Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.9% → 96.8%

Time: 27.6s

Alternatives: 14

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(\left|n - m\right| - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\\ \cos M \cdot {\left(e^{{t\_0}^{2}}\right)}^{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cbrt (- (- (fabs (- n m)) l) (pow (- (* (+ m n) 0.5) M) 2.0)))))
   (* (cos M) (pow (exp (pow t_0 2.0)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cbrt(((fabs((n - m)) - l) - pow((((m + n) * 0.5) - M), 2.0)));
	return cos(M) * pow(exp(pow(t_0, 2.0)), t_0);
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cbrt(((Math.abs((n - m)) - l) - Math.pow((((m + n) * 0.5) - M), 2.0)));
	return Math.cos(M) * Math.pow(Math.exp(Math.pow(t_0, 2.0)), t_0);
}

Julia

function code(K, m, n, M, l)
	t_0 = cbrt(Float64(Float64(abs(Float64(n - m)) - l) - (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0)))
	return Float64(cos(M) * (exp((t_0 ^ 2.0)) ^ t_0))
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Power[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Cos[M], $MachinePrecision] * N[Power[N[Exp[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.6%

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

   2. +-commutative 77.6%

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

   3. associate-/l* 77.2%

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

   4. associate-/l* 77.6%

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

   5. +-commutative 77.6%

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

   6. exp-diff 29.9%

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

   7. sub-neg 29.9%

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

   8. exp-sum 24.4%

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

   9. associate-/r* 24.4%

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

   10. exp-diff 29.9%

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

3. Simplified 77.6%

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

5. Taylor expanded in K around 0 95.4%

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

   1. cos-neg 95.4%

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

7. Simplified 95.4%

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

   1. add-sqr-sqrt 95.4%

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

   2. sqrt-unprod 94.6%

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

   3. pow-prod-up 94.6%

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

   4. div-inv 94.6%

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

   5. fma-neg 94.6%

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

   6. metadata-eval 94.6%

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

   7. metadata-eval 94.6%

      \[\leadsto \cos M \cdot e^{\left(\left(-\sqrt{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{\color{blue}{4}}}\right) - \ell\right) + \left|n - m\right|} \]

9. Applied egg-rr 94.6%

   \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{\sqrt{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{4}}}\right) - \ell\right) + \left|n - m\right|} \]
10. Step-by-step derivation

    1. fma-neg 94.6%

       \[\leadsto \cos M \cdot e^{\left(\left(-\sqrt{{\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)}}^{4}}\right) - \ell\right) + \left|n - m\right|} \]

11. Simplified 94.6%

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

    1. add-cube-cbrt 94.6%

       \[\leadsto \cos M \cdot e^{\color{blue}{\left(\sqrt[3]{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|} \cdot \sqrt[3]{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|}\right) \cdot \sqrt[3]{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|}}} \]

    2. exp-prod 94.6%

       \[\leadsto \cos M \cdot \color{blue}{{\left(e^{\sqrt[3]{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|} \cdot \sqrt[3]{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|}}\right)}^{\left(\sqrt[3]{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|}\right)}} \]

13. Applied egg-rr 95.4%

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

14. Final simplification 95.4%

    \[\leadsto \cos M \cdot {\left(e^{{\left(\sqrt[3]{\left(\left|n - m\right| - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\right)}^{2}}\right)}^{\left(\sqrt[3]{\left(\left|n - m\right| - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\right)} \]
15. Add Preprocessing

Alternative 2: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot {\left({\left(e^{\left(\left|n - m\right| - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\right)}^{3}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos M)
  (pow
   (pow (exp (- (- (fabs (- n m)) l) (pow (- (* (+ m n) 0.5) M) 2.0))) 3.0)
   0.3333333333333333)))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * pow(pow(exp(((fabs((n - m)) - l) - pow((((m + n) * 0.5) - M), 2.0))), 3.0), 0.3333333333333333);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * ((exp(((abs((n - m)) - l) - ((((m + n) * 0.5d0) - m_1) ** 2.0d0))) ** 3.0d0) ** 0.3333333333333333d0)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.pow(Math.pow(Math.exp(((Math.abs((n - m)) - l) - Math.pow((((m + n) * 0.5) - M), 2.0))), 3.0), 0.3333333333333333);
}

Python

def code(K, m, n, M, l):
	return math.cos(M) * math.pow(math.pow(math.exp(((math.fabs((n - m)) - l) - math.pow((((m + n) * 0.5) - M), 2.0))), 3.0), 0.3333333333333333)

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * ((exp(Float64(Float64(abs(Float64(n - m)) - l) - (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0))) ^ 3.0) ^ 0.3333333333333333))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * ((exp(((abs((n - m)) - l) - ((((m + n) * 0.5) - M) ^ 2.0))) ^ 3.0) ^ 0.3333333333333333);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Power[N[Power[N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.6%

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

   2. +-commutative 77.6%

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

   3. associate-/l* 77.2%

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

   4. associate-/l* 77.6%

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

   5. +-commutative 77.6%

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

   6. exp-diff 29.9%

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

   7. sub-neg 29.9%

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

   8. exp-sum 24.4%

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

   9. associate-/r* 24.4%

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

   10. exp-diff 29.9%

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

3. Simplified 77.6%

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

5. Taylor expanded in K around 0 95.4%

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

   1. cos-neg 95.4%

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

7. Simplified 95.4%

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

   1. add-sqr-sqrt 95.4%

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

   2. sqrt-unprod 94.6%

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

   3. pow-prod-up 94.6%

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

   4. div-inv 94.6%

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

   5. fma-neg 94.6%

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

   6. metadata-eval 94.6%

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

   7. metadata-eval 94.6%

      \[\leadsto \cos M \cdot e^{\left(\left(-\sqrt{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{\color{blue}{4}}}\right) - \ell\right) + \left|n - m\right|} \]

9. Applied egg-rr 94.6%

   \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{\sqrt{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{4}}}\right) - \ell\right) + \left|n - m\right|} \]
10. Step-by-step derivation

    1. fma-neg 94.6%

       \[\leadsto \cos M \cdot e^{\left(\left(-\sqrt{{\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)}}^{4}}\right) - \ell\right) + \left|n - m\right|} \]

11. Simplified 94.6%

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

    1. add-cbrt-cube 94.6%

       \[\leadsto \cos M \cdot \color{blue}{\sqrt[3]{\left(e^{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|} \cdot e^{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|}\right) \cdot e^{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|}}} \]

    2. pow1/3 94.6%

       \[\leadsto \cos M \cdot \color{blue}{{\left(\left(e^{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|} \cdot e^{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|}\right) \cdot e^{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|}\right)}^{0.3333333333333333}} \]

13. Applied egg-rr 95.4%

    \[\leadsto \cos M \cdot \color{blue}{{\left({\left(e^{\left(-{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

14. Final simplification 95.4%

    \[\leadsto \cos M \cdot {\left({\left(e^{\left(\left|n - m\right| - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\right)}^{3}\right)}^{0.3333333333333333} \]
15. Add Preprocessing

Alternative 3: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.6%

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

   2. +-commutative 77.6%

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

   3. associate-/l* 77.2%

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

   4. associate-/l* 77.6%

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

   5. +-commutative 77.6%

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

   6. exp-diff 29.9%

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

   7. sub-neg 29.9%

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

   8. exp-sum 24.4%

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

   9. associate-/r* 24.4%

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

   10. exp-diff 29.9%

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

3. Simplified 77.6%

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

5. Taylor expanded in K around 0 95.4%

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

   1. cos-neg 95.4%

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

7. Simplified 95.4%

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

   1. add-sqr-sqrt 95.4%

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

   2. sqrt-unprod 94.6%

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

   3. pow-prod-up 94.6%

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

   4. div-inv 94.6%

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

   5. fma-neg 94.6%

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

   6. metadata-eval 94.6%

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

   7. metadata-eval 94.6%

      \[\leadsto \cos M \cdot e^{\left(\left(-\sqrt{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{\color{blue}{4}}}\right) - \ell\right) + \left|n - m\right|} \]

9. Applied egg-rr 94.6%

   \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{\sqrt{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{4}}}\right) - \ell\right) + \left|n - m\right|} \]
10. Step-by-step derivation

    1. fma-neg 94.6%

       \[\leadsto \cos M \cdot e^{\left(\left(-\sqrt{{\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)}}^{4}}\right) - \ell\right) + \left|n - m\right|} \]

11. Simplified 94.6%

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

    1. fabs-sub 94.6%

       \[\leadsto \cos M \cdot e^{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \color{blue}{\left|m - n\right|}} \]

    2. *-un-lft-identity 94.6%

       \[\leadsto \cos M \cdot e^{\color{blue}{1 \cdot \left(\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|m - n\right|\right)}} \]

    3. fabs-sub 94.6%

       \[\leadsto \cos M \cdot e^{1 \cdot \left(\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \color{blue}{\left|n - m\right|}\right)} \]

    4. exp-prod 94.6%

       \[\leadsto \cos M \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|\right)}} \]

    5. exp-1-e 94.6%

       \[\leadsto \cos M \cdot {\color{blue}{e}}^{\left(\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \left|n - m\right|\right)} \]

    6. fabs-sub 94.6%

       \[\leadsto \cos M \cdot {e}^{\left(\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \ell\right) + \color{blue}{\left|m - n\right|}\right)} \]

    7. associate-+l- 94.6%

       \[\leadsto \cos M \cdot {e}^{\color{blue}{\left(\left(-\sqrt{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}}\right) - \left(\ell - \left|m - n\right|\right)\right)}} \]

    8. sqrt-pow1 95.4%

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

    9. metadata-eval 95.4%

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

    10. fabs-sub 95.4%

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

13. Applied egg-rr 95.4%

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

14. Final simplification 95.4%

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

Alternative 4: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(abs(Float64(n - m)) - Float64(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((n - m)) - (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[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.6%

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

   2. +-commutative 77.6%

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

   3. associate-/l* 77.2%

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

   4. associate-/l* 77.6%

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

   5. +-commutative 77.6%

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

   6. exp-diff 29.9%

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

   7. sub-neg 29.9%

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

   8. exp-sum 24.4%

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

   9. associate-/r* 24.4%

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

   10. exp-diff 29.9%

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

3. Simplified 77.6%

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

5. Taylor expanded in K around 0 95.4%

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

   1. cos-neg 95.4%

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

7. Simplified 95.4%

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

8. Final simplification 95.4%

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

Alternative 5: 67.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ t_1 := \cos M \cdot e^{-{M}^{2}}\\ t_2 := \left(m + n\right) \cdot 0.5 - M\\ \mathbf{if}\;M \leq -0.086:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq 2.6 \cdot 10^{-291}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;M \leq 1.45 \cdot 10^{-170}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 1.2 \cdot 10^{-59}:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(n - m\right) + \left(t\_2 \cdot t\_2 - \ell\right)}\\ \mathbf{elif}\;M \leq 26:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (* m (- M (* n 0.5))))))
        (t_1 (* (cos M) (exp (- (pow M 2.0)))))
        (t_2 (- (* (+ m n) 0.5) M)))
   (if (<= M -0.086)
     t_1
     (if (<= M 2.6e-291)
       (exp (- l))
       (if (<= M 1.45e-170)
         t_0
         (if (<= M 1.2e-59)
           (*
            (cos (- (/ K (/ 2.0 (+ m n))) M))
            (exp (+ (- n m) (- (* t_2 t_2) l))))
           (if (<= M 26.0) t_0 t_1)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp((m * (M - (n * 0.5))));
	double t_1 = cos(M) * exp(-pow(M, 2.0));
	double t_2 = ((m + n) * 0.5) - M;
	double tmp;
	if (M <= -0.086) {
		tmp = t_1;
	} else if (M <= 2.6e-291) {
		tmp = exp(-l);
	} else if (M <= 1.45e-170) {
		tmp = t_0;
	} else if (M <= 1.2e-59) {
		tmp = cos(((K / (2.0 / (m + n))) - M)) * exp(((n - m) + ((t_2 * t_2) - l)));
	} else if (M <= 26.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(m_1) * exp((m * (m_1 - (n * 0.5d0))))
    t_1 = cos(m_1) * exp(-(m_1 ** 2.0d0))
    t_2 = ((m + n) * 0.5d0) - m_1
    if (m_1 <= (-0.086d0)) then
        tmp = t_1
    else if (m_1 <= 2.6d-291) then
        tmp = exp(-l)
    else if (m_1 <= 1.45d-170) then
        tmp = t_0
    else if (m_1 <= 1.2d-59) then
        tmp = cos(((k / (2.0d0 / (m + n))) - m_1)) * exp(((n - m) + ((t_2 * t_2) - l)))
    else if (m_1 <= 26.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(M) * Math.exp((m * (M - (n * 0.5))));
	double t_1 = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	double t_2 = ((m + n) * 0.5) - M;
	double tmp;
	if (M <= -0.086) {
		tmp = t_1;
	} else if (M <= 2.6e-291) {
		tmp = Math.exp(-l);
	} else if (M <= 1.45e-170) {
		tmp = t_0;
	} else if (M <= 1.2e-59) {
		tmp = Math.cos(((K / (2.0 / (m + n))) - M)) * Math.exp(((n - m) + ((t_2 * t_2) - l)));
	} else if (M <= 26.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp((m * (M - (n * 0.5))))
	t_1 = math.cos(M) * math.exp(-math.pow(M, 2.0))
	t_2 = ((m + n) * 0.5) - M
	tmp = 0
	if M <= -0.086:
		tmp = t_1
	elif M <= 2.6e-291:
		tmp = math.exp(-l)
	elif M <= 1.45e-170:
		tmp = t_0
	elif M <= 1.2e-59:
		tmp = math.cos(((K / (2.0 / (m + n))) - M)) * math.exp(((n - m) + ((t_2 * t_2) - l)))
	elif M <= 26.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(m * Float64(M - Float64(n * 0.5)))))
	t_1 = Float64(cos(M) * exp(Float64(-(M ^ 2.0))))
	t_2 = Float64(Float64(Float64(m + n) * 0.5) - M)
	tmp = 0.0
	if (M <= -0.086)
		tmp = t_1;
	elseif (M <= 2.6e-291)
		tmp = exp(Float64(-l));
	elseif (M <= 1.45e-170)
		tmp = t_0;
	elseif (M <= 1.2e-59)
		tmp = Float64(cos(Float64(Float64(K / Float64(2.0 / Float64(m + n))) - M)) * exp(Float64(Float64(n - m) + Float64(Float64(t_2 * t_2) - l))));
	elseif (M <= 26.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp((m * (M - (n * 0.5))));
	t_1 = cos(M) * exp(-(M ^ 2.0));
	t_2 = ((m + n) * 0.5) - M;
	tmp = 0.0;
	if (M <= -0.086)
		tmp = t_1;
	elseif (M <= 2.6e-291)
		tmp = exp(-l);
	elseif (M <= 1.45e-170)
		tmp = t_0;
	elseif (M <= 1.2e-59)
		tmp = cos(((K / (2.0 / (m + n))) - M)) * exp(((n - m) + ((t_2 * t_2) - l)));
	elseif (M <= 26.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision]}, If[LessEqual[M, -0.086], t$95$1, If[LessEqual[M, 2.6e-291], N[Exp[(-l)], $MachinePrecision], If[LessEqual[M, 1.45e-170], t$95$0, If[LessEqual[M, 1.2e-59], N[(N[Cos[N[(N[(K / N[(2.0 / N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(n - m), $MachinePrecision] + N[(N[(t$95$2 * t$95$2), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 26.0], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -0.085999999999999993 or 26 < 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. Step-by-step derivation

      1. associate-/l* 81.1%

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

      2. +-commutative 81.1%

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

      3. associate-/l* 81.1%

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

      4. associate-/l* 81.1%

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

      5. +-commutative 81.1%

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

      6. exp-diff 26.5%

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

      7. sub-neg 26.5%

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

      8. exp-sum 18.9%

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

      9. associate-/r* 18.9%

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

      10. exp-diff 27.3%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in M around inf 97.0%

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

      1. mul-1-neg 72.8%

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

   10. Simplified 97.0%

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

   if -0.085999999999999993 < M < 2.5999999999999999e-291

   1. Initial program 73.2%

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

      1. associate-/l* 73.2%

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

      2. +-commutative 73.2%

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

      3. associate-/l* 73.2%

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

      4. associate-/l* 73.2%

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

      5. +-commutative 73.2%

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

      6. exp-diff 34.2%

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

      7. sub-neg 34.2%

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

      8. exp-sum 32.5%

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

      9. associate-/r* 32.5%

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

      10. exp-diff 32.5%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in K around 0 90.6%

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

      1. cos-neg 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in l around inf 46.1%

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

      1. neg-mul-1 46.1%

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

   10. Simplified 46.1%

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

   11. Taylor expanded in M around 0 46.1%

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

   if 2.5999999999999999e-291 < M < 1.45e-170 or 1.20000000000000008e-59 < M < 26

   1. Initial program 65.0%

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

      1. associate-/l* 68.0%

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

      2. +-commutative 68.0%

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

      3. associate-/l* 65.0%

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

      4. associate-/l* 68.0%

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

      5. +-commutative 68.0%

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

      6. exp-diff 16.5%

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

      7. sub-neg 16.5%

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

      8. exp-sum 13.4%

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

      9. associate-/r* 13.4%

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

      10. exp-diff 16.5%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in K around 0 92.1%

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

      1. cos-neg 92.1%

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

   7. Simplified 92.1%

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

   8. Taylor expanded in m around 0 62.2%

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

      1. +-commutative 62.2%

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

      2. unpow2 62.2%

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

      3. distribute-rgt-out 65.3%

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

      4. *-commutative 65.3%

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

      5. *-commutative 65.3%

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

   10. Simplified 65.3%

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

   11. Taylor expanded in m around inf 56.7%

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

   if 1.45e-170 < M < 1.20000000000000008e-59

   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. Step-by-step derivation

      1. associate-/l* 81.0%

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

      2. +-commutative 81.0%

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

      3. associate-/l* 81.0%

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

      4. associate-/l* 81.0%

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

      5. +-commutative 81.0%

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

      6. exp-diff 49.7%

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

      7. sub-neg 49.7%

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

      8. exp-sum 43.5%

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

      9. associate-/r* 43.5%

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

      10. exp-diff 49.7%

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

   3. Simplified 81.0%

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

      1. *-un-lft-identity 81.0%

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

      2. *-commutative 81.0%

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

   6. Applied egg-rr 53.7%

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

      1. *-rgt-identity 53.7%

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

      2. associate--r- 53.7%

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

      3. +-commutative 53.7%

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

      4. fma-neg 53.7%

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

   8. Simplified 53.7%

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

      1. clear-num 54.3%

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

      2. un-div-inv 57.4%

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

   10. Applied egg-rr 57.4%

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

       1. unpow2 57.4%

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

   12. Applied egg-rr 57.4%

       \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(n - m\right) + \left(\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right)} - \ell\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -0.086:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;M \leq 2.6 \cdot 10^{-291}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;M \leq 1.45 \cdot 10^{-170}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{elif}\;M \leq 1.2 \cdot 10^{-59}:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(n - m\right) + \left(\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) - \ell\right)}\\ \mathbf{elif}\;M \leq 26:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M - n \cdot 0.5\\ \mathbf{if}\;m \leq -27000:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + \left(\left(m - t\_0\right) \cdot t\_0 - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* n 0.5))))
   (if (<= m -27000.0)
     (* (cos M) (exp (* -0.25 (pow m 2.0))))
     (* (cos M) (exp (+ (fabs (- n m)) (- (* (- m t_0) t_0) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (n * 0.5);
	double tmp;
	if (m <= -27000.0) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp((fabs((n - m)) + (((m - t_0) * t_0) - 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) :: t_0
    real(8) :: tmp
    t_0 = m_1 - (n * 0.5d0)
    if (m <= (-27000.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp((abs((n - m)) + (((m - t_0) * t_0) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (n * 0.5);
	double tmp;
	if (m <= -27000.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp((Math.abs((n - m)) + (((m - t_0) * t_0) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M - (n * 0.5)
	tmp = 0
	if m <= -27000.0:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp((math.fabs((n - m)) + (((m - t_0) * t_0) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M - Float64(n * 0.5))
	tmp = 0.0
	if (m <= -27000.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(abs(Float64(n - m)) + Float64(Float64(Float64(m - t_0) * t_0) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = M - (n * 0.5);
	tmp = 0.0;
	if (m <= -27000.0)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp((abs((n - m)) + (((m - t_0) * t_0) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -27000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(m - t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -27000

   1. Initial program 73.6%

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

      1. associate-/l* 73.6%

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

      2. +-commutative 73.6%

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

      3. associate-/l* 73.6%

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

      4. associate-/l* 73.6%

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

      5. +-commutative 73.6%

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

      6. exp-diff 9.4%

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

      7. sub-neg 9.4%

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

      8. exp-sum 0.0%

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

      9. associate-/r* 0.0%

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

      10. exp-diff 0.0%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 100.0%

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

   if -27000 < m

   1. Initial program 78.1%

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

      1. associate-/l* 78.6%

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

      2. +-commutative 78.6%

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

      3. associate-/l* 78.1%

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

      4. associate-/l* 78.6%

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

      5. +-commutative 78.6%

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

      6. exp-diff 35.2%

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

      7. sub-neg 35.2%

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

      8. exp-sum 30.8%

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

      9. associate-/r* 30.8%

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

      10. exp-diff 37.7%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in K around 0 94.2%

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

      1. cos-neg 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in m around 0 77.1%

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

      1. +-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. distribute-rgt-out 82.6%

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

      4. *-commutative 82.6%

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

      5. *-commutative 82.6%

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

   10. Simplified 82.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -27000:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + \left(\left(m - \left(M - n \cdot 0.5\right)\right) \cdot \left(M - n \cdot 0.5\right) - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -75:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) - 0.5 \cdot \left(n \cdot \left(m + n \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -75.0)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (* (cos M) (exp (- (- (fabs (- n m)) l) (* 0.5 (* n (+ m (* n 0.5)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -75.0) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp(((fabs((n - m)) - l) - (0.5 * (n * (m + (n * 0.5))))));
	}
	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 <= (-75.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(((abs((n - m)) - l) - (0.5d0 * (n * (m + (n * 0.5d0))))))
    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 <= -75.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(((Math.abs((n - m)) - l) - (0.5 * (n * (m + (n * 0.5))))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -75.0:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(((math.fabs((n - m)) - l) - (0.5 * (n * (m + (n * 0.5))))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -75.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(0.5 * Float64(n * Float64(m + Float64(n * 0.5)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -75.0)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp(((abs((n - m)) - l) - (0.5 * (n * (m + (n * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -75.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(0.5 * N[(n * N[(m + N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -75

   1. Initial program 73.6%

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

      1. associate-/l* 73.6%

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

      2. +-commutative 73.6%

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

      3. associate-/l* 73.6%

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

      4. associate-/l* 73.6%

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

      5. +-commutative 73.6%

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

      6. exp-diff 9.4%

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

      7. sub-neg 9.4%

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

      8. exp-sum 0.0%

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

      9. associate-/r* 0.0%

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

      10. exp-diff 0.0%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 100.0%

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

   if -75 < m

   1. Initial program 78.1%

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

      1. associate-/l* 78.6%

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

      2. +-commutative 78.6%

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

      3. associate-/l* 78.1%

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

      4. associate-/l* 78.6%

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

      5. +-commutative 78.6%

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

      6. exp-diff 35.2%

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

      7. sub-neg 35.2%

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

      8. exp-sum 30.8%

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

      9. associate-/r* 30.8%

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

      10. exp-diff 37.7%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in K around 0 94.2%

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

      1. cos-neg 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in m around 0 77.1%

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

      1. +-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. distribute-rgt-out 82.6%

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

      4. *-commutative 82.6%

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

      5. *-commutative 82.6%

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

   10. Simplified 82.6%

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

   11. Taylor expanded in M around 0 64.8%

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

       1. associate--r+ 64.8%

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

       2. fabs-sub 64.8%

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

   13. Simplified 64.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -75:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) - 0.5 \cdot \left(n \cdot \left(m + n \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -75:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -3 \cdot 10^{-170}:\\ \;\;\;\;e^{-{M}^{2}} \cdot \cos \left(0.5 \cdot \left(m \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -75.0)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= m -3e-170)
     (* (exp (- (pow M 2.0))) (cos (- (* 0.5 (* m K)) M)))
     (* (cos M) (exp (* -0.25 (pow n 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -75.0) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (m <= -3e-170) {
		tmp = exp(-pow(M, 2.0)) * cos(((0.5 * (m * K)) - M));
	} else {
		tmp = cos(M) * exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-75.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= (-3d-170)) then
        tmp = exp(-(m_1 ** 2.0d0)) * cos(((0.5d0 * (m * k)) - m_1))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -75.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= -3e-170) {
		tmp = Math.exp(-Math.pow(M, 2.0)) * Math.cos(((0.5 * (m * K)) - M));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -75.0:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= -3e-170:
		tmp = math.exp(-math.pow(M, 2.0)) * math.cos(((0.5 * (m * K)) - M))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -75.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (m <= -3e-170)
		tmp = Float64(exp(Float64(-(M ^ 2.0))) * cos(Float64(Float64(0.5 * Float64(m * K)) - M)));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (n ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -75.0)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (m <= -3e-170)
		tmp = exp(-(M ^ 2.0)) * cos(((0.5 * (m * K)) - M));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -75.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -3e-170], N[(N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision] * N[Cos[N[(N[(0.5 * N[(m * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -75

   1. Initial program 73.6%

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

      1. associate-/l* 73.6%

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

      2. +-commutative 73.6%

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

      3. associate-/l* 73.6%

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

      4. associate-/l* 73.6%

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

      5. +-commutative 73.6%

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

      6. exp-diff 9.4%

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

      7. sub-neg 9.4%

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

      8. exp-sum 0.0%

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

      9. associate-/r* 0.0%

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

      10. exp-diff 0.0%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 100.0%

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

   if -75 < m < -3.00000000000000013e-170

   1. Initial program 72.8%

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

      1. associate-/l* 75.5%

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

      2. +-commutative 75.5%

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

      3. associate-/l* 72.8%

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

      4. associate-/l* 75.5%

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

      5. +-commutative 75.5%

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

      6. exp-diff 39.4%

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

      7. sub-neg 39.4%

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

      8. exp-sum 36.6%

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

      9. associate-/r* 36.6%

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

      10. exp-diff 42.2%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in m around inf 72.8%

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

      1. distribute-lft-out 72.8%

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

      2. associate-/l* 61.6%

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

   7. Simplified 61.6%

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

   8. Taylor expanded in M around inf 48.2%

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

      1. mul-1-neg 48.2%

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

   10. Simplified 48.2%

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

   11. Taylor expanded in n around 0 62.7%

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

       1. *-commutative 62.7%

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

       2. *-commutative 62.7%

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

   13. Simplified 62.7%

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

   if -3.00000000000000013e-170 < m

   1. Initial program 79.2%

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

      1. associate-/l* 79.2%

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

      2. +-commutative 79.2%

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

      3. associate-/l* 79.2%

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

      4. associate-/l* 79.2%

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

      5. +-commutative 79.2%

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

      6. exp-diff 34.3%

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

      7. sub-neg 34.3%

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

      8. exp-sum 29.5%

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

      9. associate-/r* 29.5%

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

      10. exp-diff 36.7%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in n around inf 49.5%

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

Alternative 9: 65.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.85:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -2.5 \cdot 10^{-165}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1.85)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= m -2.5e-165)
     (* (cos M) (exp (- (pow M 2.0))))
     (* (cos M) (exp (* -0.25 (pow n 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.85) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (m <= -2.5e-165) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-1.85d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= (-2.5d-165)) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.85) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= -2.5e-165) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1.85:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= -2.5e-165:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1.85)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (m <= -2.5e-165)
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (n ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1.85)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (m <= -2.5e-165)
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1.85], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -2.5e-165], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.8500000000000001

   1. Initial program 74.1%

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

      1. associate-/l* 74.1%

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

      2. +-commutative 74.1%

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

      3. associate-/l* 74.1%

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

      4. associate-/l* 74.1%

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

      5. +-commutative 74.1%

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

      6. exp-diff 11.1%

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

      7. sub-neg 11.1%

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

      8. exp-sum 1.9%

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

      9. associate-/r* 1.9%

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

      10. exp-diff 1.9%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 100.0%

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

   if -1.8500000000000001 < m < -2.4999999999999999e-165

   1. Initial program 72.0%

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

      1. associate-/l* 74.8%

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

      2. +-commutative 74.8%

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

      3. associate-/l* 72.0%

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

      4. associate-/l* 74.8%

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

      5. +-commutative 74.8%

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

      6. exp-diff 37.7%

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

      7. sub-neg 37.7%

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

      8. exp-sum 34.8%

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

      9. associate-/r* 34.8%

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

      10. exp-diff 40.6%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in K around 0 91.5%

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

      1. cos-neg 91.5%

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

   7. Simplified 91.5%

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

   8. Taylor expanded in M around inf 61.1%

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

      1. mul-1-neg 46.8%

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

   10. Simplified 61.1%

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

   if -2.4999999999999999e-165 < m

   1. Initial program 79.2%

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

      1. associate-/l* 79.2%

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

      2. +-commutative 79.2%

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

      3. associate-/l* 79.2%

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

      4. associate-/l* 79.2%

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

      5. +-commutative 79.2%

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

      6. exp-diff 34.3%

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

      7. sub-neg 34.3%

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

      8. exp-sum 29.5%

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

      9. associate-/r* 29.5%

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

      10. exp-diff 36.7%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in n around inf 49.5%

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

Alternative 10: 62.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.85:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{-170}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1.85)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= m 2.3e-170)
     (* (cos M) (exp (- (pow M 2.0))))
     (* (cos M) (exp (* m (- M (* n 0.5))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.85) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (m <= 2.3e-170) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	}
	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 <= (-1.85d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= 2.3d-170) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp((m * (m_1 - (n * 0.5d0))))
    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 <= -1.85) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= 2.3e-170) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp((m * (M - (n * 0.5))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1.85:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= 2.3e-170:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp((m * (M - (n * 0.5))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1.85)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (m <= 2.3e-170)
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1.85)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (m <= 2.3e-170)
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1.85], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.3e-170], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.8500000000000001

   1. Initial program 74.1%

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

      1. associate-/l* 74.1%

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

      2. +-commutative 74.1%

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

      3. associate-/l* 74.1%

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

      4. associate-/l* 74.1%

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

      5. +-commutative 74.1%

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

      6. exp-diff 11.1%

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

      7. sub-neg 11.1%

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

      8. exp-sum 1.9%

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

      9. associate-/r* 1.9%

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

      10. exp-diff 1.9%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 100.0%

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

   if -1.8500000000000001 < m < 2.29999999999999987e-170

   1. Initial program 80.3%

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

      1. associate-/l* 81.3%

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

      2. +-commutative 81.3%

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

      3. associate-/l* 80.3%

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

      4. associate-/l* 81.3%

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

      5. +-commutative 81.3%

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

      6. exp-diff 46.9%

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

      7. sub-neg 46.9%

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

      8. exp-sum 43.9%

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

      9. associate-/r* 43.9%

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

      10. exp-diff 53.0%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in K around 0 92.9%

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

      1. cos-neg 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in M around inf 57.7%

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

      1. mul-1-neg 44.5%

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

   10. Simplified 57.7%

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

   if 2.29999999999999987e-170 < m

   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. Step-by-step derivation

      1. associate-/l* 75.8%

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

      2. +-commutative 75.8%

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

      3. associate-/l* 75.8%

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

      4. associate-/l* 75.8%

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

      5. +-commutative 75.8%

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

      6. exp-diff 23.4%

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

      7. sub-neg 23.4%

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

      8. exp-sum 17.6%

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

      9. associate-/r* 17.6%

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

      10. exp-diff 22.4%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in K around 0 95.3%

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

      1. cos-neg 95.3%

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

   7. Simplified 95.3%

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

   8. Taylor expanded in m around 0 61.8%

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

      1. +-commutative 61.8%

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

      2. unpow2 61.8%

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

      3. distribute-rgt-out 72.5%

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

      4. *-commutative 72.5%

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

      5. *-commutative 72.5%

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

   10. Simplified 72.5%

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

   11. Taylor expanded in m around inf 41.3%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.85:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{-170}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+214} \lor \neg \left(\ell \leq 105\right):\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= l -5.2e+214) (not (<= l 105.0)))
   (exp (- l))
   (* (cos M) (exp (* m (- M (* n 0.5)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((l <= -5.2e+214) || !(l <= 105.0)) {
		tmp = exp(-l);
	} else {
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	}
	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 <= (-5.2d+214)) .or. (.not. (l <= 105.0d0))) then
        tmp = exp(-l)
    else
        tmp = cos(m_1) * exp((m * (m_1 - (n * 0.5d0))))
    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 <= -5.2e+214) || !(l <= 105.0)) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.cos(M) * Math.exp((m * (M - (n * 0.5))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (l <= -5.2e+214) or not (l <= 105.0):
		tmp = math.exp(-l)
	else:
		tmp = math.cos(M) * math.exp((m * (M - (n * 0.5))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((l <= -5.2e+214) || !(l <= 105.0))
		tmp = exp(Float64(-l));
	else
		tmp = Float64(cos(M) * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((l <= -5.2e+214) || ~((l <= 105.0)))
		tmp = exp(-l);
	else
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[l, -5.2e+214], N[Not[LessEqual[l, 105.0]], $MachinePrecision]], N[Exp[(-l)], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.19999999999999986e214 or 105 < l

   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. Step-by-step derivation

      1. associate-/l* 81.0%

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

      2. +-commutative 81.0%

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

      3. associate-/l* 81.0%

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

      4. associate-/l* 81.0%

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

      5. +-commutative 81.0%

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

      6. exp-diff 50.6%

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

      7. sub-neg 50.6%

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

      8. exp-sum 32.9%

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

      9. associate-/r* 32.9%

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

      10. exp-diff 39.2%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in K around 0 98.7%

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

      1. cos-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in l around inf 88.8%

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

      1. neg-mul-1 88.8%

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

   10. Simplified 88.8%

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

   11. Taylor expanded in M around 0 88.8%

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

   if -5.19999999999999986e214 < l < 105

   1. Initial program 75.4%

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

      1. associate-/l* 76.0%

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

      2. +-commutative 76.0%

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

      3. associate-/l* 75.4%

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

      4. associate-/l* 76.0%

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

      5. +-commutative 76.0%

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

      6. exp-diff 20.6%

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

      7. sub-neg 20.6%

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

      8. exp-sum 20.6%

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

      9. associate-/r* 20.6%

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

      10. exp-diff 25.7%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in K around 0 93.9%

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

      1. cos-neg 93.9%

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

   7. Simplified 93.9%

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

   8. Taylor expanded in m around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. unpow2 69.3%

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

      3. distribute-rgt-out 76.7%

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

      4. *-commutative 76.7%

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

      5. *-commutative 76.7%

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

   10. Simplified 76.7%

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

   11. Taylor expanded in m around inf 44.3%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+214} \lor \neg \left(\ell \leq 105\right):\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.6%

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

   2. +-commutative 77.6%

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

   3. associate-/l* 77.2%

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

   4. associate-/l* 77.6%

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

   5. +-commutative 77.6%

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

   6. exp-diff 29.9%

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

   7. sub-neg 29.9%

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

   8. exp-sum 24.4%

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

   9. associate-/r* 24.4%

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

   10. exp-diff 29.9%

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

3. Simplified 77.6%

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

5. Taylor expanded in K around 0 95.4%

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

   1. cos-neg 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in l around inf 36.9%

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

   1. neg-mul-1 36.9%

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

10. Simplified 36.9%

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

Alternative 13: 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 77.2%

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

   1. associate-/l* 77.6%

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

   2. +-commutative 77.6%

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

   3. associate-/l* 77.2%

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

   4. associate-/l* 77.6%

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

   5. +-commutative 77.6%

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

   6. exp-diff 29.9%

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

   7. sub-neg 29.9%

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

   8. exp-sum 24.4%

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

   9. associate-/r* 24.4%

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

   10. exp-diff 29.9%

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

3. Simplified 77.6%

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

5. Taylor expanded in K around 0 95.4%

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

   1. cos-neg 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in l around inf 36.9%

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

   1. neg-mul-1 36.9%

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

10. Simplified 36.9%

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

11. Taylor expanded in M around 0 36.5%

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

Alternative 14: 7.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \cos M \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(K, m, n, M, l)
	return cos(M)
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[Cos[M], $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.6%

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

   2. +-commutative 77.6%

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

   3. associate-/l* 77.2%

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

   4. associate-/l* 77.6%

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

   5. +-commutative 77.6%

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

   6. exp-diff 29.9%

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

   7. sub-neg 29.9%

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

   8. exp-sum 24.4%

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

   9. associate-/r* 24.4%

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

   10. exp-diff 29.9%

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

3. Simplified 77.6%

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

5. Taylor expanded in K around 0 95.4%

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

   1. cos-neg 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in l around inf 36.9%

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

   1. neg-mul-1 36.9%

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

10. Simplified 36.9%

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

11. Taylor expanded in l around 0 8.0%

    \[\leadsto \color{blue}{\cos M} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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)))))))