Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.0% → 96.6%

Time: 18.7s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.0% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\cos M \cdot e^{\left|n - m\right| - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)}}\right)}^{3} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (pow
  (cbrt
   (* (cos M) (exp (- (fabs (- n m)) (+ l (pow (- (* (+ m n) 0.5) M) 2.0))))))
  3.0))

C

double code(double K, double m, double n, double M, double l) {
	return pow(cbrt((cos(M) * exp((fabs((n - m)) - (l + pow((((m + n) * 0.5) - M), 2.0)))))), 3.0);
}

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.pow(Math.cbrt((Math.cos(M) * Math.exp((Math.abs((n - m)) - (l + Math.pow((((m + n) * 0.5) - M), 2.0)))))), 3.0);
}

Julia

function code(K, m, n, M, l)
	return cbrt(Float64(cos(M) * exp(Float64(abs(Float64(n - m)) - Float64(l + (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0)))))) ^ 3.0
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Power[N[Power[N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

5. Simplified 97.5%

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

   1. add-cube-cbrt 97.5%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cos M \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}} \cdot \sqrt[3]{\cos M \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}\right) \cdot \sqrt[3]{\cos M \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}} \]

   2. pow3 97.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos M \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}\right)}^{3}} \]

7. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos M \cdot e^{\left(\left(-{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}}\right)}^{3}} \]

8. Final simplification 97.5%

   \[\leadsto {\left(\sqrt[3]{\cos M \cdot e^{\left|n - m\right| - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)}}\right)}^{3} \]
9. Add Preprocessing

Alternative 2: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

5. Simplified 97.5%

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

6. Final simplification 97.5%

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

Alternative 3: 94.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.5 \cdot 10^{+20} \lor \neg \left(M \leq 1.06 \cdot 10^{+77}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1.5e+20) (not (<= M 1.06e+77)))
   (exp (- (pow M 2.0)))
   (exp (- (- (fabs (- n m)) l) (* 0.25 (pow (+ m n) 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.5e+20) || !(M <= 1.06e+77)) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = exp(((fabs((n - m)) - 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) :: tmp
    if ((m_1 <= (-1.5d+20)) .or. (.not. (m_1 <= 1.06d+77))) then
        tmp = exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((abs((n - m)) - 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 tmp;
	if ((M <= -1.5e+20) || !(M <= 1.06e+77)) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(((Math.abs((n - m)) - l) - (0.25 * Math.pow((m + n), 2.0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1.5e+20) or not (M <= 1.06e+77):
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(((math.fabs((n - m)) - l) - (0.25 * math.pow((m + n), 2.0))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1.5e+20) || !(M <= 1.06e+77))
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(0.25 * (Float64(m + n) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1.5e+20) || ~((M <= 1.06e+77)))
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp(((abs((n - m)) - l) - (0.25 * ((m + n) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1.5e+20], N[Not[LessEqual[M, 1.06e+77]], $MachinePrecision]], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1.5e20 or 1.06000000000000003e77 < M

   1. Initial program 80.0%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{1} \cdot e^{-{M}^{2}} \]

   if -1.5e20 < M < 1.06000000000000003e77

   1. Initial program 67.9%

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

   3. Taylor expanded in K around 0 95.7%

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

      1. cos-neg 95.7%

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

   5. Simplified 95.7%

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

   6. Taylor expanded in M around 0 95.0%

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

      1. associate--r+ 95.0%

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

   8. Simplified 95.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.5 \cdot 10^{+20} \lor \neg \left(M \leq 1.06 \cdot 10^{+77}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot 0.5 - M\\ \mathbf{if}\;n \leq -2.15 \cdot 10^{-162}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-218}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(m - n\right) + \left(t\_0 \cdot \left(m + t\_0\right) - \ell\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* n 0.5) M)))
   (if (<= n -2.15e-162)
     (* (cos M) (exp (* -0.25 (* m m))))
     (if (<= n 3e-218)
       (*
        (cos (- (/ (* (+ m n) K) 2.0) M))
        (exp (+ (- m n) (- (* t_0 (+ m t_0)) l))))
       (if (<= n 54.0)
         (exp (- (pow M 2.0)))
         (* (cos M) (exp (* -0.25 (pow n 2.0)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (n * 0.5) - M;
	double tmp;
	if (n <= -2.15e-162) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (n <= 3e-218) {
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(((m - n) + ((t_0 * (m + t_0)) - l)));
	} else if (n <= 54.0) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp((-0.25 * pow(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 = (n * 0.5d0) - m_1
    if (n <= (-2.15d-162)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (n <= 3d-218) then
        tmp = cos(((((m + n) * k) / 2.0d0) - m_1)) * exp(((m - n) + ((t_0 * (m + t_0)) - l)))
    else if (n <= 54.0d0) then
        tmp = exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (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 = (n * 0.5) - M;
	double tmp;
	if (n <= -2.15e-162) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (n <= 3e-218) {
		tmp = Math.cos(((((m + n) * K) / 2.0) - M)) * Math.exp(((m - n) + ((t_0 * (m + t_0)) - l)));
	} else if (n <= 54.0) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (n * 0.5) - M
	tmp = 0
	if n <= -2.15e-162:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif n <= 3e-218:
		tmp = math.cos(((((m + n) * K) / 2.0) - M)) * math.exp(((m - n) + ((t_0 * (m + t_0)) - l)))
	elif n <= 54.0:
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(n * 0.5) - M)
	tmp = 0.0
	if (n <= -2.15e-162)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 3e-218)
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) * K) / 2.0) - M)) * exp(Float64(Float64(m - n) + Float64(Float64(t_0 * Float64(m + t_0)) - l))));
	elseif (n <= 54.0)
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (n ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (n * 0.5) - M;
	tmp = 0.0;
	if (n <= -2.15e-162)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (n <= 3e-218)
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(((m - n) + ((t_0 * (m + t_0)) - l)));
	elseif (n <= 54.0)
		tmp = exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision]}, If[LessEqual[n, -2.15e-162], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3e-218], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(t$95$0 * N[(m + t$95$0), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.14999999999999998e-162

   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. Taylor expanded in K around 0 99.1%

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

      1. cos-neg 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in m around inf 58.2%

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

      1. unpow2 58.2%

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

   8. Applied egg-rr 58.2%

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

   if -2.14999999999999998e-162 < n < 2.9999999999999998e-218

   1. Initial program 89.5%

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

      1. sub-neg 89.5%

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

      2. add-sqr-sqrt 6.5%

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

      3. sqrt-unprod 29.6%

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

      4. sqr-neg 29.6%

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

      5. sqrt-unprod 29.6%

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

      6. add-sqr-sqrt 29.6%

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

      7. div-inv 29.6%

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

      8. fmm-def 29.6%

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

      9. metadata-eval 29.6%

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

      10. add-sqr-sqrt 7.2%

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

      11. fabs-sqr 7.2%

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

      12. add-sqr-sqrt 29.6%

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

   4. Applied egg-rr 29.6%

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

      1. unsub-neg 29.6%

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

      2. associate-+l- 29.6%

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

      3. +-commutative 29.6%

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

      4. fmm-undef 29.6%

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

      5. *-commutative 29.6%

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

   6. Simplified 29.6%

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

   7. Taylor expanded in m around 0 33.9%

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

      1. +-commutative 33.9%

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

      2. unpow2 33.9%

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

      3. distribute-rgt-out 33.9%

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

      4. *-commutative 33.9%

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

      5. *-commutative 33.9%

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

   9. Simplified 33.9%

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

   if 2.9999999999999998e-218 < n < 54

   1. Initial program 80.1%

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

   3. Taylor expanded in K around 0 93.5%

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

      1. cos-neg 93.5%

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

   5. Simplified 93.5%

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

   6. Taylor expanded in M around inf 67.1%

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

      1. mul-1-neg 67.1%

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

   8. Simplified 67.1%

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

   9. Taylor expanded in M around 0 67.1%

      \[\leadsto \color{blue}{1} \cdot e^{-{M}^{2}} \]

   if 54 < n

   1. Initial program 63.5%

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

   3. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in n around inf 96.9%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.15 \cdot 10^{-162}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-218}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(m - n\right) + \left(\left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right) - \ell\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot {m}^{2}}\\ t_1 := m \cdot 0.5 - M\\ t_2 := e^{-{M}^{2}}\\ \mathbf{if}\;M \leq -26:\\ \;\;\;\;\cos M \cdot t\_2\\ \mathbf{elif}\;M \leq 3.45 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 4 \cdot 10^{-105}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(m - n\right) + \left(t\_1 \cdot \left(n + t\_1\right) - \ell\right)}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (pow m 2.0))))
        (t_1 (- (* m 0.5) M))
        (t_2 (exp (- (pow M 2.0)))))
   (if (<= M -26.0)
     (* (cos M) t_2)
     (if (<= M 3.45e-189)
       t_0
       (if (<= M 4e-105)
         (*
          (cos (- (/ (* (+ m n) K) 2.0) M))
          (exp (+ (- m n) (- (* t_1 (+ n t_1)) l))))
         (if (<= M 27.0) t_0 t_2))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * pow(m, 2.0)));
	double t_1 = (m * 0.5) - M;
	double t_2 = exp(-pow(M, 2.0));
	double tmp;
	if (M <= -26.0) {
		tmp = cos(M) * t_2;
	} else if (M <= 3.45e-189) {
		tmp = t_0;
	} else if (M <= 4e-105) {
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(((m - n) + ((t_1 * (n + t_1)) - l)));
	} else if (M <= 27.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = exp(((-0.25d0) * (m ** 2.0d0)))
    t_1 = (m * 0.5d0) - m_1
    t_2 = exp(-(m_1 ** 2.0d0))
    if (m_1 <= (-26.0d0)) then
        tmp = cos(m_1) * t_2
    else if (m_1 <= 3.45d-189) then
        tmp = t_0
    else if (m_1 <= 4d-105) then
        tmp = cos(((((m + n) * k) / 2.0d0) - m_1)) * exp(((m - n) + ((t_1 * (n + t_1)) - l)))
    else if (m_1 <= 27.0d0) then
        tmp = t_0
    else
        tmp = t_2
    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.exp((-0.25 * Math.pow(m, 2.0)));
	double t_1 = (m * 0.5) - M;
	double t_2 = Math.exp(-Math.pow(M, 2.0));
	double tmp;
	if (M <= -26.0) {
		tmp = Math.cos(M) * t_2;
	} else if (M <= 3.45e-189) {
		tmp = t_0;
	} else if (M <= 4e-105) {
		tmp = Math.cos(((((m + n) * K) / 2.0) - M)) * Math.exp(((m - n) + ((t_1 * (n + t_1)) - l)));
	} else if (M <= 27.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * math.pow(m, 2.0)))
	t_1 = (m * 0.5) - M
	t_2 = math.exp(-math.pow(M, 2.0))
	tmp = 0
	if M <= -26.0:
		tmp = math.cos(M) * t_2
	elif M <= 3.45e-189:
		tmp = t_0
	elif M <= 4e-105:
		tmp = math.cos(((((m + n) * K) / 2.0) - M)) * math.exp(((m - n) + ((t_1 * (n + t_1)) - l)))
	elif M <= 27.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * (m ^ 2.0)))
	t_1 = Float64(Float64(m * 0.5) - M)
	t_2 = exp(Float64(-(M ^ 2.0)))
	tmp = 0.0
	if (M <= -26.0)
		tmp = Float64(cos(M) * t_2);
	elseif (M <= 3.45e-189)
		tmp = t_0;
	elseif (M <= 4e-105)
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) * K) / 2.0) - M)) * exp(Float64(Float64(m - n) + Float64(Float64(t_1 * Float64(n + t_1)) - l))));
	elseif (M <= 27.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (m ^ 2.0)));
	t_1 = (m * 0.5) - M;
	t_2 = exp(-(M ^ 2.0));
	tmp = 0.0;
	if (M <= -26.0)
		tmp = cos(M) * t_2;
	elseif (M <= 3.45e-189)
		tmp = t_0;
	elseif (M <= 4e-105)
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(((m - n) + ((t_1 * (n + t_1)) - l)));
	elseif (M <= 27.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]}, If[LessEqual[M, -26.0], N[(N[Cos[M], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[M, 3.45e-189], t$95$0, If[LessEqual[M, 4e-105], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(t$95$1 * N[(n + t$95$1), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 27.0], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -26

   1. Initial program 81.8%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around inf 98.7%

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

      1. mul-1-neg 98.7%

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

   8. Simplified 98.7%

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

   if -26 < M < 3.4500000000000001e-189 or 3.99999999999999986e-105 < M < 27

   1. Initial program 62.2%

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

   3. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   5. Simplified 94.7%

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

   6. Taylor expanded in m around inf 67.7%

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

   7. Taylor expanded in M around 0 67.7%

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

   if 3.4500000000000001e-189 < M < 3.99999999999999986e-105

   1. Initial program 86.2%

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

      1. sub-neg 86.2%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 48.9%

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

      4. sqr-neg 48.9%

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

      5. sqrt-unprod 48.9%

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

      6. add-sqr-sqrt 48.9%

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

      7. div-inv 48.9%

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

      8. fmm-def 48.9%

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

      9. metadata-eval 48.9%

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

      10. add-sqr-sqrt 28.0%

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

      11. fabs-sqr 28.0%

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

      12. add-sqr-sqrt 48.9%

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

   4. Applied egg-rr 48.9%

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

      1. unsub-neg 48.9%

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

      2. associate-+l- 48.9%

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

      3. +-commutative 48.9%

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

      4. fmm-undef 48.9%

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

      5. *-commutative 48.9%

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

   6. Simplified 48.9%

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

   7. Taylor expanded in n around 0 52.2%

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

      1. +-commutative 52.2%

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

      2. unpow2 52.2%

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

      3. distribute-rgt-out 55.7%

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

      4. *-commutative 55.7%

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

      5. *-commutative 55.7%

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

   9. Simplified 55.7%

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

   if 27 < M

   1. Initial program 75.4%

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

   3. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in M around inf 95.2%

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

      1. mul-1-neg 95.2%

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

   8. Simplified 95.2%

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

   9. Taylor expanded in M around 0 95.2%

      \[\leadsto \color{blue}{1} \cdot e^{-{M}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -26:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;M \leq 3.45 \cdot 10^{-189}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;M \leq 4 \cdot 10^{-105}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(m - n\right) + \left(\left(m \cdot 0.5 - M\right) \cdot \left(n + \left(m \cdot 0.5 - M\right)\right) - \ell\right)}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot {m}^{2}}\\ t_1 := m \cdot 0.5 - M\\ t_2 := e^{-{M}^{2}}\\ \mathbf{if}\;M \leq -26:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;M \leq 8.2 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 7.5 \cdot 10^{-108}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(m - n\right) + \left(t\_1 \cdot \left(n + t\_1\right) - \ell\right)}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (pow m 2.0))))
        (t_1 (- (* m 0.5) M))
        (t_2 (exp (- (pow M 2.0)))))
   (if (<= M -26.0)
     t_2
     (if (<= M 8.2e-188)
       t_0
       (if (<= M 7.5e-108)
         (*
          (cos (- (/ (* (+ m n) K) 2.0) M))
          (exp (+ (- m n) (- (* t_1 (+ n t_1)) l))))
         (if (<= M 27.0) t_0 t_2))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * pow(m, 2.0)));
	double t_1 = (m * 0.5) - M;
	double t_2 = exp(-pow(M, 2.0));
	double tmp;
	if (M <= -26.0) {
		tmp = t_2;
	} else if (M <= 8.2e-188) {
		tmp = t_0;
	} else if (M <= 7.5e-108) {
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(((m - n) + ((t_1 * (n + t_1)) - l)));
	} else if (M <= 27.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = exp(((-0.25d0) * (m ** 2.0d0)))
    t_1 = (m * 0.5d0) - m_1
    t_2 = exp(-(m_1 ** 2.0d0))
    if (m_1 <= (-26.0d0)) then
        tmp = t_2
    else if (m_1 <= 8.2d-188) then
        tmp = t_0
    else if (m_1 <= 7.5d-108) then
        tmp = cos(((((m + n) * k) / 2.0d0) - m_1)) * exp(((m - n) + ((t_1 * (n + t_1)) - l)))
    else if (m_1 <= 27.0d0) then
        tmp = t_0
    else
        tmp = t_2
    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.exp((-0.25 * Math.pow(m, 2.0)));
	double t_1 = (m * 0.5) - M;
	double t_2 = Math.exp(-Math.pow(M, 2.0));
	double tmp;
	if (M <= -26.0) {
		tmp = t_2;
	} else if (M <= 8.2e-188) {
		tmp = t_0;
	} else if (M <= 7.5e-108) {
		tmp = Math.cos(((((m + n) * K) / 2.0) - M)) * Math.exp(((m - n) + ((t_1 * (n + t_1)) - l)));
	} else if (M <= 27.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * math.pow(m, 2.0)))
	t_1 = (m * 0.5) - M
	t_2 = math.exp(-math.pow(M, 2.0))
	tmp = 0
	if M <= -26.0:
		tmp = t_2
	elif M <= 8.2e-188:
		tmp = t_0
	elif M <= 7.5e-108:
		tmp = math.cos(((((m + n) * K) / 2.0) - M)) * math.exp(((m - n) + ((t_1 * (n + t_1)) - l)))
	elif M <= 27.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * (m ^ 2.0)))
	t_1 = Float64(Float64(m * 0.5) - M)
	t_2 = exp(Float64(-(M ^ 2.0)))
	tmp = 0.0
	if (M <= -26.0)
		tmp = t_2;
	elseif (M <= 8.2e-188)
		tmp = t_0;
	elseif (M <= 7.5e-108)
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) * K) / 2.0) - M)) * exp(Float64(Float64(m - n) + Float64(Float64(t_1 * Float64(n + t_1)) - l))));
	elseif (M <= 27.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (m ^ 2.0)));
	t_1 = (m * 0.5) - M;
	t_2 = exp(-(M ^ 2.0));
	tmp = 0.0;
	if (M <= -26.0)
		tmp = t_2;
	elseif (M <= 8.2e-188)
		tmp = t_0;
	elseif (M <= 7.5e-108)
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(((m - n) + ((t_1 * (n + t_1)) - l)));
	elseif (M <= 27.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]}, If[LessEqual[M, -26.0], t$95$2, If[LessEqual[M, 8.2e-188], t$95$0, If[LessEqual[M, 7.5e-108], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(t$95$1 * N[(n + t$95$1), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 27.0], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -26 or 27 < M

   1. Initial program 78.7%

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

   3. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in M around inf 97.0%

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

      1. mul-1-neg 97.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in M around 0 97.0%

      \[\leadsto \color{blue}{1} \cdot e^{-{M}^{2}} \]

   if -26 < M < 8.19999999999999965e-188 or 7.4999999999999993e-108 < M < 27

   1. Initial program 62.2%

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

   3. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   5. Simplified 94.7%

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

   6. Taylor expanded in m around inf 67.7%

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

   7. Taylor expanded in M around 0 67.7%

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

   if 8.19999999999999965e-188 < M < 7.4999999999999993e-108

   1. Initial program 86.2%

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

      1. sub-neg 86.2%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 48.9%

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

      4. sqr-neg 48.9%

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

      5. sqrt-unprod 48.9%

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

      6. add-sqr-sqrt 48.9%

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

      7. div-inv 48.9%

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

      8. fmm-def 48.9%

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

      9. metadata-eval 48.9%

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

      10. add-sqr-sqrt 28.0%

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

      11. fabs-sqr 28.0%

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

      12. add-sqr-sqrt 48.9%

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

   4. Applied egg-rr 48.9%

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

      1. unsub-neg 48.9%

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

      2. associate-+l- 48.9%

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

      3. +-commutative 48.9%

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

      4. fmm-undef 48.9%

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

      5. *-commutative 48.9%

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

   6. Simplified 48.9%

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

   7. Taylor expanded in n around 0 52.2%

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

      1. +-commutative 52.2%

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

      2. unpow2 52.2%

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

      3. distribute-rgt-out 55.7%

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

      4. *-commutative 55.7%

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

      5. *-commutative 55.7%

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

   9. Simplified 55.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -26:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{elif}\;M \leq 8.2 \cdot 10^{-188}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;M \leq 7.5 \cdot 10^{-108}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(m - n\right) + \left(\left(m \cdot 0.5 - M\right) \cdot \left(n + \left(m \cdot 0.5 - M\right)\right) - \ell\right)}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -26 \lor \neg \left(M \leq 26.5\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -26.0) (not (<= M 26.5)))
   (exp (- (pow M 2.0)))
   (exp (* -0.25 (pow m 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -26.0) || !(M <= 26.5)) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = exp((-0.25 * 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 <= (-26.0d0)) .or. (.not. (m_1 <= 26.5d0))) then
        tmp = exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((-0.25d0) * (m ** 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 <= -26.0) || !(M <= 26.5)) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -26.0) or not (M <= 26.5):
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -26.0) || !(M <= 26.5))
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -26.0) || ~((M <= 26.5)))
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp((-0.25 * (m ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -26.0], N[Not[LessEqual[M, 26.5]], $MachinePrecision]], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -26 or 26.5 < M

   1. Initial program 78.7%

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

   3. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in M around inf 97.0%

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

      1. mul-1-neg 97.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in M around 0 97.0%

      \[\leadsto \color{blue}{1} \cdot e^{-{M}^{2}} \]

   if -26 < M < 26.5

   1. Initial program 67.6%

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

   3. Taylor expanded in K around 0 95.9%

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

      1. cos-neg 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in m around inf 62.1%

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

   7. Taylor expanded in M around 0 62.1%

      \[\leadsto \color{blue}{e^{-0.25 \cdot {m}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -26 \lor \neg \left(M \leq 26.5\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -11 \lor \neg \left(m \leq 0.000105\right):\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -11.0) (not (<= m 0.000105)))
   (exp (* -0.25 (pow m 2.0)))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -11.0) || !(m <= 0.000105)) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m <= (-11.0d0)) .or. (.not. (m <= 0.000105d0))) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -11.0) || !(m <= 0.000105)) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -11.0) or not (m <= 0.000105):
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -11.0) || !(m <= 0.000105))
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -11.0) || ~((m <= 0.000105)))
		tmp = exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -11.0], N[Not[LessEqual[m, 0.000105]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -11 or 1.05e-4 < m

   1. Initial program 63.0%

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

   3. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in m around inf 96.5%

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

   7. Taylor expanded in M around 0 96.5%

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

   if -11 < m < 1.05e-4

   1. Initial program 84.9%

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

   3. Taylor expanded in l around inf 37.9%

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

      1. mul-1-neg 37.9%

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

   5. Simplified 37.9%

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

   6. Taylor expanded in K around 0 41.0%

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

      1. cos-neg 41.0%

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

   8. Simplified 41.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -11 \lor \neg \left(m \leq 0.000105\right):\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{-0.25 \cdot {m}^{2}} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (exp (* -0.25 (pow m 2.0))))

C

double code(double K, double m, double n, double M, double l) {
	return exp((-0.25 * pow(m, 2.0)));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp(((-0.25d0) * (m ** 2.0d0)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((-0.25 * Math.pow(m, 2.0)));
}

Python

def code(K, m, n, M, l):
	return math.exp((-0.25 * math.pow(m, 2.0)))

Julia

function code(K, m, n, M, l)
	return exp(Float64(-0.25 * (m ^ 2.0)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp((-0.25 * (m ^ 2.0)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

5. Simplified 97.5%

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

6. Taylor expanded in m around inf 55.9%

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

7. Taylor expanded in M around 0 55.9%

   \[\leadsto \color{blue}{e^{-0.25 \cdot {m}^{2}}} \]
8. Add Preprocessing

Alternative 10: 7.2% 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 73.1%

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

3. Taylor expanded in l around inf 24.9%

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

   1. mul-1-neg 24.9%

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

5. Simplified 24.9%

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

6. Taylor expanded in l around 0 4.7%

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

   1. *-commutative 4.7%

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

   2. *-commutative 4.7%

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

   3. *-commutative 4.7%

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

   4. associate-*r* 4.7%

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

   5. fmm-undef 4.7%

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

   6. fmm-undef 4.7%

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

   7. associate-*r* 4.7%

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

   8. *-commutative 4.7%

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

   9. *-commutative 4.7%

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

   10. *-commutative 4.7%

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

   11. associate-*r* 4.7%

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

8. Simplified 4.7%

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

9. Taylor expanded in K around 0 5.6%

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

    1. cos-neg 5.6%

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

11. Simplified 5.6%

    \[\leadsto \color{blue}{\cos M} \]
12. Add Preprocessing

Alternative 11: 7.2% accurate, 425.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 1.0)

C

double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

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 = 1.0d0
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

Python

def code(K, m, n, M, l):
	return 1.0

Julia

function code(K, m, n, M, l)
	return 1.0
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0;
end

Wolfram

code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

5. Simplified 97.5%

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

6. Taylor expanded in M around inf 52.2%

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

   1. mul-1-neg 52.2%

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

8. Simplified 52.2%

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

9. Taylor expanded in M around 0 5.6%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024180 
(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)))))))