Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.5% → 96.4%

Time: 15.4s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t\_0 \leq \infty:\\ \;\;\;\;t\_0 \cdot \cos \left({\left(\sqrt[3]{K}\right)}^{2} \cdot \left(\sqrt[3]{K} \cdot \left(\left(m + n\right) \cdot 0.5\right)\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(m - n\right) - \ell\right) + {n}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (- (fabs (- m n)) l) (pow (- (/ (+ m n) 2.0) M) 2.0)))))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_0) INFINITY)
     (* t_0 (cos (- (* (pow (cbrt K) 2.0) (* (cbrt K) (* (+ m n) 0.5))) M)))
     (exp (+ (- (- m n) l) (* (pow n 2.0) -0.25))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((fabs((m - n)) - l) - pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_0) <= ((double) INFINITY)) {
		tmp = t_0 * cos(((pow(cbrt(K), 2.0) * (cbrt(K) * ((m + n) * 0.5))) - M));
	} else {
		tmp = exp((((m - n) - l) + (pow(n, 2.0) * -0.25)));
	}
	return tmp;
}

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((Math.abs((m - n)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * Math.cos(((Math.pow(Math.cbrt(K), 2.0) * (Math.cbrt(K) * ((m + n) * 0.5))) - M));
	} else {
		tmp = Math.exp((((m - n) - l) + (Math.pow(n, 2.0) * -0.25)));
	}
	return tmp;
}

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(abs(Float64(m - n)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_0) <= Inf)
		tmp = Float64(t_0 * cos(Float64(Float64((cbrt(K) ^ 2.0) * Float64(cbrt(K) * Float64(Float64(m + n) * 0.5))) - M)));
	else
		tmp = exp(Float64(Float64(Float64(m - n) - l) + Float64((n ^ 2.0) * -0.25)));
	end
	return tmp
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = 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]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], Infinity], N[(t$95$0 * N[Cos[N[(N[(N[Power[N[Power[K, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[K, 1/3], $MachinePrecision] * N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision] + N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 97.3%

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

      1. associate-/l* 97.3%

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

      2. add-cube-cbrt 98.2%

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

      3. associate-*l* 98.6%

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

      4. pow2 98.6%

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

      5. div-inv 98.6%

         \[\leadsto \cos \left({\left(\sqrt[3]{K}\right)}^{2} \cdot \left(\sqrt[3]{K} \cdot \color{blue}{\left(\left(m + n\right) \cdot \frac{1}{2}\right)}\right) - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      6. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in K around 0 98.0%

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

      1. cos-neg 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in M around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in m around 0 61.4%

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

       1. fabs-sub 61.4%

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

       2. sub-neg 61.4%

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

       3. mul-1-neg 61.4%

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

       4. fabs-neg 61.4%

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

       5. associate--r+ 61.4%

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

       6. cancel-sign-sub-inv 61.4%

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

       7. fabs-neg 61.4%

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

       8. mul-1-neg 61.4%

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

       9. sub-neg 61.4%

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

       10. fabs-sub 61.4%

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

       11. rem-square-sqrt 39.5%

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

       12. fabs-sqr 39.5%

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

       13. rem-square-sqrt 84.6%

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

       14. metadata-eval 84.6%

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

       15. *-commutative 84.6%

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

   11. Simplified 84.6%

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

4. Final simplification 95.8%

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

Alternative 2: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t\_0 \leq 2:\\ \;\;\;\;t\_0 \cdot \cos \left(\frac{1}{\frac{\frac{2}{K}}{m + n}} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(m - n\right) - \ell\right) + {n}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (- (fabs (- m n)) l) (pow (- (/ (+ m n) 2.0) M) 2.0)))))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_0) 2.0)
     (* t_0 (cos (- (/ 1.0 (/ (/ 2.0 K) (+ m n))) M)))
     (exp (+ (- (- m n) l) (* (pow n 2.0) -0.25))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((fabs((m - n)) - l) - pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_0) <= 2.0) {
		tmp = t_0 * cos(((1.0 / ((2.0 / K) / (m + n))) - M));
	} else {
		tmp = exp((((m - n) - l) + (pow(n, 2.0) * -0.25)));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((abs((m - n)) - l) - ((((m + n) / 2.0d0) - m_1) ** 2.0d0)))
    if ((cos((((k * (m + n)) / 2.0d0) - m_1)) * t_0) <= 2.0d0) then
        tmp = t_0 * cos(((1.0d0 / ((2.0d0 / k) / (m + n))) - m_1))
    else
        tmp = exp((((m - n) - l) + ((n ** 2.0d0) * (-0.25d0))))
    end if
    code = tmp
end function

Java

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

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.exp(((math.fabs((m - n)) - l) - math.pow((((m + n) / 2.0) - M), 2.0)))
	tmp = 0
	if (math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= 2.0:
		tmp = t_0 * math.cos(((1.0 / ((2.0 / K) / (m + n))) - M))
	else:
		tmp = math.exp((((m - n) - l) + (math.pow(n, 2.0) * -0.25)))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(abs(Float64(m - n)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_0) <= 2.0)
		tmp = Float64(t_0 * cos(Float64(Float64(1.0 / Float64(Float64(2.0 / K) / Float64(m + n))) - M)));
	else
		tmp = exp(Float64(Float64(Float64(m - n) - l) + Float64((n ^ 2.0) * -0.25)));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((abs((m - n)) - l) - ((((m + n) / 2.0) - M) ^ 2.0)));
	tmp = 0.0;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_0) <= 2.0)
		tmp = t_0 * cos(((1.0 / ((2.0 / K) / (m + n))) - M));
	else
		tmp = exp((((m - n) - l) + ((n ^ 2.0) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = 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]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(t$95$0 * N[Cos[N[(N[(1.0 / N[(N[(2.0 / K), $MachinePrecision] / N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision] + N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < 2

   1. Initial program 98.7%

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

      1. clear-num 99.1%

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

      2. inv-pow 99.1%

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

      3. associate-/r* 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. unpow-1 99.1%

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

   6. Simplified 99.1%

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

   if 2 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 6.9%

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

   3. Taylor expanded in K around 0 93.1%

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

      1. cos-neg 93.1%

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

   5. Simplified 93.1%

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

   6. Taylor expanded in M around 0 94.8%

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

      1. *-commutative 94.8%

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

   8. Simplified 94.8%

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

   9. Taylor expanded in m around 0 60.9%

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

       1. fabs-sub 60.9%

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

       2. sub-neg 60.9%

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

       3. mul-1-neg 60.9%

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

       4. fabs-neg 60.9%

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

       5. associate--r+ 60.9%

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

       6. cancel-sign-sub-inv 60.9%

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

       7. fabs-neg 60.9%

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

       8. mul-1-neg 60.9%

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

       9. sub-neg 60.9%

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

       10. fabs-sub 60.9%

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

       11. rem-square-sqrt 36.4%

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

       12. fabs-sqr 36.4%

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

       13. rem-square-sqrt 81.3%

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

       14. metadata-eval 81.3%

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

       15. *-commutative 81.3%

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

   11. Simplified 81.3%

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

4. Final simplification 95.0%

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

Alternative 3: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t\_0 \leq \infty:\\ \;\;\;\;t\_0 \cdot \cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(m - n\right) - \ell\right) + {n}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (- (fabs (- m n)) l) (pow (- (/ (+ m n) 2.0) M) 2.0)))))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_0) INFINITY)
     (* t_0 (cos (* K (* (+ m n) 0.5))))
     (exp (+ (- (- m n) l) (* (pow n 2.0) -0.25))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((fabs((m - n)) - l) - pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_0) <= ((double) INFINITY)) {
		tmp = t_0 * cos((K * ((m + n) * 0.5)));
	} else {
		tmp = exp((((m - n) - l) + (pow(n, 2.0) * -0.25)));
	}
	return tmp;
}

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((Math.abs((m - n)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * Math.cos((K * ((m + n) * 0.5)));
	} else {
		tmp = Math.exp((((m - n) - l) + (Math.pow(n, 2.0) * -0.25)));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.exp(((math.fabs((m - n)) - l) - math.pow((((m + n) / 2.0) - M), 2.0)))
	tmp = 0
	if (math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= math.inf:
		tmp = t_0 * math.cos((K * ((m + n) * 0.5)))
	else:
		tmp = math.exp((((m - n) - l) + (math.pow(n, 2.0) * -0.25)))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(abs(Float64(m - n)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_0) <= Inf)
		tmp = Float64(t_0 * cos(Float64(K * Float64(Float64(m + n) * 0.5))));
	else
		tmp = exp(Float64(Float64(Float64(m - n) - l) + Float64((n ^ 2.0) * -0.25)));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((abs((m - n)) - l) - ((((m + n) / 2.0) - M) ^ 2.0)));
	tmp = 0.0;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_0) <= Inf)
		tmp = t_0 * cos((K * ((m + n) * 0.5)));
	else
		tmp = exp((((m - n) - l) + ((n ^ 2.0) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = 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]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], Infinity], N[(t$95$0 * N[Cos[N[(K * N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision] + N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 97.3%

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

   3. Taylor expanded in M around 0 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-*r* 97.6%

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

   5. Simplified 97.6%

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

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in K around 0 98.0%

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

      1. cos-neg 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in M around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in m around 0 61.4%

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

       1. fabs-sub 61.4%

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

       2. sub-neg 61.4%

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

       3. mul-1-neg 61.4%

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

       4. fabs-neg 61.4%

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

       5. associate--r+ 61.4%

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

       6. cancel-sign-sub-inv 61.4%

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

       7. fabs-neg 61.4%

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

       8. mul-1-neg 61.4%

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

       9. sub-neg 61.4%

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

       10. fabs-sub 61.4%

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

       11. rem-square-sqrt 39.5%

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

       12. fabs-sqr 39.5%

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

       13. rem-square-sqrt 84.6%

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

       14. metadata-eval 84.6%

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

       15. *-commutative 84.6%

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

   11. Simplified 84.6%

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

4. Final simplification 95.0%

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

Alternative 4: 96.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (* (exp (- (- (fabs (- m n)) l) (pow (- (/ (+ m n) 2.0) M) 2.0))) (cos M)))

C

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

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp(((abs((m - n)) - l) - ((((m + n) / 2.0d0) - m_1) ** 2.0d0))) * cos(m_1)
end function

Java

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

Python

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

Julia

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

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = exp(((abs((m - n)) - l) - ((((m + n) / 2.0) - M) ^ 2.0))) * cos(M);
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[(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] * N[Cos[M], $MachinePrecision]), $MachinePrecision]

Derivation

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 K around 0 96.2%

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

   1. cos-neg 96.2%

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

5. Simplified 96.2%

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

6. Final simplification 96.2%

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

Alternative 5: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -8.5 \cdot 10^{-5} \lor \neg \left(M \leq 700\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + \left(\left(m + n\right) \cdot \left(m + n\right)\right) \cdot 0.25\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -8.5e-5) (not (<= M 700.0)))
   (* (cos M) (exp (- (pow M 2.0))))
   (exp (- (fabs (- m n)) (+ l (* (* (+ m n) (+ m n)) 0.25))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -8.5e-5) || !(M <= 700.0)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = exp((fabs((m - n)) - (l + (((m + n) * (m + n)) * 0.25))));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 <= (-8.5d-5)) .or. (.not. (m_1 <= 700.0d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = exp((abs((m - n)) - (l + (((m + n) * (m + n)) * 0.25d0))))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -8.5e-5) || !(M <= 700.0)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp((Math.abs((m - n)) - (l + (((m + n) * (m + n)) * 0.25))));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if (M <= -8.5e-5) or not (M <= 700.0):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp((math.fabs((m - n)) - (l + (((m + n) * (m + n)) * 0.25))))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -8.5e-5) || !(M <= 700.0))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(Float64(Float64(m + n) * Float64(m + n)) * 0.25))));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -8.5e-5) || ~((M <= 700.0)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = exp((abs((m - n)) - (l + (((m + n) * (m + n)) * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -8.5e-5], N[Not[LessEqual[M, 700.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -8.500000000000001e-5 or 700 < M

   1. Initial program 81.0%

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

   3. Taylor expanded in K around 0 97.6%

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

      1. cos-neg 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in M around inf 97.7%

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

      1. mul-1-neg 97.7%

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

   8. Simplified 97.7%

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

   if -8.500000000000001e-5 < M < 700

   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. Add Preprocessing

   3. Taylor expanded in K around 0 94.9%

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

      1. cos-neg 94.9%

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

   5. Simplified 94.9%

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

   6. Taylor expanded in M around 0 94.9%

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

      1. *-commutative 94.9%

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

   8. Simplified 94.9%

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

      1. unpow2 94.9%

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

   10. Applied egg-rr 94.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -8.5 \cdot 10^{-5} \lor \neg \left(M \leq 700\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + \left(\left(m + n\right) \cdot \left(m + n\right)\right) \cdot 0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ e^{\left|m - n\right| - \left(\ell + \left(\left(m + n\right) \cdot \left(m + n\right)\right) \cdot 0.25\right)} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (exp (- (fabs (- m n)) (+ l (* (* (+ m n) (+ m n)) 0.25)))))

C

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

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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) * (m + n)) * 0.25d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = exp((abs((m - n)) - (l + (((m + n) * (m + n)) * 0.25))));
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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 K around 0 96.2%

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

   1. cos-neg 96.2%

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

5. Simplified 96.2%

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

6. Taylor expanded in M around 0 86.2%

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

   1. *-commutative 86.2%

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

8. Simplified 86.2%

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

   1. unpow2 86.2%

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

10. Applied egg-rr 86.2%

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

Alternative 7: 84.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ e^{\left(\left(m - n\right) - \ell\right) + {n}^{2} \cdot -0.25} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (exp (+ (- (- m n) l) (* (pow n 2.0) -0.25))))

C

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

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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((((m - n) - l) + ((n ** 2.0d0) * (-0.25d0))))
end function

Java

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

Python

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

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return exp(Float64(Float64(Float64(m - n) - l) + Float64((n ^ 2.0) * -0.25)))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = exp((((m - n) - l) + ((n ^ 2.0) * -0.25)));
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[Exp[N[(N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision] + N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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 K around 0 96.2%

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

   1. cos-neg 96.2%

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

5. Simplified 96.2%

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

6. Taylor expanded in M around 0 86.2%

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

   1. *-commutative 86.2%

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

8. Simplified 86.2%

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

9. Taylor expanded in m around 0 62.3%

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

    1. fabs-sub 62.3%

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

    2. sub-neg 62.3%

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

    3. mul-1-neg 62.3%

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

    4. fabs-neg 62.3%

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

    5. associate--r+ 62.3%

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

    6. cancel-sign-sub-inv 62.3%

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

    7. fabs-neg 62.3%

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

    8. mul-1-neg 62.3%

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

    9. sub-neg 62.3%

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

    10. fabs-sub 62.3%

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

    11. rem-square-sqrt 34.4%

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

    12. fabs-sqr 34.4%

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

    13. rem-square-sqrt 75.0%

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

    14. metadata-eval 75.0%

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

    15. *-commutative 75.0%

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

11. Simplified 75.0%

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

Alternative 8: 76.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;m \leq -52:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{{n}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= m -52.0) (exp (* -0.25 (pow m 2.0))) (exp (* (pow n 2.0) -0.25))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -52.0) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = exp((pow(n, 2.0) * -0.25));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 <= (-52.0d0)) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = exp(((n ** 2.0d0) * (-0.25d0)))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -52.0) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.exp((Math.pow(n, 2.0) * -0.25));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if m <= -52.0:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.exp((math.pow(n, 2.0) * -0.25))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -52.0)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	else
		tmp = exp(Float64((n ^ 2.0) * -0.25));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -52.0)
		tmp = exp((-0.25 * (m ^ 2.0)));
	else
		tmp = exp(((n ^ 2.0) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[m, -52.0], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -52

   1. Initial program 69.1%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in m around inf 98.6%

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

       1. *-commutative 98.6%

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

   11. Simplified 98.6%

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

   if -52 < m

   1. Initial program 81.1%

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

   3. Taylor expanded in K around 0 94.9%

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

      1. cos-neg 94.9%

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

   5. Simplified 94.9%

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

   6. Taylor expanded in M around 0 81.2%

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

      1. *-commutative 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in n around inf 62.1%

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

       1. *-commutative 62.1%

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

   11. Simplified 62.1%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -52:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{{n}^{2} \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.00072:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= l 0.00072) (exp (* -0.25 (pow m 2.0))) (exp (- l))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.00072) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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 <= 0.00072d0) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.00072) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if l <= 0.00072:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 0.00072)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 0.00072)
		tmp = exp((-0.25 * (m ^ 2.0)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[l, 0.00072], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.20000000000000045e-4

   1. Initial program 76.9%

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

   3. Taylor expanded in K around 0 95.3%

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

      1. cos-neg 95.3%

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

   5. Simplified 95.3%

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

   6. Taylor expanded in M around 0 82.9%

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

      1. *-commutative 82.9%

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

   8. Simplified 82.9%

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

   9. Taylor expanded in m around inf 57.2%

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

       1. *-commutative 57.2%

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

   11. Simplified 57.2%

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

   if 7.20000000000000045e-4 < l

   1. Initial program 82.0%

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

   3. Taylor expanded in l around inf 82.0%

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

      1. mul-1-neg 82.0%

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

   5. Simplified 82.0%

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

   6. Taylor expanded in m around inf 86.0%

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

      1. associate-*r* 86.0%

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

   8. Simplified 86.0%

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

   9. Taylor expanded in K around 0 100.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.00072:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ e^{-\ell} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l) :precision binary64 (exp (- l)))

C

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

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return exp(Float64(-l))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = exp(-l);
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[Exp[(-l)], $MachinePrecision]

Derivation

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 l around inf 25.8%

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

   1. mul-1-neg 25.8%

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

5. Simplified 25.8%

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

6. Taylor expanded in m around inf 27.2%

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

   1. associate-*r* 27.2%

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

8. Simplified 27.2%

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

9. Taylor expanded in K around 0 29.3%

   \[\leadsto \color{blue}{e^{-\ell}} \]
10. Add Preprocessing

Alternative 11: 7.1% accurate, 425.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ 1 \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l) :precision binary64 1.0)

C

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

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0
end function

Java

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

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return 1.0

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return 1.0
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = 1.0;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := 1.0

Derivation

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 l around inf 25.8%

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

   1. mul-1-neg 25.8%

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

5. Simplified 25.8%

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

6. Taylor expanded in l around 0 8.0%

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

7. Taylor expanded in K around 0 8.5%

   \[\leadsto \cos \color{blue}{\left(-1 \cdot M\right)} \]
8. Step-by-step derivation

   1. neg-mul-1 8.5%

      \[\leadsto \cos \color{blue}{\left(-M\right)} \]

9. Simplified 8.5%

   \[\leadsto \cos \color{blue}{\left(-M\right)} \]

10. Taylor expanded in M around 0 8.6%

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

Reproduce

herbie shell --seed 2024113 
(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)))))))