Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.2% → 96.9%

Time: 20.0s

Alternatives: 7

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: 76.2% 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: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. associate-/l* 74.6%

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

   2. associate--r- 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in K around 0 98.3%

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

   1. cos-neg 98.3%

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

6. Simplified 98.3%

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

7. Final simplification 98.3%

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

Alternative 2: 90.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ t_1 := e^{M \cdot M + \left(\ell - t_0\right)}\\ \mathbf{if}\;M \leq -3.3 \cdot 10^{+163}:\\ \;\;\;\;\frac{1}{t_1}\\ \mathbf{elif}\;M \leq 1.48 \cdot 10^{+29}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))) (t_1 (exp (+ (* M M) (- l t_0)))))
   (if (<= M -3.3e+163)
     (/ 1.0 t_1)
     (if (<= M 1.48e+29)
       (exp (- t_0 (+ l (* 0.25 (pow (+ m n) 2.0)))))
       (/ (cos M) t_1)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double t_1 = exp(((M * M) + (l - t_0)));
	double tmp;
	if (M <= -3.3e+163) {
		tmp = 1.0 / t_1;
	} else if (M <= 1.48e+29) {
		tmp = exp((t_0 - (l + (0.25 * pow((m + n), 2.0)))));
	} else {
		tmp = cos(M) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((m - n))
    t_1 = exp(((m_1 * m_1) + (l - t_0)))
    if (m_1 <= (-3.3d+163)) then
        tmp = 1.0d0 / t_1
    else if (m_1 <= 1.48d+29) then
        tmp = exp((t_0 - (l + (0.25d0 * ((m + n) ** 2.0d0)))))
    else
        tmp = cos(m_1) / t_1
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((m - n));
	double t_1 = Math.exp(((M * M) + (l - t_0)));
	double tmp;
	if (M <= -3.3e+163) {
		tmp = 1.0 / t_1;
	} else if (M <= 1.48e+29) {
		tmp = Math.exp((t_0 - (l + (0.25 * Math.pow((m + n), 2.0)))));
	} else {
		tmp = Math.cos(M) / t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	t_1 = math.exp(((M * M) + (l - t_0)))
	tmp = 0
	if M <= -3.3e+163:
		tmp = 1.0 / t_1
	elif M <= 1.48e+29:
		tmp = math.exp((t_0 - (l + (0.25 * math.pow((m + n), 2.0)))))
	else:
		tmp = math.cos(M) / t_1
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	t_1 = exp(Float64(Float64(M * M) + Float64(l - t_0)))
	tmp = 0.0
	if (M <= -3.3e+163)
		tmp = Float64(1.0 / t_1);
	elseif (M <= 1.48e+29)
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * (Float64(m + n) ^ 2.0)))));
	else
		tmp = Float64(cos(M) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	t_1 = exp(((M * M) + (l - t_0)));
	tmp = 0.0;
	if (M <= -3.3e+163)
		tmp = 1.0 / t_1;
	elseif (M <= 1.48e+29)
		tmp = exp((t_0 - (l + (0.25 * ((m + n) ^ 2.0)))));
	else
		tmp = cos(M) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(M * M), $MachinePrecision] + N[(l - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -3.3e+163], N[(1.0 / t$95$1), $MachinePrecision], If[LessEqual[M, 1.48e+29], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -3.3e163

   1. Initial program 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in M around inf 73.8%

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

      1. unpow2 73.8%

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

   6. Simplified 73.8%

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

   7. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in M around 0 100.0%

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

   if -3.3e163 < M < 1.48e29

   1. Initial program 74.9%

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

      1. associate-/l* 75.5%

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

      2. associate--r- 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in K around 0 97.5%

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

      1. cos-neg 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in M around 0 94.2%

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

      1. fabs-sub 94.2%

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

      2. +-commutative 94.2%

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

   9. Simplified 94.2%

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

   if 1.48e29 < M

   1. Initial program 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in M around inf 67.5%

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

      1. unpow2 67.5%

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

   6. Simplified 67.5%

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

   7. Taylor expanded in K around 0 90.8%

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

      1. cos-neg 90.8%

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

   9. Simplified 90.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3.3 \cdot 10^{+163}:\\ \;\;\;\;\frac{1}{e^{M \cdot M + \left(\ell - \left|m - n\right|\right)}}\\ \mathbf{elif}\;M \leq 1.48 \cdot 10^{+29}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M + \left(\ell - \left|m - n\right|\right)}}\\ \end{array} \]

Alternative 3: 69.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;m \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{elif}\;m \leq -3.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M + \left(\ell - t_0\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (<= m -3.8e-7)
     (exp (- t_0 (+ l (* 0.25 (* m m)))))
     (if (<= m -3.5e-224)
       (/ (cos M) (exp (+ (* M M) (- l t_0))))
       (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((m - n));
	double tmp;
	if (m <= -3.8e-7) {
		tmp = exp((t_0 - (l + (0.25 * (m * m)))));
	} else if (m <= -3.5e-224) {
		tmp = cos(M) / exp(((M * M) + (l - t_0)));
	} 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((m - n))
    if (m <= (-3.8d-7)) then
        tmp = exp((t_0 - (l + (0.25d0 * (m * m)))))
    else if (m <= (-3.5d-224)) then
        tmp = cos(m_1) / exp(((m_1 * m_1) + (l - t_0)))
    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((m - n));
	double tmp;
	if (m <= -3.8e-7) {
		tmp = Math.exp((t_0 - (l + (0.25 * (m * m)))));
	} else if (m <= -3.5e-224) {
		tmp = Math.cos(M) / Math.exp(((M * M) + (l - t_0)));
	} 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((m - n))
	tmp = 0
	if m <= -3.8e-7:
		tmp = math.exp((t_0 - (l + (0.25 * (m * m)))))
	elif m <= -3.5e-224:
		tmp = math.cos(M) / math.exp(((M * M) + (l - t_0)))
	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(m - n))
	tmp = 0.0
	if (m <= -3.8e-7)
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * Float64(m * m)))));
	elseif (m <= -3.5e-224)
		tmp = Float64(cos(M) / exp(Float64(Float64(M * M) + Float64(l - t_0))));
	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((m - n));
	tmp = 0.0;
	if (m <= -3.8e-7)
		tmp = exp((t_0 - (l + (0.25 * (m * m)))));
	elseif (m <= -3.5e-224)
		tmp = cos(M) / exp(((M * M) + (l - t_0)));
	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[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -3.8e-7], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -3.5e-224], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(M * M), $MachinePrecision] + N[(l - t$95$0), $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 m < -3.80000000000000015e-7

   1. Initial program 67.8%

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

      1. associate-/l* 67.8%

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

      2. associate--r- 67.8%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 98.3%

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

      1. fabs-sub 98.3%

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

      2. +-commutative 98.3%

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

   9. Simplified 98.3%

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

   10. Taylor expanded in m around inf 88.3%

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

       1. unpow2 88.3%

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

   12. Simplified 88.3%

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

   if -3.80000000000000015e-7 < m < -3.50000000000000019e-224

   1. Initial program 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in M around inf 59.0%

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

      1. unpow2 59.0%

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

   6. Simplified 59.0%

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

   7. Taylor expanded in K around 0 79.1%

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

      1. cos-neg 79.1%

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

   9. Simplified 79.1%

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

   if -3.50000000000000019e-224 < m

   1. Initial program 77.4%

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

      1. associate-/l* 77.4%

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

      2. associate--r- 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in K around 0 98.8%

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

      1. cos-neg 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in M around 0 86.7%

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

      1. fabs-sub 86.7%

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

      2. +-commutative 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in m around 0 70.5%

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

       1. unpow2 70.5%

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

   12. Simplified 70.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{elif}\;m \leq -3.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M + \left(\ell - \left|m - n\right|\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \]

Alternative 4: 69.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;m \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{elif}\;m \leq -4.6 \cdot 10^{-225}:\\ \;\;\;\;\frac{1}{e^{M \cdot M + \left(\ell - t_0\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (<= m -3.8e-7)
     (exp (- t_0 (+ l (* 0.25 (* m m)))))
     (if (<= m -4.6e-225)
       (/ 1.0 (exp (+ (* M M) (- l t_0))))
       (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((m - n));
	double tmp;
	if (m <= -3.8e-7) {
		tmp = exp((t_0 - (l + (0.25 * (m * m)))));
	} else if (m <= -4.6e-225) {
		tmp = 1.0 / exp(((M * M) + (l - t_0)));
	} 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((m - n))
    if (m <= (-3.8d-7)) then
        tmp = exp((t_0 - (l + (0.25d0 * (m * m)))))
    else if (m <= (-4.6d-225)) then
        tmp = 1.0d0 / exp(((m_1 * m_1) + (l - t_0)))
    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((m - n));
	double tmp;
	if (m <= -3.8e-7) {
		tmp = Math.exp((t_0 - (l + (0.25 * (m * m)))));
	} else if (m <= -4.6e-225) {
		tmp = 1.0 / Math.exp(((M * M) + (l - t_0)));
	} 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((m - n))
	tmp = 0
	if m <= -3.8e-7:
		tmp = math.exp((t_0 - (l + (0.25 * (m * m)))))
	elif m <= -4.6e-225:
		tmp = 1.0 / math.exp(((M * M) + (l - t_0)))
	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(m - n))
	tmp = 0.0
	if (m <= -3.8e-7)
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * Float64(m * m)))));
	elseif (m <= -4.6e-225)
		tmp = Float64(1.0 / exp(Float64(Float64(M * M) + Float64(l - t_0))));
	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((m - n));
	tmp = 0.0;
	if (m <= -3.8e-7)
		tmp = exp((t_0 - (l + (0.25 * (m * m)))));
	elseif (m <= -4.6e-225)
		tmp = 1.0 / exp(((M * M) + (l - t_0)));
	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[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -3.8e-7], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -4.6e-225], N[(1.0 / N[Exp[N[(N[(M * M), $MachinePrecision] + N[(l - t$95$0), $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 m < -3.80000000000000015e-7

   1. Initial program 67.8%

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

      1. associate-/l* 67.8%

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

      2. associate--r- 67.8%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 98.3%

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

      1. fabs-sub 98.3%

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

      2. +-commutative 98.3%

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

   9. Simplified 98.3%

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

   10. Taylor expanded in m around inf 88.3%

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

       1. unpow2 88.3%

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

   12. Simplified 88.3%

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

   if -3.80000000000000015e-7 < m < -4.5999999999999998e-225

   1. Initial program 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in M around inf 59.0%

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

      1. unpow2 59.0%

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

   6. Simplified 59.0%

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

   7. Taylor expanded in K around 0 79.1%

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

      1. cos-neg 79.1%

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

   9. Simplified 79.1%

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

   10. Taylor expanded in M around 0 79.1%

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

   if -4.5999999999999998e-225 < m

   1. Initial program 77.4%

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

      1. associate-/l* 77.4%

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

      2. associate--r- 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in K around 0 98.8%

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

      1. cos-neg 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in M around 0 86.7%

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

      1. fabs-sub 86.7%

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

      2. +-commutative 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in m around 0 70.5%

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

       1. unpow2 70.5%

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

   12. Simplified 70.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{elif}\;m \leq -4.6 \cdot 10^{-225}:\\ \;\;\;\;\frac{1}{e^{M \cdot M + \left(\ell - \left|m - n\right|\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \]

Alternative 5: 70.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;m \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (<= m -3.8e-7)
     (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((m - n));
	double tmp;
	if (m <= -3.8e-7) {
		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((m - n))
    if (m <= (-3.8d-7)) 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((m - n));
	double tmp;
	if (m <= -3.8e-7) {
		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((m - n))
	tmp = 0
	if m <= -3.8e-7:
		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(m - n))
	tmp = 0.0
	if (m <= -3.8e-7)
		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((m - n));
	tmp = 0.0;
	if (m <= -3.8e-7)
		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[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -3.8e-7], 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 m < -3.80000000000000015e-7

   1. Initial program 67.8%

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

      1. associate-/l* 67.8%

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

      2. associate--r- 67.8%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 98.3%

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

      1. fabs-sub 98.3%

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

      2. +-commutative 98.3%

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

   9. Simplified 98.3%

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

   10. Taylor expanded in m around inf 88.3%

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

       1. unpow2 88.3%

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

   12. Simplified 88.3%

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

   if -3.80000000000000015e-7 < m

   1. Initial program 76.2%

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

      1. associate-/l* 76.7%

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

      2. associate--r- 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in K around 0 97.8%

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

      1. cos-neg 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in M around 0 81.6%

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

      1. fabs-sub 81.6%

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

      2. +-commutative 81.6%

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

   9. Simplified 81.6%

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

   10. Taylor expanded in m around 0 69.1%

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

       1. unpow2 69.1%

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

   12. Simplified 69.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \]

Alternative 6: 60.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. associate-/l* 74.6%

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

   2. associate--r- 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in K around 0 98.3%

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

   1. cos-neg 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in M around 0 85.4%

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

   1. fabs-sub 85.4%

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

   2. +-commutative 85.4%

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

9. Simplified 85.4%

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

10. Taylor expanded in m around inf 61.2%

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

    1. unpow2 61.2%

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

12. Simplified 61.2%

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

13. Final simplification 61.2%

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

Alternative 7: 24.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\left|m - n\right| - \ell} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

3. Simplified 74.3%

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

4. Taylor expanded in M around inf 46.1%

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

   1. unpow2 46.1%

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

6. Simplified 46.1%

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

7. Taylor expanded in K around 0 58.7%

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

   1. cos-neg 58.7%

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

9. Simplified 58.7%

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

10. Taylor expanded in M around 0 25.4%

    \[\leadsto \color{blue}{\frac{1}{e^{\ell - \left|n - m\right|}}} \]
11. Step-by-step derivation

    1. rec-exp 25.4%

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

    2. fabs-sub 25.4%

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

    3. sub-neg 25.4%

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

    4. +-commutative 25.4%

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

    5. distribute-neg-in 25.4%

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

    6. remove-double-neg 25.4%

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

    7. sub-neg 25.4%

       \[\leadsto e^{\color{blue}{\left|m - n\right| - \ell}} \]

    8. fabs-sub 25.4%

       \[\leadsto e^{\color{blue}{\left|n - m\right|} - \ell} \]

12. Simplified 25.4%

    \[\leadsto \color{blue}{e^{\left|n - m\right| - \ell}} \]

13. Final simplification 25.4%

    \[\leadsto e^{\left|m - n\right| - \ell} \]

Reproduce

herbie shell --seed 2023255 
(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)))))))