Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.9% → 96.8%

Time: 20.2s

Alternatives: 10

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in K around 0 97.0%

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

   1. cos-neg 97.0%

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

5. Simplified 97.0%

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

6. Final simplification 97.0%

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

Alternative 2: 96.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(K, m, n, M, l)
	return exp(1) ^ Float64(Float64(abs(Float64(m - n)) - l) - (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 2.71828182845904523536 ^ ((abs((m - n)) - l) - ((((m + n) * 0.5) - M) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in n around inf 84.7%

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

   1. associate-*r* 84.7%

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

5. Simplified 84.7%

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

6. Taylor expanded in K around 0 96.2%

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

   1. *-un-lft-identity 96.2%

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

   2. exp-prod 96.2%

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

   3. associate--r+ 96.2%

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

   4. fabs-sub 96.2%

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

   5. fma-neg 96.2%

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

8. Applied egg-rr 96.2%

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

   1. exp-1-e 96.2%

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

   2. fabs-sub 96.2%

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

   3. unpow2 96.2%

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

   4. cancel-sign-sub-inv 96.2%

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

   5. +-commutative 96.2%

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

   6. *-lft-identity 96.2%

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

   7. metadata-eval 96.2%

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

   8. cancel-sign-sub-inv 96.2%

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

   9. fma-neg 96.2%

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

   10. +-commutative 96.2%

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

   11. *-lft-identity 96.2%

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

   12. metadata-eval 96.2%

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

   13. cancel-sign-sub-inv 96.2%

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

   14. fma-neg 96.2%

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

10. Simplified 96.2%

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

11. Final simplification 96.2%

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

Alternative 3: 96.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp((abs((m - n)) - (l + ((((m + n) * 0.5d0) - m_1) ** 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in n around inf 84.7%

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

   1. associate-*r* 84.7%

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

5. Simplified 84.7%

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

6. Taylor expanded in K around 0 96.2%

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

7. Final simplification 96.2%

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

Alternative 4: 87.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1000000000:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \left(\ell - \left|m - n\right|\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1000000000.0)
   (exp (* (pow m 2.0) -0.25))
   (pow E (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1000000000.0) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = pow(((double) M_E), ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - fabs((m - n)))));
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1000000000.0) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.pow(Math.E, ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - Math.abs((m - n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1000000000.0:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.pow(math.e, ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - math.fabs((m - n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1000000000.0)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	else
		tmp = exp(1) ^ Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - Float64(l - abs(Float64(m - n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1000000000.0)
		tmp = exp(((m ^ 2.0) * -0.25));
	else
		tmp = 2.71828182845904523536 ^ ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - abs((m - n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1000000000.0], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[Power[E, N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1e9

   1. Initial program 68.9%

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

   3. Taylor expanded in n around inf 91.8%

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

      1. associate-*r* 91.8%

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

   5. Simplified 91.8%

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

   6. Taylor expanded in K around 0 100.0%

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

   7. Taylor expanded in n around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. unpow2 91.9%

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

      3. distribute-rgt-out 95.2%

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

   9. Simplified 95.2%

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

   10. Taylor expanded in m around inf 100.0%

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

       1. *-commutative 100.0%

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

   12. Simplified 100.0%

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

   if -1e9 < m

   1. Initial program 77.9%

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

   3. Taylor expanded in n around inf 82.4%

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

      1. associate-*r* 82.4%

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

   5. Simplified 82.4%

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

   6. Taylor expanded in K around 0 95.0%

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

      1. *-un-lft-identity 95.0%

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

      2. exp-prod 95.0%

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

      3. associate--r+ 95.0%

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

      4. fabs-sub 95.0%

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

      5. fma-neg 95.0%

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

   8. Applied egg-rr 95.0%

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

      1. exp-1-e 95.0%

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

      2. fabs-sub 95.0%

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

      3. unpow2 95.0%

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

      4. cancel-sign-sub-inv 95.0%

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

      5. +-commutative 95.0%

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

      6. *-lft-identity 95.0%

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

      7. metadata-eval 95.0%

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

      8. cancel-sign-sub-inv 95.0%

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

      9. fma-neg 95.0%

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

      10. +-commutative 95.0%

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

      11. *-lft-identity 95.0%

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

      12. metadata-eval 95.0%

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

      13. cancel-sign-sub-inv 95.0%

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

      14. fma-neg 95.0%

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

   10. Simplified 95.0%

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

   11. Taylor expanded in m around 0 79.3%

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

       1. +-commutative 79.3%

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

       2. unpow2 79.3%

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

       3. distribute-rgt-out 81.4%

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

   13. Simplified 81.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1000000000:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \left(\ell - \left|m - n\right|\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{-13}:\\ \;\;\;\;e^{\left|m - n\right| + \left(\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-{M}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M 1.1e-13)
   (exp (+ (fabs (- m n)) (- (* (+ n (- (* m 0.5) M)) (- M (* m 0.5))) l)))
   (exp (- (pow M 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= 1.1e-13) {
		tmp = exp((fabs((m - n)) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - l)));
	} else {
		tmp = exp(-pow(M, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m_1 <= 1.1d-13) then
        tmp = exp((abs((m - n)) + (((n + ((m * 0.5d0) - m_1)) * (m_1 - (m * 0.5d0))) - l)))
    else
        tmp = exp(-(m_1 ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= 1.1e-13) {
		tmp = Math.exp((Math.abs((m - n)) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - l)));
	} else {
		tmp = Math.exp(-Math.pow(M, 2.0));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= 1.1e-13:
		tmp = math.exp((math.fabs((m - n)) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - l)))
	else:
		tmp = math.exp(-math.pow(M, 2.0))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= 1.1e-13)
		tmp = exp(Float64(abs(Float64(m - n)) + Float64(Float64(Float64(n + Float64(Float64(m * 0.5) - M)) * Float64(M - Float64(m * 0.5))) - l)));
	else
		tmp = exp(Float64(-(M ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= 1.1e-13)
		tmp = exp((abs((m - n)) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - l)));
	else
		tmp = exp(-(M ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, 1.1e-13], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(n + N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.09999999999999998e-13

   1. Initial program 74.7%

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

   3. Taylor expanded in n around inf 83.7%

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

      1. associate-*r* 83.7%

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

   5. Simplified 83.7%

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

   6. Taylor expanded in K around 0 95.3%

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

   7. Taylor expanded in n around 0 70.0%

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

      1. +-commutative 70.0%

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

      2. unpow2 70.0%

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

      3. distribute-rgt-out 75.9%

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

   9. Simplified 75.9%

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

   if 1.09999999999999998e-13 < M

   1. Initial program 78.6%

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

   3. Taylor expanded in n around inf 87.1%

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

      1. associate-*r* 87.1%

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

   5. Simplified 87.1%

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

   6. Taylor expanded in K around 0 98.6%

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

   7. Taylor expanded in n around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. unpow2 77.3%

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

      3. distribute-rgt-out 87.3%

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

   9. Simplified 87.3%

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

   10. Taylor expanded in M around inf 95.8%

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

       1. mul-1-neg 95.8%

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

   12. Simplified 95.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{-13}:\\ \;\;\;\;e^{\left|m - n\right| + \left(\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -110:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -110.0)
   (exp (* (pow m 2.0) -0.25))
   (exp (+ (fabs (- m n)) (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -110.0) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = exp((fabs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-110.0d0)) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = exp((abs((m - n)) + ((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -110.0) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.exp((Math.abs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -110.0:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.exp((math.fabs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -110.0)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	else
		tmp = exp(Float64(abs(Float64(m - n)) + Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -110.0)
		tmp = exp(((m ^ 2.0) * -0.25));
	else
		tmp = exp((abs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -110.0], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -110

   1. Initial program 68.9%

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

   3. Taylor expanded in n around inf 91.8%

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

      1. associate-*r* 91.8%

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

   5. Simplified 91.8%

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

   6. Taylor expanded in K around 0 100.0%

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

   7. Taylor expanded in n around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. unpow2 91.9%

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

      3. distribute-rgt-out 95.2%

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

   9. Simplified 95.2%

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

   10. Taylor expanded in m around inf 100.0%

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

       1. *-commutative 100.0%

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

   12. Simplified 100.0%

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

   if -110 < m

   1. Initial program 77.9%

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

   3. Taylor expanded in n around inf 82.4%

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

      1. associate-*r* 82.4%

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

   5. Simplified 82.4%

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

   6. Taylor expanded in K around 0 95.0%

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

   7. Taylor expanded in m around 0 79.3%

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

      1. +-commutative 79.3%

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

      2. unpow2 79.3%

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

      3. distribute-rgt-out 81.4%

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

   9. Simplified 81.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -110:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -21000000000000 \lor \neg \left(M \leq 1.1 \cdot 10^{-13}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -21000000000000.0) (not (<= M 1.1e-13)))
   (exp (- (pow M 2.0)))
   (exp (* (pow m 2.0) -0.25))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -21000000000000.0) || !(M <= 1.1e-13)) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = exp((pow(m, 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) :: tmp
    if ((m_1 <= (-21000000000000.0d0)) .or. (.not. (m_1 <= 1.1d-13))) then
        tmp = exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((m ** 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 tmp;
	if ((M <= -21000000000000.0) || !(M <= 1.1e-13)) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -21000000000000.0) or not (M <= 1.1e-13):
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -21000000000000.0) || !(M <= 1.1e-13))
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -21000000000000.0) || ~((M <= 1.1e-13)))
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp(((m ^ 2.0) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -21000000000000.0], N[Not[LessEqual[M, 1.1e-13]], $MachinePrecision]], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -2.1e13 or 1.09999999999999998e-13 < 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. Add Preprocessing

   3. Taylor expanded in n around inf 86.9%

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

      1. associate-*r* 86.9%

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

   5. Simplified 86.9%

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

   6. Taylor expanded in K around 0 98.5%

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

   7. Taylor expanded in n around 0 78.2%

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

      1. +-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. distribute-rgt-out 89.2%

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

   9. Simplified 89.2%

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

   10. Taylor expanded in M around inf 97.1%

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

       1. mul-1-neg 97.1%

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

   12. Simplified 97.1%

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

   if -2.1e13 < M < 1.09999999999999998e-13

   1. Initial program 73.9%

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

   3. Taylor expanded in n around inf 82.1%

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

      1. associate-*r* 82.1%

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

   5. Simplified 82.1%

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

   6. Taylor expanded in K around 0 93.5%

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

   7. Taylor expanded in n around 0 64.7%

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

      1. +-commutative 64.7%

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

      2. unpow2 64.7%

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

      3. distribute-rgt-out 67.3%

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

   9. Simplified 67.3%

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

   10. Taylor expanded in m around inf 59.5%

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

       1. *-commutative 59.5%

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

   12. Simplified 59.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -21000000000000 \lor \neg \left(M \leq 1.1 \cdot 10^{-13}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -27 \lor \neg \left(M \leq 1.75 \cdot 10^{-18}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -27.0) (not (<= M 1.75e-18)))
   (exp (- (pow M 2.0)))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -27.0) || !(M <= 1.75e-18)) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-27.0d0)) .or. (.not. (m_1 <= 1.75d-18))) then
        tmp = exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -27.0) || !(M <= 1.75e-18)) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -27.0) or not (M <= 1.75e-18):
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -27.0) || !(M <= 1.75e-18))
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -27.0) || ~((M <= 1.75e-18)))
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -27.0], N[Not[LessEqual[M, 1.75e-18]], $MachinePrecision]], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -27 or 1.7499999999999999e-18 < M

   1. Initial program 77.3%

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

   3. Taylor expanded in n around inf 87.2%

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

      1. associate-*r* 87.2%

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

   5. Simplified 87.2%

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

   6. Taylor expanded in K around 0 98.6%

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

   7. Taylor expanded in n around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. unpow2 77.4%

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

      3. distribute-rgt-out 88.1%

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

   9. Simplified 88.1%

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

   10. Taylor expanded in M around inf 96.5%

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

       1. mul-1-neg 96.5%

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

   12. Simplified 96.5%

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

   if -27 < M < 1.7499999999999999e-18

   1. Initial program 73.9%

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

   3. Taylor expanded in n around inf 81.5%

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

      1. associate-*r* 81.5%

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

   5. Simplified 81.5%

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

   6. Taylor expanded in K around 0 93.3%

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

   7. Taylor expanded in n around 0 65.2%

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

      1. +-commutative 65.2%

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

      2. unpow2 65.2%

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

      3. distribute-rgt-out 67.9%

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

   9. Simplified 67.9%

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

   10. Taylor expanded in l around inf 35.9%

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

       1. mul-1-neg 35.9%

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

   12. Simplified 35.9%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -27 \lor \neg \left(M \leq 1.75 \cdot 10^{-18}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 720.0) (exp (* n (- M (* m 0.5)))) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = exp((n * (M - (m * 0.5))));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 720.0d0) then
        tmp = exp((n * (m_1 - (m * 0.5d0))))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = Math.exp((n * (M - (m * 0.5))));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 720.0:
		tmp = math.exp((n * (M - (m * 0.5))))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 720.0)
		tmp = exp(Float64(n * Float64(M - Float64(m * 0.5))));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 720.0)
		tmp = exp((n * (M - (m * 0.5))));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 720.0], N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 720

   1. Initial program 75.0%

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

   3. Taylor expanded in n around inf 83.2%

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

      1. associate-*r* 83.2%

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

   5. Simplified 83.2%

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

   6. Taylor expanded in K around 0 95.2%

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

   7. Taylor expanded in n around 0 68.7%

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

      1. +-commutative 68.7%

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

      2. unpow2 68.7%

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

      3. distribute-rgt-out 76.6%

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

   9. Simplified 76.6%

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

   10. Taylor expanded in n around inf 43.1%

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

   if 720 < l

   1. Initial program 78.8%

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

   3. Taylor expanded in n around inf 90.4%

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

      1. associate-*r* 90.4%

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

   5. Simplified 90.4%

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

   6. Taylor expanded in K around 0 100.0%

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

   7. Taylor expanded in n around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. unpow2 84.7%

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

      3. distribute-rgt-out 88.6%

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

   9. Simplified 88.6%

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

   10. Taylor expanded in l around inf 100.0%

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

       1. mul-1-neg 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in n around inf 84.7%

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

   1. associate-*r* 84.7%

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

5. Simplified 84.7%

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

6. Taylor expanded in K around 0 96.2%

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

7. Taylor expanded in n around 0 72.0%

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

   1. +-commutative 72.0%

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

   2. unpow2 72.0%

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

   3. distribute-rgt-out 79.0%

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

9. Simplified 79.0%

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

10. Taylor expanded in l around inf 29.4%

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

    1. mul-1-neg 29.4%

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

12. Simplified 29.4%

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

13. Final simplification 29.4%

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

Reproduce

herbie shell --seed 2024039 
(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)))))))