Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.5% → 96.5%

Time: 13.4s

Alternatives: 5

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Step-by-step derivation

   1. +-commutative 75.1%

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

   2. +-commutative 75.1%

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

   3. fabs-sub 75.1%

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

   4. associate-/l* 75.0%

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

   5. +-commutative 75.0%

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

3. Simplified 75.0%

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

4. Taylor expanded in K around 0 97.8%

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

   1. cos-neg 97.8%

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

6. Simplified 97.8%

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

7. Final simplification 97.8%

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

Alternative 2: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;M \leq -5.2 \cdot 10^{+124} \lor \neg \left(M \leq 3.1 \cdot 10^{+137}\right):\\ \;\;\;\;e^{t_0 - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (or (<= M -5.2e+124) (not (<= M 3.1e+137)))
     (exp (- t_0 (+ l (* M M))))
     (exp (- t_0 (+ l (* (pow (+ m n) 2.0) 0.25)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if ((M <= -5.2e+124) || !(M <= 3.1e+137)) {
		tmp = exp((t_0 - (l + (M * M))));
	} else {
		tmp = exp((t_0 - (l + (pow((m + n), 2.0) * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if ((m_1 <= (-5.2d+124)) .or. (.not. (m_1 <= 3.1d+137))) then
        tmp = exp((t_0 - (l + (m_1 * m_1))))
    else
        tmp = exp((t_0 - (l + (((m + n) ** 2.0d0) * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if ((M <= -5.2e+124) || !(M <= 3.1e+137)) {
		tmp = Math.exp((t_0 - (l + (M * M))));
	} else {
		tmp = Math.exp((t_0 - (l + (Math.pow((m + n), 2.0) * 0.25))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if (M <= -5.2e+124) or not (M <= 3.1e+137):
		tmp = math.exp((t_0 - (l + (M * M))))
	else:
		tmp = math.exp((t_0 - (l + (math.pow((m + n), 2.0) * 0.25))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if ((M <= -5.2e+124) || !(M <= 3.1e+137))
		tmp = exp(Float64(t_0 - Float64(l + Float64(M * M))));
	else
		tmp = exp(Float64(t_0 - Float64(l + Float64((Float64(m + n) ^ 2.0) * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if ((M <= -5.2e+124) || ~((M <= 3.1e+137)))
		tmp = exp((t_0 - (l + (M * M))));
	else
		tmp = exp((t_0 - (l + (((m + n) ^ 2.0) * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[M, -5.2e+124], N[Not[LessEqual[M, 3.1e+137]], $MachinePrecision]], N[Exp[N[(t$95$0 - N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - N[(l + N[(N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -5.2000000000000001e124 or 3.0999999999999999e137 < M

   1. Initial program 71.4%

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

      1. +-commutative 71.4%

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

      2. +-commutative 71.4%

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

      3. fabs-sub 71.4%

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

      4. associate-/l* 71.4%

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

      5. +-commutative 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in M around inf 71.4%

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

      1. unpow2 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in n around inf 84.4%

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

   8. Taylor expanded in K around 0 100.0%

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

   if -5.2000000000000001e124 < M < 3.0999999999999999e137

   1. Initial program 76.7%

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

      1. +-commutative 76.7%

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

      2. +-commutative 76.7%

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

      3. fabs-sub 76.7%

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

      4. associate-/l* 76.6%

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

      5. +-commutative 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in K around 0 84.6%

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

      1. cos-neg 84.6%

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

      2. sin-neg 84.6%

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

   6. Simplified 84.6%

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

   7. Taylor expanded in M around 0 91.4%

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

      1. *-commutative 91.4%

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

   9. Simplified 91.4%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -5.2 \cdot 10^{+124} \lor \neg \left(M \leq 3.1 \cdot 10^{+137}\right):\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)}\\ \end{array} \]

Alternative 3: 68.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq -3 \cdot 10^{-265}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;e^{t_0 - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n -3e-265)
     (exp (- t_0 (+ l (* 0.25 (* m m)))))
     (if (<= n 1.0)
       (exp (- t_0 (+ l (* M M))))
       (exp (- t_0 (+ l (* 0.25 (* n n)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= -3e-265) {
		tmp = exp((t_0 - (l + (0.25 * (m * m)))));
	} else if (n <= 1.0) {
		tmp = exp((t_0 - (l + (M * M))));
	} else {
		tmp = exp((t_0 - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (n <= (-3d-265)) then
        tmp = exp((t_0 - (l + (0.25d0 * (m * m)))))
    else if (n <= 1.0d0) then
        tmp = exp((t_0 - (l + (m_1 * m_1))))
    else
        tmp = exp((t_0 - (l + (0.25d0 * (n * n)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (n <= -3e-265) {
		tmp = Math.exp((t_0 - (l + (0.25 * (m * m)))));
	} else if (n <= 1.0) {
		tmp = Math.exp((t_0 - (l + (M * M))));
	} else {
		tmp = Math.exp((t_0 - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= -3e-265:
		tmp = math.exp((t_0 - (l + (0.25 * (m * m)))))
	elif n <= 1.0:
		tmp = math.exp((t_0 - (l + (M * M))))
	else:
		tmp = math.exp((t_0 - (l + (0.25 * (n * n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= -3e-265)
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * Float64(m * m)))));
	elseif (n <= 1.0)
		tmp = exp(Float64(t_0 - Float64(l + Float64(M * M))));
	else
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (n <= -3e-265)
		tmp = exp((t_0 - (l + (0.25 * (m * m)))));
	elseif (n <= 1.0)
		tmp = exp((t_0 - (l + (M * M))));
	else
		tmp = exp((t_0 - (l + (0.25 * (n * n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -3e-265], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.0], N[Exp[N[(t$95$0 - N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.9999999999999998e-265

   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. Step-by-step derivation

      1. +-commutative 75.9%

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

      2. +-commutative 75.9%

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

      3. fabs-sub 75.9%

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

      4. associate-/l* 75.9%

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

      5. +-commutative 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in K around 0 80.5%

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

      1. cos-neg 80.5%

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

      2. sin-neg 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in M around 0 84.3%

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

      1. *-commutative 84.3%

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

   9. Simplified 84.3%

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

   10. Taylor expanded in m around inf 59.7%

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

       1. unpow2 59.7%

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

   12. Simplified 59.7%

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

   if -2.9999999999999998e-265 < n < 1

   1. Initial program 82.7%

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

      1. +-commutative 82.7%

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

      2. +-commutative 82.7%

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

      3. fabs-sub 82.7%

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

      4. associate-/l* 82.5%

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

      5. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in M around inf 70.2%

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

      1. unpow2 70.2%

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

   6. Simplified 70.2%

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

   7. Taylor expanded in n around inf 80.5%

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

   8. Taylor expanded in K around 0 80.6%

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

   if 1 < n

   1. Initial program 64.1%

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

      1. +-commutative 64.1%

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

      2. +-commutative 64.1%

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

      3. fabs-sub 64.1%

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

      4. associate-/l* 64.1%

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

      5. +-commutative 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in K around 0 71.9%

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

      1. cos-neg 71.9%

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

      2. sin-neg 71.9%

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

   6. Simplified 71.9%

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

   7. Taylor expanded in M around 0 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in m around 0 84.6%

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

       1. unpow2 84.6%

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

   12. Simplified 84.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-265}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \]

Alternative 4: 71.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 9 \cdot 10^{+24}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 9e+24)
     (exp (- t_0 (+ l (* 0.25 (* m m)))))
     (exp (- t_0 (+ l (* 0.25 (* n n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= 9e+24) {
		tmp = exp((t_0 - (l + (0.25 * (m * m)))));
	} else {
		tmp = exp((t_0 - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (n <= 9d+24) then
        tmp = exp((t_0 - (l + (0.25d0 * (m * m)))))
    else
        tmp = exp((t_0 - (l + (0.25d0 * (n * n)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (n <= 9e+24) {
		tmp = Math.exp((t_0 - (l + (0.25 * (m * m)))));
	} else {
		tmp = Math.exp((t_0 - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 9e+24:
		tmp = math.exp((t_0 - (l + (0.25 * (m * m)))))
	else:
		tmp = math.exp((t_0 - (l + (0.25 * (n * n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 9e+24)
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * Float64(m * m)))));
	else
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (n <= 9e+24)
		tmp = exp((t_0 - (l + (0.25 * (m * m)))));
	else
		tmp = exp((t_0 - (l + (0.25 * (n * n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 9e+24], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 9.00000000000000039e24

   1. Initial program 79.4%

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

      1. +-commutative 79.4%

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

      2. +-commutative 79.4%

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

      3. fabs-sub 79.4%

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

      4. associate-/l* 79.3%

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

      5. +-commutative 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in K around 0 84.0%

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

      1. cos-neg 84.0%

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

      2. sin-neg 84.0%

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

   6. Simplified 84.0%

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

   7. Taylor expanded in M around 0 82.4%

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

      1. *-commutative 82.4%

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

   9. Simplified 82.4%

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

   10. Taylor expanded in m around inf 68.0%

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

       1. unpow2 68.0%

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

   12. Simplified 68.0%

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

   if 9.00000000000000039e24 < n

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

      2. +-commutative 60.3%

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

      3. fabs-sub 60.3%

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

      4. associate-/l* 60.3%

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

      5. +-commutative 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in K around 0 69.0%

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

      1. cos-neg 69.0%

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

      2. sin-neg 69.0%

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

   6. Simplified 69.0%

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

   7. Taylor expanded in M around 0 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in m around 0 89.8%

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

       1. unpow2 89.8%

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

   12. Simplified 89.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 9 \cdot 10^{+24}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \]

Alternative 5: 61.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (fabs (- n m)) (+ l (* 0.25 (* m m))))))

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - (l + (0.25 * (m * 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 = exp((abs((n - m)) - (l + (0.25d0 * (m * m)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.25 * N[(m * m), $MachinePrecision]), $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. Step-by-step derivation

   1. +-commutative 75.1%

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

   2. +-commutative 75.1%

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

   3. fabs-sub 75.1%

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

   4. associate-/l* 75.0%

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

   5. +-commutative 75.0%

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

3. Simplified 75.0%

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

4. Taylor expanded in K around 0 80.6%

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

   1. cos-neg 80.6%

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

   2. sin-neg 80.6%

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

6. Simplified 80.6%

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

7. Taylor expanded in M around 0 86.4%

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

   1. *-commutative 86.4%

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

9. Simplified 86.4%

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

10. Taylor expanded in m around inf 63.7%

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

    1. unpow2 63.7%

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

12. Simplified 63.7%

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

13. Final simplification 63.7%

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

Reproduce

herbie shell --seed 2023292 
(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)))))))