Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 97.0%

Time: 15.8s

Alternatives: 10

Speedup: 1.9×

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: 75.9% 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: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

3. Taylor expanded in K around 0 97.3%

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

   1. cos-neg 97.3%

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

5. Simplified 97.3%

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

6. Final simplification 97.3%

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

Alternative 2: 87.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.9 \cdot 10^{+33}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + e^{\left(\left|m - n\right| - \ell\right) - {\left(n \cdot 0.5 - M\right)}^{2}}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -3.9e+33)
   (exp (* -0.25 (pow m 2.0)))
   (+ (+ 1.0 (exp (- (- (fabs (- m n)) l) (pow (- (* n 0.5) M) 2.0)))) -1.0)))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -3.9e+33) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = (1.0 + exp(((fabs((m - n)) - l) - pow(((n * 0.5) - M), 2.0)))) + -1.0;
	}
	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 <= (-3.9d+33)) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = (1.0d0 + exp(((abs((m - n)) - l) - (((n * 0.5d0) - m_1) ** 2.0d0)))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -3.9e+33) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = (1.0 + Math.exp(((Math.abs((m - n)) - l) - Math.pow(((n * 0.5) - M), 2.0)))) + -1.0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -3.9e+33:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = (1.0 + math.exp(((math.fabs((m - n)) - l) - math.pow(((n * 0.5) - M), 2.0)))) + -1.0
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -3.9e+33)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(abs(Float64(m - n)) - l) - (Float64(Float64(n * 0.5) - M) ^ 2.0)))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -3.9e+33)
		tmp = exp((-0.25 * (m ^ 2.0)));
	else
		tmp = (1.0 + exp(((abs((m - n)) - l) - (((n * 0.5) - M) ^ 2.0)))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -3.9e+33], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 + N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.9000000000000002e33

   1. Initial program 70.4%

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

   3. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 0 98.1%

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

   7. Taylor expanded in m around inf 98.2%

      \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]

   if -3.9000000000000002e33 < m

   1. Initial program 76.3%

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

   3. Taylor expanded in K around 0 96.6%

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

      1. cos-neg 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in M around 0 96.2%

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

      1. *-un-lft-identity 96.2%

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

      2. expm1-log1p-u 96.2%

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

      3. expm1-undefine 96.2%

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

   8. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\left(\left(-{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}\right) - \ell\right) + \left|m - n\right|}\right)} - 1} \]

   9. Taylor expanded in m around 0 86.9%

      \[\leadsto \color{blue}{\left(1 + e^{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot n - M\right)}^{2}\right)}\right)} - 1 \]
   10. Step-by-step derivation

       1. +-commutative 86.9%

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

       2. associate--r+ 86.9%

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

       3. *-commutative 86.9%

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

   11. Simplified 86.9%

       \[\leadsto \color{blue}{\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(n \cdot 0.5 - M\right)}^{2}} + 1\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.9 \cdot 10^{+33}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + e^{\left(\left|m - n\right| - \ell\right) - {\left(n \cdot 0.5 - M\right)}^{2}}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

3. Taylor expanded in K around 0 97.3%

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

   1. cos-neg 97.3%

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

5. Simplified 97.3%

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

6. Taylor expanded in M around 0 96.6%

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

7. Final simplification 96.6%

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

Alternative 4: 95.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.92 \cdot 10^{+27} \lor \neg \left(M \leq 1.35 \cdot 10^{+40}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(m - n\right) - \ell\right) + -0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1.92e+27) (not (<= M 1.35e+40)))
   (exp (- (pow M 2.0)))
   (exp (+ (- (- m n) 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 <= -1.92e+27) || !(M <= 1.35e+40)) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = exp((((m - n) - 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 <= (-1.92d+27)) .or. (.not. (m_1 <= 1.35d+40))) then
        tmp = exp(-(m_1 ** 2.0d0))
    else
        tmp = exp((((m - n) - 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 <= -1.92e+27) || !(M <= 1.35e+40)) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp((((m - n) - l) + (-0.25 * Math.pow((m + n), 2.0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1.92e+27) or not (M <= 1.35e+40):
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp((((m - n) - l) + (-0.25 * math.pow((m + n), 2.0))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1.92e+27) || !(M <= 1.35e+40))
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64(Float64(Float64(m - n) - 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 <= -1.92e+27) || ~((M <= 1.35e+40)))
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp((((m - n) - l) + (-0.25 * ((m + n) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1.92e+27], N[Not[LessEqual[M, 1.35e+40]], $MachinePrecision]], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[N[(N[(N[(m - n), $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 < -1.92000000000000004e27 or 1.35000000000000005e40 < 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. 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 0 99.1%

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

   7. Taylor expanded in M around inf 99.2%

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

      1. mul-1-neg 99.2%

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

   9. Simplified 99.2%

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

   if -1.92000000000000004e27 < M < 1.35000000000000005e40

   1. Initial program 75.7%

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

   3. Taylor expanded in K around 0 95.0%

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

      1. cos-neg 95.0%

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

   5. Simplified 95.0%

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

   6. Taylor expanded in M around 0 93.0%

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

      1. associate--r+ 93.0%

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

      2. cancel-sign-sub-inv 93.0%

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

      3. sub-neg 93.0%

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

      4. neg-mul-1 93.0%

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

      5. neg-mul-1 93.0%

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

      6. sub-neg 93.0%

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

      7. rem-square-sqrt 51.8%

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

      8. fabs-sqr 51.8%

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

      9. rem-square-sqrt 93.0%

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

      10. metadata-eval 93.0%

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

   8. Simplified 93.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.92 \cdot 10^{+27} \lor \neg \left(M \leq 1.35 \cdot 10^{+40}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(m - n\right) - \ell\right) + -0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-{M}^{2}}\\ \mathbf{if}\;M \leq -300000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq -3.4 \cdot 10^{-65}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (pow M 2.0)))))
   (if (<= M -300000.0)
     t_0
     (if (<= M -3.4e-65)
       (exp (- l))
       (if (<= M 27.0) (exp (* -0.25 (pow m 2.0))) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-pow(M, 2.0));
	double tmp;
	if (M <= -300000.0) {
		tmp = t_0;
	} else if (M <= -3.4e-65) {
		tmp = exp(-l);
	} else if (M <= 27.0) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-(m_1 ** 2.0d0))
    if (m_1 <= (-300000.0d0)) then
        tmp = t_0
    else if (m_1 <= (-3.4d-65)) then
        tmp = exp(-l)
    else if (m_1 <= 27.0d0) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(-Math.pow(M, 2.0));
	double tmp;
	if (M <= -300000.0) {
		tmp = t_0;
	} else if (M <= -3.4e-65) {
		tmp = Math.exp(-l);
	} else if (M <= 27.0) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-math.pow(M, 2.0))
	tmp = 0
	if M <= -300000.0:
		tmp = t_0
	elif M <= -3.4e-65:
		tmp = math.exp(-l)
	elif M <= 27.0:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-(M ^ 2.0)))
	tmp = 0.0
	if (M <= -300000.0)
		tmp = t_0;
	elseif (M <= -3.4e-65)
		tmp = exp(Float64(-l));
	elseif (M <= 27.0)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-(M ^ 2.0));
	tmp = 0.0;
	if (M <= -300000.0)
		tmp = t_0;
	elseif (M <= -3.4e-65)
		tmp = exp(-l);
	elseif (M <= 27.0)
		tmp = exp((-0.25 * (m ^ 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]}, If[LessEqual[M, -300000.0], t$95$0, If[LessEqual[M, -3.4e-65], N[Exp[(-l)], $MachinePrecision], If[LessEqual[M, 27.0], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if M < -3e5 or 27 < M

   1. Initial program 74.6%

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

   3. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 0 98.5%

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

   7. Taylor expanded in M around inf 96.3%

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

      1. mul-1-neg 96.3%

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

   9. Simplified 96.3%

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

   if -3e5 < M < -3.39999999999999987e-65

   1. Initial program 72.9%

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

   3. Taylor expanded in K around 0 91.1%

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

      1. cos-neg 91.1%

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

   5. Simplified 91.1%

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

   6. Taylor expanded in M around 0 92.8%

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

   7. Taylor expanded in l around inf 74.5%

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

      1. mul-1-neg 74.5%

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

   9. Simplified 74.5%

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

   if -3.39999999999999987e-65 < M < 27

   1. Initial program 75.9%

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

   3. Taylor expanded in K around 0 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 0 94.7%

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

   7. Taylor expanded in m around inf 59.8%

      \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -300000:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{elif}\;M \leq -3.4 \cdot 10^{-65}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.46 \cdot 10^{+30}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-187}:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -2.46e+30)
   (exp (* -0.25 (pow m 2.0)))
   (if (<= m 2e-187) (exp (- (pow M 2.0))) (exp (* -0.25 (pow n 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.46e+30) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else if (m <= 2e-187) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = 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) :: tmp
    if (m <= (-2.46d+30)) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= 2d-187) then
        tmp = exp(-(m_1 ** 2.0d0))
    else
        tmp = 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 tmp;
	if (m <= -2.46e+30) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= 2e-187) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -2.46e+30:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= 2e-187:
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -2.46e+30)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	elseif (m <= 2e-187)
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -2.46e+30)
		tmp = exp((-0.25 * (m ^ 2.0)));
	elseif (m <= 2e-187)
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -2.46e+30], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 2e-187], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.4599999999999999e30

   1. Initial program 70.4%

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

   3. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 0 98.1%

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

   7. Taylor expanded in m around inf 98.2%

      \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]

   if -2.4599999999999999e30 < m < 2e-187

   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. Add Preprocessing

   3. Taylor expanded in K around 0 96.1%

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

      1. cos-neg 96.1%

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

   5. Simplified 96.1%

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

   6. Taylor expanded in M around 0 95.3%

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

   7. Taylor expanded in M around inf 61.7%

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

      1. mul-1-neg 61.7%

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

   9. Simplified 61.7%

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

   if 2e-187 < m

   1. Initial program 72.3%

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

   3. Taylor expanded in K around 0 97.1%

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

      1. cos-neg 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in M around 0 97.1%

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

   7. Taylor expanded in n around inf 51.0%

      \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.46 \cdot 10^{+30}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-187}:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -300000 \lor \neg \left(M \leq 1.95 \cdot 10^{-14}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -300000.0) (not (<= M 1.95e-14)))
   (exp (- (pow M 2.0)))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -300000.0) || !(M <= 1.95e-14)) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-300000.0d0)) .or. (.not. (m_1 <= 1.95d-14))) then
        tmp = exp(-(m_1 ** 2.0d0))
    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 <= -300000.0) || !(M <= 1.95e-14)) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -300000.0) or not (M <= 1.95e-14):
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -300000.0) || !(M <= 1.95e-14))
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -300000.0) || ~((M <= 1.95e-14)))
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -300000.0], N[Not[LessEqual[M, 1.95e-14]], $MachinePrecision]], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -3e5 or 1.9499999999999999e-14 < M

   1. Initial program 74.5%

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

   3. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 0 98.5%

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

   7. Taylor expanded in M around inf 94.3%

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

      1. mul-1-neg 94.3%

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

   9. Simplified 94.3%

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

   if -3e5 < M < 1.9499999999999999e-14

   1. Initial program 75.8%

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

   3. Taylor expanded in K around 0 94.2%

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

      1. cos-neg 94.2%

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

   5. Simplified 94.2%

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

   6. Taylor expanded in M around 0 94.3%

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

   7. Taylor expanded in l around inf 44.4%

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

      1. mul-1-neg 44.4%

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

   9. Simplified 44.4%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -300000 \lor \neg \left(M \leq 1.95 \cdot 10^{-14}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

3. Taylor expanded in l around inf 30.2%

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

   1. mul-1-neg 35.7%

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

5. Simplified 30.2%

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

6. Taylor expanded in K around 0 36.5%

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

   1. cos-neg 36.5%

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

8. Simplified 36.5%

   \[\leadsto \color{blue}{\cos M \cdot e^{-\ell}} \]
9. Add Preprocessing

Alternative 9: 35.7% 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 75.1%

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

3. Taylor expanded in K around 0 97.3%

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

   1. cos-neg 97.3%

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

5. Simplified 97.3%

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

6. Taylor expanded in M around 0 96.6%

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

7. Taylor expanded in l around inf 35.7%

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

   1. mul-1-neg 35.7%

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

9. Simplified 35.7%

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

10. Final simplification 35.7%

    \[\leadsto e^{-\ell} \]
11. Add Preprocessing

Alternative 10: 6.9% 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 75.1%

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

3. Taylor expanded in l around inf 30.2%

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

   1. mul-1-neg 35.7%

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

5. Simplified 30.2%

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

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. Taylor expanded in K around 0 5.3%

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

   1. cos-neg 5.3%

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

9. Simplified 5.3%

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

10. Taylor expanded in M around 0 5.4%

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

Reproduce

herbie shell --seed 2024157 
(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)))))))