Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.3% → 96.6%

Time: 17.5s

Alternatives: 13

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: 76.3% 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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

   1. associate-/l* 79.4%

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

   2. exp-diff 24.7%

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

   3. sub-neg 24.7%

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

   4. exp-sum 18.0%

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

   5. associate-/r* 18.0%

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

   6. exp-diff 24.7%

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

   7. exp-diff 79.4%

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

   8. sub-neg 79.4%

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

   9. remove-double-neg 79.4%

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

   10. fabs-sub 79.4%

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

3. Simplified 79.4%

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

4. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

6. Simplified 98.1%

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

7. Final simplification 98.1%

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

Alternative 2: 84.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\cos M \cdot e^{t_0 + \left(\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(\left(n \cdot 0.5\right) \cdot \left(m + n \cdot 0.5\right) + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 5e+123)
     (* (cos M) (exp (+ t_0 (- (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) l))))
     (* (cos M) (exp (- t_0 (+ (* (* n 0.5) (+ m (* n 0.5))) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= 5e+123) {
		tmp = cos(M) * exp((t_0 + ((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l)));
	} else {
		tmp = cos(M) * exp((t_0 - (((n * 0.5) * (m + (n * 0.5))) + 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 = abs((n - m))
    if (n <= 5d+123) then
        tmp = cos(m_1) * exp((t_0 + ((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) - l)))
    else
        tmp = cos(m_1) * exp((t_0 - (((n * 0.5d0) * (m + (n * 0.5d0))) + 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 = Math.abs((n - m));
	double tmp;
	if (n <= 5e+123) {
		tmp = Math.cos(M) * Math.exp((t_0 + ((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - (((n * 0.5) * (m + (n * 0.5))) + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 5e+123:
		tmp = math.cos(M) * math.exp((t_0 + ((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l)))
	else:
		tmp = math.cos(M) * math.exp((t_0 - (((n * 0.5) * (m + (n * 0.5))) + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 5e+123)
		tmp = Float64(cos(M) * exp(Float64(t_0 + Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) - l))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(Float64(n * 0.5) * Float64(m + Float64(n * 0.5))) + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (n <= 5e+123)
		tmp = cos(M) * exp((t_0 + ((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l)));
	else
		tmp = cos(M) * exp((t_0 - (((n * 0.5) * (m + (n * 0.5))) + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 5e+123], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(N[(n * 0.5), $MachinePrecision] * N[(m + N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 4.99999999999999974e123

   1. Initial program 80.1%

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

      1. associate-/l* 80.5%

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

      2. exp-diff 28.9%

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

      3. sub-neg 28.9%

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

      4. exp-sum 21.5%

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

      5. associate-/r* 21.5%

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

      6. exp-diff 29.4%

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

      7. exp-diff 80.5%

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

      8. sub-neg 80.5%

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

      9. remove-double-neg 80.5%

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

      10. fabs-sub 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in K around 0 97.7%

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

      1. cos-neg 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in n around 0 84.3%

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

      1. +-commutative 84.3%

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

      2. unpow2 84.3%

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

      3. distribute-rgt-out 87.6%

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

      4. *-commutative 87.6%

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

      5. *-commutative 87.6%

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

   9. Simplified 87.6%

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

   if 4.99999999999999974e123 < n

   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%

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

      2. exp-diff 2.4%

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

      3. sub-neg 2.4%

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

      4. exp-sum 0.0%

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

      5. associate-/r* 0.0%

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

      6. exp-diff 0.0%

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

      7. exp-diff 73.2%

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

      8. sub-neg 73.2%

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

      9. remove-double-neg 73.2%

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

      10. fabs-sub 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 82.9%

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

      1. +-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. distribute-rgt-out 97.6%

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

      4. *-commutative 97.6%

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

      5. *-commutative 97.6%

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

   9. Simplified 97.6%

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

   10. Taylor expanded in M around 0 97.6%

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

       1. associate-*r* 97.6%

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

       2. *-commutative 97.6%

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

       3. *-commutative 97.6%

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

   12. Simplified 97.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + \left(\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\left(n \cdot 0.5\right) \cdot \left(m + n \cdot 0.5\right) + \ell\right)}\\ \end{array} \]

Alternative 3: 86.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.3 \cdot 10^{+73}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + \left(\left(m + \left(n \cdot 0.5 - M\right)\right) \cdot \left(M - n \cdot 0.5\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1.3e+73)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (*
    (cos M)
    (exp (+ (fabs (- n m)) (- (* (+ m (- (* n 0.5) M)) (- M (* n 0.5))) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.3e+73) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp((fabs((n - m)) + (((m + ((n * 0.5) - M)) * (M - (n * 0.5))) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-1.3d+73)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp((abs((n - m)) + (((m + ((n * 0.5d0) - m_1)) * (m_1 - (n * 0.5d0))) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.3e+73) {
		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 + ((n * 0.5) - M)) * (M - (n * 0.5))) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1.3e+73:
		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 + ((n * 0.5) - M)) * (M - (n * 0.5))) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1.3e+73)
		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 + Float64(Float64(n * 0.5) - M)) * Float64(M - Float64(n * 0.5))) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1.3e+73)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp((abs((n - m)) + (((m + ((n * 0.5) - M)) * (M - (n * 0.5))) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1.3e+73], 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 + N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.3e73

   1. Initial program 78.6%

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

      1. associate-/l* 78.6%

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

      2. exp-diff 11.9%

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

      3. sub-neg 11.9%

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

      4. exp-sum 0.0%

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

      5. associate-/r* 0.0%

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

      6. exp-diff 0.0%

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

      7. exp-diff 78.6%

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

      8. sub-neg 78.6%

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

      9. remove-double-neg 78.6%

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

      10. fabs-sub 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. distribute-rgt-out 88.3%

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

      4. *-commutative 88.3%

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

      5. *-commutative 88.3%

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

   9. Simplified 88.3%

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

   10. Taylor expanded in m around inf 100.0%

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

   if -1.3e73 < m

   1. Initial program 79.1%

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

      1. associate-/l* 79.5%

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

      2. exp-diff 27.2%

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

      3. sub-neg 27.2%

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

      4. exp-sum 21.6%

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

      5. associate-/r* 21.6%

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

      6. exp-diff 29.5%

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

      7. exp-diff 79.5%

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

      8. sub-neg 79.5%

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

      9. remove-double-neg 79.5%

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

      10. fabs-sub 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in K around 0 97.7%

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

      1. cos-neg 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in m around 0 79.6%

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

      1. +-commutative 79.6%

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

      2. unpow2 79.6%

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

      3. distribute-rgt-out 84.8%

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

      4. *-commutative 84.8%

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

      5. *-commutative 84.8%

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

   9. Simplified 84.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.3 \cdot 10^{+73}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + \left(\left(m + \left(n \cdot 0.5 - M\right)\right) \cdot \left(M - n \cdot 0.5\right) - \ell\right)}\\ \end{array} \]

Alternative 4: 74.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -13200000:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\left(n \cdot 0.5\right) \cdot \left(m + n \cdot 0.5\right) + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -13200000.0)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (* (cos M) (exp (- (fabs (- n m)) (+ (* (* n 0.5) (+ m (* n 0.5))) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -13200000.0) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp((fabs((n - m)) - (((n * 0.5) * (m + (n * 0.5))) + l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-13200000.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp((abs((n - m)) - (((n * 0.5d0) * (m + (n * 0.5d0))) + l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -13200000.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp((Math.abs((n - m)) - (((n * 0.5) * (m + (n * 0.5))) + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -13200000.0:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp((math.fabs((n - m)) - (((n * 0.5) * (m + (n * 0.5))) + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -13200000.0)
		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(n * 0.5) * Float64(m + Float64(n * 0.5))) + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -13200000.0)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp((abs((n - m)) - (((n * 0.5) * (m + (n * 0.5))) + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -13200000.0], 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[(n * 0.5), $MachinePrecision] * N[(m + N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.32e7

   1. Initial program 79.3%

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

      1. associate-/l* 79.3%

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

      2. exp-diff 12.1%

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

      3. sub-neg 12.1%

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

      4. exp-sum 0.0%

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

      5. associate-/r* 0.0%

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

      6. exp-diff 3.4%

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

      7. exp-diff 79.3%

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

      8. sub-neg 79.3%

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

      9. remove-double-neg 79.3%

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

      10. fabs-sub 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 86.3%

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

      1. +-commutative 86.3%

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

      2. unpow2 86.3%

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

      3. distribute-rgt-out 89.8%

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

      4. *-commutative 89.8%

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

      5. *-commutative 89.8%

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

   9. Simplified 89.8%

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

   10. Taylor expanded in m around inf 94.9%

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

   if -1.32e7 < m

   1. Initial program 78.9%

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

      1. associate-/l* 79.4%

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

      2. exp-diff 28.4%

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

      3. sub-neg 28.4%

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

      4. exp-sum 23.3%

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

      5. associate-/r* 23.3%

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

      6. exp-diff 30.9%

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

      7. exp-diff 79.4%

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

      8. sub-neg 79.4%

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

      9. remove-double-neg 79.4%

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

      10. fabs-sub 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in K around 0 97.5%

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

      1. cos-neg 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in m around 0 80.0%

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

      1. +-commutative 80.0%

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

      2. unpow2 80.0%

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

      3. distribute-rgt-out 84.6%

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

      4. *-commutative 84.6%

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

      5. *-commutative 84.6%

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

   9. Simplified 84.6%

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

   10. Taylor expanded in M around 0 62.8%

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

       1. associate-*r* 62.8%

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

       2. *-commutative 62.8%

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

       3. *-commutative 62.8%

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

   12. Simplified 62.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -13200000:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\left(n \cdot 0.5\right) \cdot \left(m + n \cdot 0.5\right) + \ell\right)}\\ \end{array} \]

Alternative 5: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ t_1 := \cos M \cdot e^{-{M}^{2}}\\ t_2 := e^{-\ell}\\ \mathbf{if}\;M \leq -27:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \leq 1.8 \cdot 10^{-272}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 5.8 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;M \leq 4.6 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;t_2 \cdot \cos \left(\frac{K}{\frac{2}{n}} - M\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (* -0.25 (pow m 2.0)))))
        (t_1 (* (cos M) (exp (- (pow M 2.0)))))
        (t_2 (exp (- l))))
   (if (<= M -27.0)
     t_1
     (if (<= M 1.8e-272)
       t_0
       (if (<= M 5.8e-133)
         t_2
         (if (<= M 4.6e-61)
           t_0
           (if (<= M 27.0) (* t_2 (cos (- (/ K (/ 2.0 n)) M))) t_1)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp((-0.25 * pow(m, 2.0)));
	double t_1 = cos(M) * exp(-pow(M, 2.0));
	double t_2 = exp(-l);
	double tmp;
	if (M <= -27.0) {
		tmp = t_1;
	} else if (M <= 1.8e-272) {
		tmp = t_0;
	} else if (M <= 5.8e-133) {
		tmp = t_2;
	} else if (M <= 4.6e-61) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = t_2 * cos(((K / (2.0 / n)) - M));
	} 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(((-0.25d0) * (m ** 2.0d0)))
    t_1 = cos(m_1) * exp(-(m_1 ** 2.0d0))
    t_2 = exp(-l)
    if (m_1 <= (-27.0d0)) then
        tmp = t_1
    else if (m_1 <= 1.8d-272) then
        tmp = t_0
    else if (m_1 <= 5.8d-133) then
        tmp = t_2
    else if (m_1 <= 4.6d-61) then
        tmp = t_0
    else if (m_1 <= 27.0d0) then
        tmp = t_2 * cos(((k / (2.0d0 / n)) - m_1))
    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((-0.25 * Math.pow(m, 2.0)));
	double t_1 = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	double t_2 = Math.exp(-l);
	double tmp;
	if (M <= -27.0) {
		tmp = t_1;
	} else if (M <= 1.8e-272) {
		tmp = t_0;
	} else if (M <= 5.8e-133) {
		tmp = t_2;
	} else if (M <= 4.6e-61) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = t_2 * Math.cos(((K / (2.0 / n)) - M));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	t_1 = math.cos(M) * math.exp(-math.pow(M, 2.0))
	t_2 = math.exp(-l)
	tmp = 0
	if M <= -27.0:
		tmp = t_1
	elif M <= 1.8e-272:
		tmp = t_0
	elif M <= 5.8e-133:
		tmp = t_2
	elif M <= 4.6e-61:
		tmp = t_0
	elif M <= 27.0:
		tmp = t_2 * math.cos(((K / (2.0 / n)) - M))
	else:
		tmp = t_1
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))))
	t_1 = Float64(cos(M) * exp(Float64(-(M ^ 2.0))))
	t_2 = exp(Float64(-l))
	tmp = 0.0
	if (M <= -27.0)
		tmp = t_1;
	elseif (M <= 1.8e-272)
		tmp = t_0;
	elseif (M <= 5.8e-133)
		tmp = t_2;
	elseif (M <= 4.6e-61)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = Float64(t_2 * cos(Float64(Float64(K / Float64(2.0 / n)) - M)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp((-0.25 * (m ^ 2.0)));
	t_1 = cos(M) * exp(-(M ^ 2.0));
	t_2 = exp(-l);
	tmp = 0.0;
	if (M <= -27.0)
		tmp = t_1;
	elseif (M <= 1.8e-272)
		tmp = t_0;
	elseif (M <= 5.8e-133)
		tmp = t_2;
	elseif (M <= 4.6e-61)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = t_2 * cos(((K / (2.0 / n)) - M));
	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[(-0.25 * N[Power[m, 2.0], $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[Exp[(-l)], $MachinePrecision]}, If[LessEqual[M, -27.0], t$95$1, If[LessEqual[M, 1.8e-272], t$95$0, If[LessEqual[M, 5.8e-133], t$95$2, If[LessEqual[M, 4.6e-61], t$95$0, If[LessEqual[M, 27.0], N[(t$95$2 * N[Cos[N[(N[(K / N[(2.0 / n), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -27 or 27 < M

   1. Initial program 83.1%

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

      1. associate-/l* 83.1%

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

      2. exp-diff 25.4%

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

      3. sub-neg 25.4%

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

      4. exp-sum 20.0%

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

      5. associate-/r* 20.0%

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

      6. exp-diff 28.5%

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

      7. exp-diff 83.1%

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

      8. sub-neg 83.1%

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

      9. remove-double-neg 83.1%

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

      10. fabs-sub 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in n around 0 89.3%

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

      1. +-commutative 89.3%

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

      2. unpow2 89.3%

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

      3. distribute-rgt-out 94.7%

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

      4. *-commutative 94.7%

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

      5. *-commutative 94.7%

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

   9. Simplified 94.7%

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

   10. Taylor expanded in M around inf 96.2%

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

       1. mul-1-neg 96.2%

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

   12. Simplified 96.2%

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

   if -27 < M < 1.79999999999999984e-272 or 5.7999999999999997e-133 < M < 4.59999999999999984e-61

   1. Initial program 74.0%

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

      1. associate-/l* 74.0%

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

      2. exp-diff 18.0%

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

      3. sub-neg 18.0%

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

      4. exp-sum 10.9%

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

      5. associate-/r* 10.9%

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

      6. exp-diff 13.3%

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

      7. exp-diff 74.0%

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

      8. sub-neg 74.0%

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

      9. remove-double-neg 74.0%

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

      10. fabs-sub 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in K around 0 97.6%

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

      1. cos-neg 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in n around 0 64.8%

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

      1. +-commutative 64.8%

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

      2. unpow2 64.8%

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

      3. distribute-rgt-out 68.3%

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

      4. *-commutative 68.3%

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

      5. *-commutative 68.3%

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

   9. Simplified 68.3%

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

   10. Taylor expanded in m around inf 65.3%

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

   if 1.79999999999999984e-272 < M < 5.7999999999999997e-133

   1. Initial program 76.9%

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

      1. associate-/l* 80.8%

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

      2. exp-diff 34.6%

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

      3. sub-neg 34.6%

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

      4. exp-sum 30.8%

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

      5. associate-/r* 30.8%

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

      6. exp-diff 38.5%

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

      7. exp-diff 80.8%

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

      8. sub-neg 80.8%

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

      9. remove-double-neg 80.8%

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

      10. fabs-sub 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 58.3%

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

      1. +-commutative 58.3%

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

      2. unpow2 58.3%

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

      3. distribute-rgt-out 58.4%

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

      4. *-commutative 58.4%

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

      5. *-commutative 58.4%

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

   9. Simplified 58.4%

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

   10. Taylor expanded in l around inf 73.9%

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

       1. neg-mul-1 73.9%

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

   12. Simplified 73.9%

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

   13. Taylor expanded in M around 0 73.9%

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

   if 4.59999999999999984e-61 < M < 27

   1. Initial program 75.1%

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

      1. associate-/l* 75.1%

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

      2. exp-diff 37.6%

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

      3. sub-neg 37.6%

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

      4. exp-sum 18.8%

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

      5. associate-/r* 18.8%

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

      6. exp-diff 31.3%

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

      7. exp-diff 75.1%

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

      8. sub-neg 75.1%

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

      9. remove-double-neg 75.1%

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

      10. fabs-sub 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in n around 0 56.6%

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

      1. +-commutative 63.0%

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

      2. unpow2 63.0%

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

      3. distribute-rgt-out 63.0%

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

      4. *-commutative 63.0%

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

      5. *-commutative 63.0%

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

   6. Simplified 56.6%

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

   7. Taylor expanded in m around 0 50.5%

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

      1. associate-*r* 50.5%

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

      2. neg-mul-1 50.5%

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

   9. Simplified 50.5%

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

   10. Taylor expanded in m around 0 50.7%

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

   11. Taylor expanded in l around inf 63.3%

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

       1. neg-mul-1 63.3%

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

   13. Simplified 63.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -27:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;M \leq 1.8 \cdot 10^{-272}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;M \leq 5.8 \cdot 10^{-133}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;M \leq 4.6 \cdot 10^{-61}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{-\ell} \cdot \cos \left(\frac{K}{\frac{2}{n}} - M\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \]

Alternative 6: 64.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.8 \cdot 10^{-119}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{elif}\;M \leq 1.65 \cdot 10^{+20}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -1.8e-119)
   (* (cos M) (exp (* M (- n M))))
   (if (<= M 1.65e+20) (/ (cos M) (exp l)) (* (cos M) (exp (- (pow M 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -1.8e-119) {
		tmp = cos(M) * exp((M * (n - M)));
	} else if (M <= 1.65e+20) {
		tmp = cos(M) / exp(l);
	} else {
		tmp = cos(M) * exp(-pow(M, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m_1 <= (-1.8d-119)) then
        tmp = cos(m_1) * exp((m_1 * (n - m_1)))
    else if (m_1 <= 1.65d+20) then
        tmp = cos(m_1) / exp(l)
    else
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -1.8e-119) {
		tmp = Math.cos(M) * Math.exp((M * (n - M)));
	} else if (M <= 1.65e+20) {
		tmp = Math.cos(M) / Math.exp(l);
	} else {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -1.8e-119:
		tmp = math.cos(M) * math.exp((M * (n - M)))
	elif M <= 1.65e+20:
		tmp = math.cos(M) / math.exp(l)
	else:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -1.8e-119)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(n - M))));
	elseif (M <= 1.65e+20)
		tmp = Float64(cos(M) / exp(l));
	else
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -1.8e-119)
		tmp = cos(M) * exp((M * (n - M)));
	elseif (M <= 1.65e+20)
		tmp = cos(M) / exp(l);
	else
		tmp = cos(M) * exp(-(M ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -1.8e-119], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 1.65e+20], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -1.8e-119

   1. Initial program 80.0%

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

      1. associate-/l* 80.0%

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

      2. exp-diff 18.8%

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

      3. sub-neg 18.8%

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

      4. exp-sum 13.8%

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

      5. associate-/r* 13.8%

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

      6. exp-diff 21.3%

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

      7. exp-diff 80.0%

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

      8. sub-neg 80.0%

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

      9. remove-double-neg 80.0%

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

      10. fabs-sub 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in K around 0 97.5%

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

      1. cos-neg 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in n around 0 83.9%

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

      1. +-commutative 83.9%

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

      2. unpow2 83.9%

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

      3. distribute-rgt-out 88.9%

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

      4. *-commutative 88.9%

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

      5. *-commutative 88.9%

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

   9. Simplified 88.9%

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

   10. Taylor expanded in M around -inf 63.0%

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

       1. +-commutative 63.0%

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

       2. mul-1-neg 63.0%

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

       3. unsub-neg 63.0%

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

       4. +-commutative 63.0%

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

   12. Simplified 63.0%

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

   13. Taylor expanded in m around 0 69.4%

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

       1. unpow2 69.4%

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

       2. distribute-lft-out-- 71.9%

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

   15. Simplified 71.9%

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

   if -1.8e-119 < M < 1.65e20

   1. Initial program 76.3%

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

      1. associate-/l* 77.2%

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

      2. exp-diff 26.8%

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

      3. sub-neg 26.8%

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

      4. exp-sum 17.6%

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

      5. associate-/r* 17.6%

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

      6. exp-diff 24.0%

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

      7. exp-diff 77.2%

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

      8. sub-neg 77.2%

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

      9. remove-double-neg 77.2%

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

      10. fabs-sub 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in K around 0 97.3%

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

      1. cos-neg 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in n around 0 63.8%

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

      1. +-commutative 63.8%

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

      2. unpow2 63.8%

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

      3. distribute-rgt-out 64.8%

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

      4. *-commutative 64.8%

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

      5. *-commutative 64.8%

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

   9. Simplified 64.8%

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

   10. Taylor expanded in l around inf 53.5%

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

       1. neg-mul-1 53.5%

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

   12. Simplified 53.5%

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

   13. Taylor expanded in l around -inf 53.5%

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

       1. rem-exp-log 53.5%

          \[\leadsto \color{blue}{e^{\log \cos M}} \cdot e^{-1 \cdot \ell} \]

       2. neg-mul-1 53.5%

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

       3. exp-sum 53.5%

          \[\leadsto \color{blue}{e^{\log \cos M + \left(-\ell\right)}} \]

       4. unsub-neg 53.5%

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

       5. exp-diff 53.5%

          \[\leadsto \color{blue}{\frac{e^{\log \cos M}}{e^{\ell}}} \]

       6. rem-exp-log 53.5%

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

   15. Simplified 53.5%

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

   if 1.65e20 < M

   1. Initial program 82.1%

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

      1. associate-/l* 82.1%

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

      2. exp-diff 28.4%

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

      3. sub-neg 28.4%

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

      4. exp-sum 23.9%

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

      5. associate-/r* 23.9%

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

      6. exp-diff 29.9%

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

      7. exp-diff 82.1%

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

      8. sub-neg 82.1%

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

      9. remove-double-neg 82.1%

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

      10. fabs-sub 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 88.1%

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

      1. +-commutative 88.1%

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

      2. unpow2 88.1%

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

      3. distribute-rgt-out 95.6%

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

      4. *-commutative 95.6%

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

      5. *-commutative 95.6%

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

   9. Simplified 95.6%

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

   10. Taylor expanded in M around inf 98.5%

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

       1. mul-1-neg 98.5%

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

   12. Simplified 98.5%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.8 \cdot 10^{-119}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{elif}\;M \leq 1.65 \cdot 10^{+20}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \]

Alternative 7: 61.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -7.6 \cdot 10^{-119}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{elif}\;M \leq 8 \cdot 10^{+29}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -7.6e-119)
   (* (cos M) (exp (* M (- n M))))
   (if (<= M 8e+29) (/ (cos M) (exp l)) (* (cos M) (exp (* M (- m M)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -7.6e-119) {
		tmp = cos(M) * exp((M * (n - M)));
	} else if (M <= 8e+29) {
		tmp = cos(M) / exp(l);
	} else {
		tmp = cos(M) * exp((M * (m - M)));
	}
	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 <= (-7.6d-119)) then
        tmp = cos(m_1) * exp((m_1 * (n - m_1)))
    else if (m_1 <= 8d+29) then
        tmp = cos(m_1) / exp(l)
    else
        tmp = cos(m_1) * exp((m_1 * (m - m_1)))
    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 <= -7.6e-119) {
		tmp = Math.cos(M) * Math.exp((M * (n - M)));
	} else if (M <= 8e+29) {
		tmp = Math.cos(M) / Math.exp(l);
	} else {
		tmp = Math.cos(M) * Math.exp((M * (m - M)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -7.6e-119:
		tmp = math.cos(M) * math.exp((M * (n - M)))
	elif M <= 8e+29:
		tmp = math.cos(M) / math.exp(l)
	else:
		tmp = math.cos(M) * math.exp((M * (m - M)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -7.6e-119)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(n - M))));
	elseif (M <= 8e+29)
		tmp = Float64(cos(M) / exp(l));
	else
		tmp = Float64(cos(M) * exp(Float64(M * Float64(m - M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -7.6e-119)
		tmp = cos(M) * exp((M * (n - M)));
	elseif (M <= 8e+29)
		tmp = cos(M) / exp(l);
	else
		tmp = cos(M) * exp((M * (m - M)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -7.6e-119], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 8e+29], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -7.59999999999999949e-119

   1. Initial program 80.0%

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

      1. associate-/l* 80.0%

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

      2. exp-diff 18.8%

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

      3. sub-neg 18.8%

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

      4. exp-sum 13.8%

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

      5. associate-/r* 13.8%

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

      6. exp-diff 21.3%

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

      7. exp-diff 80.0%

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

      8. sub-neg 80.0%

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

      9. remove-double-neg 80.0%

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

      10. fabs-sub 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in K around 0 97.5%

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

      1. cos-neg 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in n around 0 83.9%

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

      1. +-commutative 83.9%

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

      2. unpow2 83.9%

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

      3. distribute-rgt-out 88.9%

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

      4. *-commutative 88.9%

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

      5. *-commutative 88.9%

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

   9. Simplified 88.9%

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

   10. Taylor expanded in M around -inf 63.0%

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

       1. +-commutative 63.0%

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

       2. mul-1-neg 63.0%

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

       3. unsub-neg 63.0%

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

       4. +-commutative 63.0%

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

   12. Simplified 63.0%

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

   13. Taylor expanded in m around 0 69.4%

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

       1. unpow2 69.4%

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

       2. distribute-lft-out-- 71.9%

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

   15. Simplified 71.9%

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

   if -7.59999999999999949e-119 < M < 7.99999999999999931e29

   1. Initial program 75.6%

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

      1. associate-/l* 76.5%

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

      2. exp-diff 28.2%

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

      3. sub-neg 28.2%

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

      4. exp-sum 17.7%

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

      5. associate-/r* 17.7%

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

      6. exp-diff 23.8%

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

      7. exp-diff 76.5%

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

      8. sub-neg 76.5%

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

      9. remove-double-neg 76.5%

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

      10. fabs-sub 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in K around 0 97.4%

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

      1. cos-neg 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in n around 0 65.4%

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

      1. +-commutative 65.4%

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

      2. unpow2 65.4%

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

      3. distribute-rgt-out 66.3%

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

      4. *-commutative 66.3%

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

      5. *-commutative 66.3%

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

   9. Simplified 66.3%

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

   10. Taylor expanded in l around inf 54.7%

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

       1. neg-mul-1 54.7%

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

   12. Simplified 54.7%

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

   13. Taylor expanded in l around -inf 54.7%

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

       1. rem-exp-log 52.9%

          \[\leadsto \color{blue}{e^{\log \cos M}} \cdot e^{-1 \cdot \ell} \]

       2. neg-mul-1 52.9%

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

       3. exp-sum 52.9%

          \[\leadsto \color{blue}{e^{\log \cos M + \left(-\ell\right)}} \]

       4. unsub-neg 52.9%

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

       5. exp-diff 52.9%

          \[\leadsto \color{blue}{\frac{e^{\log \cos M}}{e^{\ell}}} \]

       6. rem-exp-log 54.7%

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

   15. Simplified 54.7%

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

   if 7.99999999999999931e29 < M

   1. Initial program 83.9%

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

      1. associate-/l* 83.9%

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

      2. exp-diff 25.8%

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

      3. sub-neg 25.8%

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

      4. exp-sum 24.2%

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

      5. associate-/r* 24.2%

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

      6. exp-diff 30.6%

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

      7. exp-diff 83.9%

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

      8. sub-neg 83.9%

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

      9. remove-double-neg 83.9%

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

      10. fabs-sub 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 87.2%

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

      1. +-commutative 87.2%

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

      2. unpow2 87.2%

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

      3. distribute-rgt-out 95.2%

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

      4. *-commutative 95.2%

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

      5. *-commutative 95.2%

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

   9. Simplified 95.2%

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

   10. Taylor expanded in M around -inf 66.3%

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

       1. +-commutative 66.3%

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

       2. mul-1-neg 66.3%

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

       3. unsub-neg 66.3%

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

       4. +-commutative 66.3%

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

   12. Simplified 66.3%

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

   13. Taylor expanded in n around 0 75.9%

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

       1. unpow2 75.9%

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

       2. distribute-lft-out-- 84.1%

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

   15. Simplified 84.1%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7.6 \cdot 10^{-119}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{elif}\;M \leq 8 \cdot 10^{+29}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \end{array} \]

Alternative 8: 39.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -4.55 \cdot 10^{-119}:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \mathbf{elif}\;M \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -4.55e-119)
   (* (cos M) (exp (* M n)))
   (if (<= M 2.1e+31) (/ (cos M) (exp l)) (* (cos M) (exp (* M m))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -4.55e-119) {
		tmp = cos(M) * exp((M * n));
	} else if (M <= 2.1e+31) {
		tmp = cos(M) / exp(l);
	} else {
		tmp = cos(M) * exp((M * m));
	}
	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 <= (-4.55d-119)) then
        tmp = cos(m_1) * exp((m_1 * n))
    else if (m_1 <= 2.1d+31) then
        tmp = cos(m_1) / exp(l)
    else
        tmp = cos(m_1) * exp((m_1 * m))
    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 <= -4.55e-119) {
		tmp = Math.cos(M) * Math.exp((M * n));
	} else if (M <= 2.1e+31) {
		tmp = Math.cos(M) / Math.exp(l);
	} else {
		tmp = Math.cos(M) * Math.exp((M * m));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -4.55e-119:
		tmp = math.cos(M) * math.exp((M * n))
	elif M <= 2.1e+31:
		tmp = math.cos(M) / math.exp(l)
	else:
		tmp = math.cos(M) * math.exp((M * m))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -4.55e-119)
		tmp = Float64(cos(M) * exp(Float64(M * n)));
	elseif (M <= 2.1e+31)
		tmp = Float64(cos(M) / exp(l));
	else
		tmp = Float64(cos(M) * exp(Float64(M * m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -4.55e-119)
		tmp = cos(M) * exp((M * n));
	elseif (M <= 2.1e+31)
		tmp = cos(M) / exp(l);
	else
		tmp = cos(M) * exp((M * m));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -4.55e-119], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 2.1e+31], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -4.55e-119

   1. Initial program 80.0%

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

      1. associate-/l* 80.0%

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

      2. exp-diff 18.8%

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

      3. sub-neg 18.8%

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

      4. exp-sum 13.8%

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

      5. associate-/r* 13.8%

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

      6. exp-diff 21.3%

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

      7. exp-diff 80.0%

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

      8. sub-neg 80.0%

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

      9. remove-double-neg 80.0%

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

      10. fabs-sub 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in K around 0 97.5%

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

      1. cos-neg 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in n around 0 83.9%

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

      1. +-commutative 83.9%

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

      2. unpow2 83.9%

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

      3. distribute-rgt-out 88.9%

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

      4. *-commutative 88.9%

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

      5. *-commutative 88.9%

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

   9. Simplified 88.9%

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

   10. Taylor expanded in M around -inf 63.0%

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

       1. +-commutative 63.0%

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

       2. mul-1-neg 63.0%

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

       3. unsub-neg 63.0%

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

       4. +-commutative 63.0%

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

   12. Simplified 63.0%

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

   13. Taylor expanded in n around inf 35.3%

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

   if -4.55e-119 < M < 2.09999999999999979e31

   1. Initial program 75.6%

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

      1. associate-/l* 76.5%

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

      2. exp-diff 28.2%

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

      3. sub-neg 28.2%

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

      4. exp-sum 17.7%

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

      5. associate-/r* 17.7%

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

      6. exp-diff 23.8%

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

      7. exp-diff 76.5%

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

      8. sub-neg 76.5%

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

      9. remove-double-neg 76.5%

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

      10. fabs-sub 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in K around 0 97.4%

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

      1. cos-neg 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in n around 0 65.4%

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

      1. +-commutative 65.4%

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

      2. unpow2 65.4%

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

      3. distribute-rgt-out 66.3%

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

      4. *-commutative 66.3%

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

      5. *-commutative 66.3%

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

   9. Simplified 66.3%

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

   10. Taylor expanded in l around inf 54.7%

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

       1. neg-mul-1 54.7%

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

   12. Simplified 54.7%

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

   13. Taylor expanded in l around -inf 54.7%

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

       1. rem-exp-log 52.9%

          \[\leadsto \color{blue}{e^{\log \cos M}} \cdot e^{-1 \cdot \ell} \]

       2. neg-mul-1 52.9%

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

       3. exp-sum 52.9%

          \[\leadsto \color{blue}{e^{\log \cos M + \left(-\ell\right)}} \]

       4. unsub-neg 52.9%

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

       5. exp-diff 52.9%

          \[\leadsto \color{blue}{\frac{e^{\log \cos M}}{e^{\ell}}} \]

       6. rem-exp-log 54.7%

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

   15. Simplified 54.7%

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

   if 2.09999999999999979e31 < M

   1. Initial program 83.9%

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

      1. associate-/l* 83.9%

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

      2. exp-diff 25.8%

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

      3. sub-neg 25.8%

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

      4. exp-sum 24.2%

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

      5. associate-/r* 24.2%

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

      6. exp-diff 30.6%

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

      7. exp-diff 83.9%

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

      8. sub-neg 83.9%

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

      9. remove-double-neg 83.9%

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

      10. fabs-sub 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 87.2%

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

      1. +-commutative 87.2%

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

      2. unpow2 87.2%

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

      3. distribute-rgt-out 95.2%

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

      4. *-commutative 95.2%

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

      5. *-commutative 95.2%

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

   9. Simplified 95.2%

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

   10. Taylor expanded in M around -inf 66.3%

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

       1. +-commutative 66.3%

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

       2. mul-1-neg 66.3%

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

       3. unsub-neg 66.3%

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

       4. +-commutative 66.3%

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

   12. Simplified 66.3%

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

   13. Taylor expanded in m around inf 29.1%

       \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot m}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4.55 \cdot 10^{-119}:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \mathbf{elif}\;M \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \end{array} \]

Alternative 9: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 700:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 700.0) (* (cos M) (exp (* M (- m M)))) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 700.0) {
		tmp = cos(M) * exp((M * (m - M)));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 700.0d0) then
        tmp = cos(m_1) * exp((m_1 * (m - m_1)))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 700.0) {
		tmp = Math.cos(M) * Math.exp((M * (m - M)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 700.0:
		tmp = math.cos(M) * math.exp((M * (m - M)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 700.0)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(m - M))));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 700.0)
		tmp = cos(M) * exp((M * (m - M)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 700.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 700

   1. Initial program 77.4%

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

      1. associate-/l* 77.9%

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

      2. exp-diff 14.7%

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

      3. sub-neg 14.7%

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

      4. exp-sum 14.7%

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

      5. associate-/r* 14.7%

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

      6. exp-diff 23.9%

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

      7. exp-diff 77.9%

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

      8. sub-neg 77.9%

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

      9. remove-double-neg 77.9%

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

      10. fabs-sub 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in K around 0 97.3%

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

      1. cos-neg 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in n around 0 74.9%

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

      1. +-commutative 74.9%

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

      2. unpow2 74.9%

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

      3. distribute-rgt-out 78.7%

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

      4. *-commutative 78.7%

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

      5. *-commutative 78.7%

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

   9. Simplified 78.7%

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

   10. Taylor expanded in M around -inf 48.5%

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

       1. +-commutative 48.5%

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

       2. mul-1-neg 48.5%

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

       3. unsub-neg 48.5%

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

       4. +-commutative 48.5%

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

   12. Simplified 48.5%

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

   13. Taylor expanded in n around 0 50.3%

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

       1. unpow2 50.3%

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

       2. distribute-lft-out-- 54.7%

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

   15. Simplified 54.7%

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

   if 700 < l

   1. Initial program 83.1%

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

      1. associate-/l* 83.1%

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

      2. exp-diff 50.7%

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

      3. sub-neg 50.7%

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

      4. exp-sum 26.8%

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

      5. associate-/r* 26.8%

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

      6. exp-diff 26.8%

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

      7. exp-diff 83.1%

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

      8. sub-neg 83.1%

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

      9. remove-double-neg 83.1%

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

      10. fabs-sub 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. unpow2 80.5%

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

      3. distribute-rgt-out 84.7%

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

      4. *-commutative 84.7%

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

      5. *-commutative 84.7%

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

   9. Simplified 84.7%

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

   10. Taylor expanded in l around inf 100.0%

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

       1. neg-mul-1 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in M around 0 100.0%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 700:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 10: 48.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 700:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 700.0) (* (cos M) (exp (* M m))) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 700.0) {
		tmp = cos(M) * exp((M * m));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 700.0d0) then
        tmp = cos(m_1) * exp((m_1 * m))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 700.0) {
		tmp = Math.cos(M) * Math.exp((M * m));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 700.0:
		tmp = math.cos(M) * math.exp((M * m))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 700.0)
		tmp = Float64(cos(M) * exp(Float64(M * m)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 700.0)
		tmp = cos(M) * exp((M * m));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 700.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 700

   1. Initial program 77.4%

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

      1. associate-/l* 77.9%

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

      2. exp-diff 14.7%

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

      3. sub-neg 14.7%

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

      4. exp-sum 14.7%

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

      5. associate-/r* 14.7%

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

      6. exp-diff 23.9%

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

      7. exp-diff 77.9%

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

      8. sub-neg 77.9%

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

      9. remove-double-neg 77.9%

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

      10. fabs-sub 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in K around 0 97.3%

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

      1. cos-neg 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in n around 0 74.9%

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

      1. +-commutative 74.9%

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

      2. unpow2 74.9%

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

      3. distribute-rgt-out 78.7%

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

      4. *-commutative 78.7%

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

      5. *-commutative 78.7%

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

   9. Simplified 78.7%

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

   10. Taylor expanded in M around -inf 48.5%

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

       1. +-commutative 48.5%

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

       2. mul-1-neg 48.5%

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

       3. unsub-neg 48.5%

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

       4. +-commutative 48.5%

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

   12. Simplified 48.5%

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

   13. Taylor expanded in m around inf 25.7%

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

   if 700 < l

   1. Initial program 83.1%

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

      1. associate-/l* 83.1%

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

      2. exp-diff 50.7%

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

      3. sub-neg 50.7%

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

      4. exp-sum 26.8%

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

      5. associate-/r* 26.8%

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

      6. exp-diff 26.8%

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

      7. exp-diff 83.1%

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

      8. sub-neg 83.1%

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

      9. remove-double-neg 83.1%

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

      10. fabs-sub 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. unpow2 80.5%

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

      3. distribute-rgt-out 84.7%

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

      4. *-commutative 84.7%

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

      5. *-commutative 84.7%

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

   9. Simplified 84.7%

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

   10. Taylor expanded in l around inf 100.0%

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

       1. neg-mul-1 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in M around 0 100.0%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 700:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 11: 36.0% accurate, 2.1× speedup

Math

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

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(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 79.0%

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

   1. associate-/l* 79.4%

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

   2. exp-diff 24.7%

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

   3. sub-neg 24.7%

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

   4. exp-sum 18.0%

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

   5. associate-/r* 18.0%

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

   6. exp-diff 24.7%

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

   7. exp-diff 79.4%

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

   8. sub-neg 79.4%

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

   9. remove-double-neg 79.4%

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

   10. fabs-sub 79.4%

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

3. Simplified 79.4%

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

4. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

6. Simplified 98.1%

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

7. Taylor expanded in n around 0 76.4%

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

   1. +-commutative 76.4%

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

   2. unpow2 76.4%

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

   3. distribute-rgt-out 80.4%

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

   4. *-commutative 80.4%

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

   5. *-commutative 80.4%

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

9. Simplified 80.4%

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

10. Taylor expanded in l around inf 36.8%

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

    1. neg-mul-1 36.8%

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

12. Simplified 36.8%

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

13. Taylor expanded in l around -inf 36.8%

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

    1. rem-exp-log 30.1%

       \[\leadsto \color{blue}{e^{\log \cos M}} \cdot e^{-1 \cdot \ell} \]

    2. neg-mul-1 30.1%

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

    3. exp-sum 30.1%

       \[\leadsto \color{blue}{e^{\log \cos M + \left(-\ell\right)}} \]

    4. unsub-neg 30.1%

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

    5. exp-diff 30.1%

       \[\leadsto \color{blue}{\frac{e^{\log \cos M}}{e^{\ell}}} \]

    6. rem-exp-log 36.8%

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

15. Simplified 36.8%

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

16. Final simplification 36.8%

    \[\leadsto \frac{\cos M}{e^{\ell}} \]

Alternative 12: 35.7% 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 79.0%

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

   1. associate-/l* 79.4%

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

   2. exp-diff 24.7%

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

   3. sub-neg 24.7%

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

   4. exp-sum 18.0%

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

   5. associate-/r* 18.0%

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

   6. exp-diff 24.7%

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

   7. exp-diff 79.4%

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

   8. sub-neg 79.4%

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

   9. remove-double-neg 79.4%

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

   10. fabs-sub 79.4%

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

3. Simplified 79.4%

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

4. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

6. Simplified 98.1%

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

7. Taylor expanded in n around 0 76.4%

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

   1. +-commutative 76.4%

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

   2. unpow2 76.4%

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

   3. distribute-rgt-out 80.4%

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

   4. *-commutative 80.4%

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

   5. *-commutative 80.4%

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

9. Simplified 80.4%

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

10. Taylor expanded in l around inf 36.8%

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

    1. neg-mul-1 36.8%

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

12. Simplified 36.8%

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

13. Taylor expanded in M around 0 36.1%

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

14. Final simplification 36.1%

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

Alternative 13: 7.4% 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 79.0%

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

   1. associate-/l* 79.4%

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

   2. exp-diff 24.7%

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

   3. sub-neg 24.7%

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

   4. exp-sum 18.0%

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

   5. associate-/r* 18.0%

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

   6. exp-diff 24.7%

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

   7. exp-diff 79.4%

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

   8. sub-neg 79.4%

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

   9. remove-double-neg 79.4%

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

   10. fabs-sub 79.4%

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

3. Simplified 79.4%

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

4. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

6. Simplified 98.1%

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

7. Taylor expanded in n around 0 76.4%

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

   1. +-commutative 76.4%

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

   2. unpow2 76.4%

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

   3. distribute-rgt-out 80.4%

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

   4. *-commutative 80.4%

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

   5. *-commutative 80.4%

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

9. Simplified 80.4%

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

10. Taylor expanded in l around inf 36.8%

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

    1. neg-mul-1 36.8%

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

12. Simplified 36.8%

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

13. Taylor expanded in l around 0 5.4%

    \[\leadsto \color{blue}{\cos M} \]

14. Final simplification 5.4%

    \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023305 
(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)))))))