.25)))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -1.28e+20) || !(m <= 3.2e-48)) {
		tmp = exp((m * (m * -0.25)));
	} 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) :: tmp
    if ((m <= (-1.28d+20)) .or. (.not. (m <= 3.2d-48))) then
        tmp = exp((m * (m * (-0.25d0))))
    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 tmp;
	if ((m <= -1.28e+20) || !(m <= 3.2e-48)) {
		tmp = Math.exp((m * (m * -0.25)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -1.28e+20) or not (m <= 3.2e-48):
		tmp = math.exp((m * (m * -0.25)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -1.28e+20) || !(m <= 3.2e-48))
		tmp = exp(Float64(m * Float64(m * -0.25)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -1.28e+20) || ~((m <= 3.2e-48)))
		tmp = exp((m * (m * -0.25)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -1.28e+20], N[Not[LessEqual[m, 3.2e-48]], $MachinePrecision]], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.28e20 or 3.1999999999999998e-48 < m

   1. Initial program 74.4%

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

      1. associate-/l* 73.6%

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

      2. associate--r- 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in K around 0 97.6%

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

      1. cos-neg 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in M around 0 96.8%

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

   8. Taylor expanded in m around inf 88.3%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 88.3%

         \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

      2. unpow2 88.3%

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

      3. associate-*l* 88.3%

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

   10. Simplified 88.3%

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

   if -1.28e20 < m < 3.1999999999999998e-48

   1. Initial program 83.8%

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

      1. associate-/l* 83.7%

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

      2. associate--r- 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in K around 0 94.5%

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

      1. cos-neg 94.5%

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

   6. Simplified 94.5%

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

   7. Taylor expanded in M around 0 72.5%

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

   8. Taylor expanded in l around inf 46.5%

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

      1. neg-mul-1 46.5%

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

   10. Simplified 46.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.28 \cdot 10^{+20} \lor \neg \left(m \leq 3.2 \cdot 10^{-48}\right):\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 9: 60.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.28 \cdot 10^{+20}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 8.4 \cdot 10^{-237}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1.28e+20)
   (exp (* m (* m -0.25)))
   (if (<= m 8.4e-237) (exp (- l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.28e+20) {
		tmp = exp((m * (m * -0.25)));
	} else if (m <= 8.4e-237) {
		tmp = exp(-l);
	} else {
		tmp = exp((-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) :: tmp
    if (m <= (-1.28d+20)) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (m <= 8.4d-237) then
        tmp = exp(-l)
    else
        tmp = exp(((-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 tmp;
	if (m <= -1.28e+20) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (m <= 8.4e-237) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1.28e+20:
		tmp = math.exp((m * (m * -0.25)))
	elif m <= 8.4e-237:
		tmp = math.exp(-l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1.28e+20)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (m <= 8.4e-237)
		tmp = exp(Float64(-l));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1.28e+20)
		tmp = exp((m * (m * -0.25)));
	elseif (m <= 8.4e-237)
		tmp = exp(-l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1.28e+20], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 8.4e-237], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.28e20

   1. Initial program 71.0%

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

      1. associate-/l* 71.0%

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

      2. associate--r- 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 100.0%

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

   8. Taylor expanded in m around inf 98.4%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

      2. unpow2 98.4%

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

      3. associate-*l* 98.4%

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

   10. Simplified 98.4%

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

   if -1.28e20 < m < 8.4000000000000005e-237

   1. Initial program 82.9%

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

      1. associate-/l* 82.6%

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

      2. associate--r- 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in K around 0 93.2%

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

      1. cos-neg 93.2%

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

   6. Simplified 93.2%

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

   7. Taylor expanded in M around 0 70.9%

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

   8. Taylor expanded in l around inf 45.6%

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

      1. neg-mul-1 45.6%

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

   10. Simplified 45.6%

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

   if 8.4000000000000005e-237 < m

   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. associate-/l* 80.4%

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

      2. associate--r- 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in K around 0 95.8%

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

      1. cos-neg 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in M around 0 84.9%

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

   8. Taylor expanded in n around inf 49.4%

      \[\leadsto e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 49.4%

         \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

      2. unpow2 49.4%

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

   10. Simplified 49.4%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.28 \cdot 10^{+20}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 8.4 \cdot 10^{-237}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 10: 35.6% 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 79.2%

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

   1. associate-/l* 78.8%

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

   2. associate--r- 78.8%

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

3. Simplified 78.8%

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

4. Taylor expanded in K around 0 96.0%

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

   1. cos-neg 96.0%

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

6. Simplified 96.0%

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

7. Taylor expanded in M around 0 84.4%

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

8. Taylor expanded in l around inf 39.5%

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

   1. neg-mul-1 39.5%

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

10. Simplified 39.5%

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

11. Final simplification 39.5%

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

Alternative 11: 6.7% 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 79.2%

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

3. Simplified 79.2%

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

4. Taylor expanded in M around inf 54.6%

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

   1. unpow2 54.6%

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

6. Simplified 54.6%

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

7. Taylor expanded in l around inf 35.7%

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

8. Taylor expanded in l around 0 7.9%

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

9. Taylor expanded in K around 0 8.4%

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

    1. cos-neg 8.4%

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

11. Simplified 8.4%

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

12. Final simplification 8.4%

    \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023213 
(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)))))))