Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.6% → 96.5%

Time: 6.5s

Alternatives: 9

Speedup: 2.8×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    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: 76.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    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: 96.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

3. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. fabs-sub N/A

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

   7. lower-fabs.f64 N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

5. Applied rewrites 96.9%

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

6. Final simplification 96.9%

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

Alternative 2: 67.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-245}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 0.005:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left(M \cdot M + \left(\ell - \left|m - n\right|\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -3.3e-245)
   (* (cos M) (exp (* (* m m) -0.25)))
   (if (<= n 0.005)
     (*
      (cos (- (/ (* K (+ m n)) 2.0) M))
      (exp (- (+ (* M M) (- l (fabs (- m n)))))))
     (* (cos M) (exp (* -0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -3.3e-245) {
		tmp = cos(M) * exp(((m * m) * -0.25));
	} else if (n <= 0.005) {
		tmp = cos((((K * (m + n)) / 2.0) - M)) * exp(-((M * M) + (l - fabs((m - n)))));
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    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 (n <= (-3.3d-245)) then
        tmp = cos(m_1) * exp(((m * m) * (-0.25d0)))
    else if (n <= 0.005d0) then
        tmp = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp(-((m_1 * m_1) + (l - abs((m - n)))))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -3.3e-245) {
		tmp = Math.cos(M) * Math.exp(((m * m) * -0.25));
	} else if (n <= 0.005) {
		tmp = Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp(-((M * M) + (l - Math.abs((m - n)))));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -3.3e-245:
		tmp = math.cos(M) * math.exp(((m * m) * -0.25))
	elif n <= 0.005:
		tmp = math.cos((((K * (m + n)) / 2.0) - M)) * math.exp(-((M * M) + (l - math.fabs((m - n)))))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -3.3e-245)
		tmp = Float64(cos(M) * exp(Float64(Float64(m * m) * -0.25)));
	elseif (n <= 0.005)
		tmp = Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(-Float64(Float64(M * M) + Float64(l - abs(Float64(m - n)))))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -3.3e-245)
		tmp = cos(M) * exp(((m * m) * -0.25));
	elseif (n <= 0.005)
		tmp = cos((((K * (m + n)) / 2.0) - M)) * exp(-((M * M) + (l - abs((m - n)))));
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -3.3e-245], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.005], N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[(-N[(N[(M * M), $MachinePrecision] + N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.3000000000000001e-245

   1. Initial program 76.1%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(m \cdot m\right) \cdot \frac{-1}{4}} \]

      4. lower-*.f64 42.4

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   5. Applied rewrites 42.4%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(m \cdot m\right) \cdot \frac{-1}{4}} \]
   7. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \cos M \cdot e^{\left(m \cdot m\right) \cdot \frac{-1}{4}} \]

      2. lift-cos.f64 55.8

         \[\leadsto \cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   8. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   if -3.3000000000000001e-245 < n < 0.0050000000000000001

   1. Initial program 82.2%

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

   3. Taylor expanded in M around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

   if 0.0050000000000000001 < 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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around inf

      \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot \left(n \cdot n\right)} \]

      3. lower-*.f64 98.6

         \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   8. Applied rewrites 98.6%

      \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-245}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 0.005:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left(M \cdot M + \left(\ell - \left|m - n\right|\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-57}:\\ \;\;\;\;e^{-\ell} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.5 \cdot \left(K \cdot m\right)\right) - M\right)\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 5e-105)
   (* (cos M) (exp (* (* m m) -0.25)))
   (if (<= n 4e-57)
     (* (exp (- l)) (sin (- (fma 0.5 PI (* 0.5 (* K m))) M)))
     (if (<= n 55.0)
       (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (* (- M) M)))
       (* 1.0 (exp (* -0.25 (* n n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 5e-105) {
		tmp = cos(M) * exp(((m * m) * -0.25));
	} else if (n <= 4e-57) {
		tmp = exp(-l) * sin((fma(0.5, ((double) M_PI), (0.5 * (K * m))) - M));
	} else if (n <= 55.0) {
		tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-M * M));
	} else {
		tmp = 1.0 * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 5e-105)
		tmp = Float64(cos(M) * exp(Float64(Float64(m * m) * -0.25)));
	elseif (n <= 4e-57)
		tmp = Float64(exp(Float64(-l)) * sin(Float64(fma(0.5, pi, Float64(0.5 * Float64(K * m))) - M)));
	elseif (n <= 55.0)
		tmp = Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-M) * M)));
	else
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 5e-105], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4e-57], N[(N[Exp[(-l)], $MachinePrecision] * N[Sin[N[(N[(0.5 * Pi + N[(0.5 * N[(K * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 55.0], N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < 4.99999999999999963e-105

   1. Initial program 77.1%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(m \cdot m\right) \cdot \frac{-1}{4}} \]

      4. lower-*.f64 41.2

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(m \cdot m\right) \cdot \frac{-1}{4}} \]
   7. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \cos M \cdot e^{\left(m \cdot m\right) \cdot \frac{-1}{4}} \]

      2. lift-cos.f64 56.1

         \[\leadsto \cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   8. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   if 4.99999999999999963e-105 < n < 3.99999999999999982e-57

   1. Initial program 84.7%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sin-+PI/2-rev N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 86.4%

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

   8. Taylor expanded in l around inf

      \[\leadsto e^{-1 \cdot \ell} \cdot \sin \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{1}{2} \cdot \left(K \cdot m\right)\right)} - M\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 63.9

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

   10. Applied rewrites 63.9%

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

   if 3.99999999999999982e-57 < n < 55

   1. Initial program 90.9%

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

   3. Taylor expanded in M around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if 55 < 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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around inf

      \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot \left(n \cdot n\right)} \]

      3. lower-*.f64 98.6

         \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   8. Applied rewrites 98.6%

      \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot \left(n \cdot n\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.6%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(n \cdot n\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-57}:\\ \;\;\;\;e^{-\ell} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.5 \cdot \left(K \cdot m\right)\right) - M\right)\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-57}:\\ \;\;\;\;e^{-\ell} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.5 \cdot \left(K \cdot m\right)\right) - M\right)\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 5e-105)
   (* (cos M) (exp (* (* m m) -0.25)))
   (if (<= n 1.65e-57)
     (* (exp (- l)) (sin (- (fma 0.5 PI (* 0.5 (* K m))) M)))
     (if (<= n 55.0)
       (* (cos M) (exp (* (- M) M)))
       (* 1.0 (exp (* -0.25 (* n n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 5e-105) {
		tmp = cos(M) * exp(((m * m) * -0.25));
	} else if (n <= 1.65e-57) {
		tmp = exp(-l) * sin((fma(0.5, ((double) M_PI), (0.5 * (K * m))) - M));
	} else if (n <= 55.0) {
		tmp = cos(M) * exp((-M * M));
	} else {
		tmp = 1.0 * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 5e-105)
		tmp = Float64(cos(M) * exp(Float64(Float64(m * m) * -0.25)));
	elseif (n <= 1.65e-57)
		tmp = Float64(exp(Float64(-l)) * sin(Float64(fma(0.5, pi, Float64(0.5 * Float64(K * m))) - M)));
	elseif (n <= 55.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(-M) * M)));
	else
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 5e-105], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.65e-57], N[(N[Exp[(-l)], $MachinePrecision] * N[Sin[N[(N[(0.5 * Pi + N[(0.5 * N[(K * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 55.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < 4.99999999999999963e-105

   1. Initial program 77.1%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(m \cdot m\right) \cdot \frac{-1}{4}} \]

      4. lower-*.f64 41.2

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(m \cdot m\right) \cdot \frac{-1}{4}} \]
   7. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \cos M \cdot e^{\left(m \cdot m\right) \cdot \frac{-1}{4}} \]

      2. lift-cos.f64 56.1

         \[\leadsto \cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   8. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   if 4.99999999999999963e-105 < n < 1.6499999999999999e-57

   1. Initial program 84.7%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sin-+PI/2-rev N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 86.4%

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

   8. Taylor expanded in l around inf

      \[\leadsto e^{-1 \cdot \ell} \cdot \sin \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{1}{2} \cdot \left(K \cdot m\right)\right)} - M\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 63.9

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

   10. Applied rewrites 63.9%

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

   if 1.6499999999999999e-57 < n < 55

   1. Initial program 90.9%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \cos M \cdot e^{-1 \cdot {M}^{2}} \]

      2. pow2 N/A

         \[\leadsto \cos M \cdot e^{-1 \cdot \left(M \cdot M\right)} \]

      3. lift-*.f64 73.1

         \[\leadsto \cos M \cdot e^{-1 \cdot \left(M \cdot M\right)} \]

   8. Applied rewrites 73.1%

      \[\leadsto \cos M \cdot e^{-1 \cdot \left(M \cdot M\right)} \]

   if 55 < 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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around inf

      \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot \left(n \cdot n\right)} \]

      3. lower-*.f64 98.6

         \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   8. Applied rewrites 98.6%

      \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot \left(n \cdot n\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.6%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(n \cdot n\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-57}:\\ \;\;\;\;e^{-\ell} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.5 \cdot \left(K \cdot m\right)\right) - M\right)\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-143}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -1.2e-143)
   (* 1.0 (exp (* -0.25 (* m m))))
   (if (<= n 55.0)
     (* (cos M) (exp (* (- M) M)))
     (* 1.0 (exp (* -0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -1.2e-143) {
		tmp = 1.0 * exp((-0.25 * (m * m)));
	} else if (n <= 55.0) {
		tmp = cos(M) * exp((-M * M));
	} else {
		tmp = 1.0 * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    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 (n <= (-1.2d-143)) then
        tmp = 1.0d0 * exp(((-0.25d0) * (m * m)))
    else if (n <= 55.0d0) then
        tmp = cos(m_1) * exp((-m_1 * m_1))
    else
        tmp = 1.0d0 * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -1.2e-143) {
		tmp = 1.0 * Math.exp((-0.25 * (m * m)));
	} else if (n <= 55.0) {
		tmp = Math.cos(M) * Math.exp((-M * M));
	} else {
		tmp = 1.0 * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -1.2e-143:
		tmp = 1.0 * math.exp((-0.25 * (m * m)))
	elif n <= 55.0:
		tmp = math.cos(M) * math.exp((-M * M))
	else:
		tmp = 1.0 * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -1.2e-143)
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 55.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(-M) * M)));
	else
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -1.2e-143)
		tmp = 1.0 * exp((-0.25 * (m * m)));
	elseif (n <= 55.0)
		tmp = cos(M) * exp((-M * M));
	else
		tmp = 1.0 * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -1.2e-143], N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 55.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.1999999999999999e-143

   1. Initial program 76.4%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 23.9

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

   5. Applied rewrites 23.9%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lift-+.f64 24.1

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

   8. Applied rewrites 24.1%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 28.8%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in m around inf

       \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot {m}^{2}}} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\frac{-1}{4} \cdot \color{blue}{{m}^{2}}} \]

       2. pow2 N/A

          \[\leadsto 1 \cdot e^{\frac{-1}{4} \cdot \left(m \cdot \color{blue}{m}\right)} \]

       3. lift-*.f64 52.0

          \[\leadsto 1 \cdot e^{-0.25 \cdot \left(m \cdot \color{blue}{m}\right)} \]

   14. Applied rewrites 52.0%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(m \cdot m\right)}} \]

   if -1.1999999999999999e-143 < n < 55

   1. Initial program 80.4%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \cos M \cdot e^{-1 \cdot {M}^{2}} \]

      2. pow2 N/A

         \[\leadsto \cos M \cdot e^{-1 \cdot \left(M \cdot M\right)} \]

      3. lift-*.f64 59.7

         \[\leadsto \cos M \cdot e^{-1 \cdot \left(M \cdot M\right)} \]

   8. Applied rewrites 59.7%

      \[\leadsto \cos M \cdot e^{-1 \cdot \left(M \cdot M\right)} \]

   if 55 < 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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around inf

      \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot \left(n \cdot n\right)} \]

      3. lower-*.f64 98.6

         \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   8. Applied rewrites 98.6%

      \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot \left(n \cdot n\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.6%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(n \cdot n\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-143}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -54 \lor \neg \left(n \leq 55\right):\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{\left(-M\right) \cdot M}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -54.0) (not (<= n 55.0)))
   (* 1.0 (exp (* -0.25 (* n n))))
   (* 1.0 (exp (* (- M) M)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -54.0) || !(n <= 55.0)) {
		tmp = 1.0 * exp((-0.25 * (n * n)));
	} else {
		tmp = 1.0 * exp((-M * M));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    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 ((n <= (-54.0d0)) .or. (.not. (n <= 55.0d0))) then
        tmp = 1.0d0 * exp(((-0.25d0) * (n * n)))
    else
        tmp = 1.0d0 * exp((-m_1 * 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 ((n <= -54.0) || !(n <= 55.0)) {
		tmp = 1.0 * Math.exp((-0.25 * (n * n)));
	} else {
		tmp = 1.0 * Math.exp((-M * M));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -54.0) or not (n <= 55.0):
		tmp = 1.0 * math.exp((-0.25 * (n * n)))
	else:
		tmp = 1.0 * math.exp((-M * M))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -54.0) || !(n <= 55.0))
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(n * n))));
	else
		tmp = Float64(1.0 * exp(Float64(Float64(-M) * M)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -54.0) || ~((n <= 55.0)))
		tmp = 1.0 * exp((-0.25 * (n * n)));
	else
		tmp = 1.0 * exp((-M * M));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -54.0], N[Not[LessEqual[n, 55.0]], $MachinePrecision]], N[(1.0 * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -54 or 55 < n

   1. Initial program 73.3%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in n around inf

      \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot \left(n \cdot n\right)} \]

      3. lower-*.f64 97.8

         \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   8. Applied rewrites 97.8%

      \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot \left(n \cdot n\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.8%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(n \cdot n\right)}} \]

   if -54 < n < 55

   1. Initial program 81.2%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 38.5

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lift-+.f64 36.9

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

   8. Applied rewrites 36.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 39.5%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in M around inf

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

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{-1 \cdot \color{blue}{{M}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{-1 \cdot \left(M \cdot \color{blue}{M}\right)} \]

       3. lower-*.f64 59.2

          \[\leadsto 1 \cdot e^{-1 \cdot \left(M \cdot \color{blue}{M}\right)} \]

   14. Applied rewrites 59.2%

       \[\leadsto 1 \cdot e^{\color{blue}{-1 \cdot \left(M \cdot M\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -54 \lor \neg \left(n \leq 55\right):\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{\left(-M\right) \cdot M}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+19} \lor \neg \left(n \leq 5.5 \cdot 10^{-24}\right):\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -1.4e+19) (not (<= n 5.5e-24)))
   (* 1.0 (exp (* -0.25 (* n n))))
   (* 1.0 (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -1.4e+19) || !(n <= 5.5e-24)) {
		tmp = 1.0 * exp((-0.25 * (n * n)));
	} else {
		tmp = 1.0 * exp(-l);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    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 ((n <= (-1.4d+19)) .or. (.not. (n <= 5.5d-24))) then
        tmp = 1.0d0 * exp(((-0.25d0) * (n * n)))
    else
        tmp = 1.0d0 * 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 ((n <= -1.4e+19) || !(n <= 5.5e-24)) {
		tmp = 1.0 * Math.exp((-0.25 * (n * n)));
	} else {
		tmp = 1.0 * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -1.4e+19) or not (n <= 5.5e-24):
		tmp = 1.0 * math.exp((-0.25 * (n * n)))
	else:
		tmp = 1.0 * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -1.4e+19) || !(n <= 5.5e-24))
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(n * n))));
	else
		tmp = Float64(1.0 * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -1.4e+19) || ~((n <= 5.5e-24)))
		tmp = 1.0 * exp((-0.25 * (n * n)));
	else
		tmp = 1.0 * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -1.4e+19], N[Not[LessEqual[n, 5.5e-24]], $MachinePrecision]], N[(1.0 * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.4e19 or 5.4999999999999999e-24 < n

   1. Initial program 73.9%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in n around inf

      \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot \left(n \cdot n\right)} \]

      3. lower-*.f64 95.0

         \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   8. Applied rewrites 95.0%

      \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot \left(n \cdot n\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 95.0%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(n \cdot n\right)}} \]

   if -1.4e19 < n < 5.4999999999999999e-24

   1. Initial program 80.7%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.5

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

   5. Applied rewrites 39.5%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lift-+.f64 37.8

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 40.5%

       \[\leadsto 1 \cdot e^{-\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+19} \lor \neg \left(n \leq 5.5 \cdot 10^{-24}\right):\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-143}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;1 \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -1.2e-143)
   (* 1.0 (exp (* -0.25 (* m m))))
   (if (<= n 55.0) (* 1.0 (exp (* (- M) M))) (* 1.0 (exp (* -0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -1.2e-143) {
		tmp = 1.0 * exp((-0.25 * (m * m)));
	} else if (n <= 55.0) {
		tmp = 1.0 * exp((-M * M));
	} else {
		tmp = 1.0 * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    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 (n <= (-1.2d-143)) then
        tmp = 1.0d0 * exp(((-0.25d0) * (m * m)))
    else if (n <= 55.0d0) then
        tmp = 1.0d0 * exp((-m_1 * m_1))
    else
        tmp = 1.0d0 * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -1.2e-143) {
		tmp = 1.0 * Math.exp((-0.25 * (m * m)));
	} else if (n <= 55.0) {
		tmp = 1.0 * Math.exp((-M * M));
	} else {
		tmp = 1.0 * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -1.2e-143:
		tmp = 1.0 * math.exp((-0.25 * (m * m)))
	elif n <= 55.0:
		tmp = 1.0 * math.exp((-M * M))
	else:
		tmp = 1.0 * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -1.2e-143)
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 55.0)
		tmp = Float64(1.0 * exp(Float64(Float64(-M) * M)));
	else
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -1.2e-143)
		tmp = 1.0 * exp((-0.25 * (m * m)));
	elseif (n <= 55.0)
		tmp = 1.0 * exp((-M * M));
	else
		tmp = 1.0 * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -1.2e-143], N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 55.0], N[(1.0 * N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.1999999999999999e-143

   1. Initial program 76.4%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 23.9

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

   5. Applied rewrites 23.9%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lift-+.f64 24.1

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

   8. Applied rewrites 24.1%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 28.8%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in m around inf

       \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot {m}^{2}}} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\frac{-1}{4} \cdot \color{blue}{{m}^{2}}} \]

       2. pow2 N/A

          \[\leadsto 1 \cdot e^{\frac{-1}{4} \cdot \left(m \cdot \color{blue}{m}\right)} \]

       3. lift-*.f64 52.0

          \[\leadsto 1 \cdot e^{-0.25 \cdot \left(m \cdot \color{blue}{m}\right)} \]

   14. Applied rewrites 52.0%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(m \cdot m\right)}} \]

   if -1.1999999999999999e-143 < n < 55

   1. Initial program 80.4%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.9

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lift-+.f64 37.8

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 39.0%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in M around inf

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

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{-1 \cdot \color{blue}{{M}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{-1 \cdot \left(M \cdot \color{blue}{M}\right)} \]

       3. lower-*.f64 59.7

          \[\leadsto 1 \cdot e^{-1 \cdot \left(M \cdot \color{blue}{M}\right)} \]

   14. Applied rewrites 59.7%

       \[\leadsto 1 \cdot e^{\color{blue}{-1 \cdot \left(M \cdot M\right)}} \]

   if 55 < 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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around inf

      \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot {n}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \cos M \cdot e^{\frac{-1}{4} \cdot \left(n \cdot n\right)} \]

      3. lower-*.f64 98.6

         \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   8. Applied rewrites 98.6%

      \[\leadsto \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot \left(n \cdot n\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.6%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot \left(n \cdot n\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-143}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;1 \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot e^{-\ell} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (* 1.0 (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 * exp(-l);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0 * exp(-l)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 * Math.exp(-l);
}

Python

def code(K, m, n, M, l):
	return 1.0 * math.exp(-l)

Julia

function code(K, m, n, M, l)
	return Float64(1.0 * exp(Float64(-l)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 * exp(-l);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

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

3. Taylor expanded in l around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 28.0

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

5. Applied rewrites 28.0%

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

6. Taylor expanded in K around 0

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

   1. cos-neg-rev N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. lower--.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. sin-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lift-+.f64 28.1

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

8. Applied rewrites 28.1%

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

9. Taylor expanded in M around 0

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

11. Applied rewrites 31.9%

    \[\leadsto 1 \cdot e^{-\ell} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025072 
(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)))))))