Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.8% → 96.6%

Time: 6.0s

Alternatives: 8

Speedup: 1.5×

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: 76.8% 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: 96.6% 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 78.4%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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. Final simplification 94.8%

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

Alternative 2: 95.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -7.2 \cdot 10^{+67} \lor \neg \left(M \leq 180000\right):\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -7.2e+67) (not (<= M 180000.0)))
   (* (cos M) (exp (* (- M) M)))
   (exp (- (fabs (- m n)) (+ l (* 0.25 (pow (+ m n) 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -7.2e+67) || !(M <= 180000.0)) {
		tmp = cos(M) * exp((-M * M));
	} else {
		tmp = exp((fabs((m - n)) - (l + (0.25 * pow((m + n), 2.0)))));
	}
	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 ((m_1 <= (-7.2d+67)) .or. (.not. (m_1 <= 180000.0d0))) then
        tmp = cos(m_1) * exp((-m_1 * m_1))
    else
        tmp = exp((abs((m - n)) - (l + (0.25d0 * ((m + n) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -7.2e+67) || !(M <= 180000.0)) {
		tmp = Math.cos(M) * Math.exp((-M * M));
	} else {
		tmp = Math.exp((Math.abs((m - n)) - (l + (0.25 * Math.pow((m + n), 2.0)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -7.2e+67) or not (M <= 180000.0):
		tmp = math.cos(M) * math.exp((-M * M))
	else:
		tmp = math.exp((math.fabs((m - n)) - (l + (0.25 * math.pow((m + n), 2.0)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -7.2e+67) || !(M <= 180000.0))
		tmp = Float64(cos(M) * exp(Float64(Float64(-M) * M)));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(0.25 * (Float64(m + n) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -7.2e+67) || ~((M <= 180000.0)))
		tmp = cos(M) * exp((-M * M));
	else
		tmp = exp((abs((m - n)) - (l + (0.25 * ((m + n) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -7.2e+67], N[Not[LessEqual[M, 180000.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -7.1999999999999998e67 or 1.8e5 < M

   1. Initial program 82.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 99.2%

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

   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. unpow2 N/A

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

      3. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

   if -7.1999999999999998e67 < M < 1.8e5

   1. Initial program 74.9%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 90.7%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 90.7

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

   8. Applied rewrites 90.7%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7.2 \cdot 10^{+67} \lor \neg \left(M \leq 180000\right):\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-208}:\\ \;\;\;\;e^{\left|m - n\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 5.6e-208)
   (exp (- (fabs (- m n)) (* 0.25 (* m m))))
   (if (<= n 54.0) (* (cos M) (exp (* (- M) M))) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 5.6e-208) {
		tmp = exp((fabs((m - n)) - (0.25 * (m * m))));
	} else if (n <= 54.0) {
		tmp = cos(M) * exp((-M * M));
	} else {
		tmp = 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 <= 5.6d-208) then
        tmp = exp((abs((m - n)) - (0.25d0 * (m * m))))
    else if (n <= 54.0d0) then
        tmp = cos(m_1) * exp((-m_1 * m_1))
    else
        tmp = 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 <= 5.6e-208) {
		tmp = Math.exp((Math.abs((m - n)) - (0.25 * (m * m))));
	} else if (n <= 54.0) {
		tmp = Math.cos(M) * Math.exp((-M * M));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 5.6e-208:
		tmp = math.exp((math.fabs((m - n)) - (0.25 * (m * m))))
	elif n <= 54.0:
		tmp = math.cos(M) * math.exp((-M * M))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 5.6e-208)
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(0.25 * Float64(m * m))));
	elseif (n <= 54.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(-M) * M)));
	else
		tmp = 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 <= 5.6e-208)
		tmp = exp((abs((m - n)) - (0.25 * (m * m))));
	elseif (n <= 54.0)
		tmp = cos(M) * exp((-M * M));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 5.6e-208], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 54.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 5.60000000000000003e-208

   1. Initial program 81.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 93.9%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 81.5

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

   8. Applied rewrites 81.5%

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

   9. Taylor expanded in m around inf

      \[\leadsto e^{\left|n - m\right| - \frac{1}{4} \cdot {m}^{2}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \frac{1}{4} \cdot {m}^{2}} \]

       2. pow2 N/A

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

       3. lift-*.f64 49.8

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

   11. Applied rewrites 49.8%

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

   if 5.60000000000000003e-208 < n < 54

   1. Initial program 79.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 92.6%

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

   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. unpow2 N/A

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

      3. lower-*.f64 73.6

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

   8. Applied rewrites 73.6%

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

   if 54 < n

   1. Initial program 69.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 98.4%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 92.2

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

   8. Applied rewrites 92.2%

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

   9. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 98.4

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

   11. Applied rewrites 98.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-208}:\\ \;\;\;\;e^{\left|m - n\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;e^{\left|m - n\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 53:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.25e-213)
   (exp (- (fabs (- m n)) (* 0.25 (* m m))))
   (if (<= n 53.0) (* (cos M) (exp (- l))) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.25e-213) {
		tmp = exp((fabs((m - n)) - (0.25 * (m * m))));
	} else if (n <= 53.0) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = 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 <= 1.25d-213) then
        tmp = exp((abs((m - n)) - (0.25d0 * (m * m))))
    else if (n <= 53.0d0) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = 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 <= 1.25e-213) {
		tmp = Math.exp((Math.abs((m - n)) - (0.25 * (m * m))));
	} else if (n <= 53.0) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.25e-213:
		tmp = math.exp((math.fabs((m - n)) - (0.25 * (m * m))))
	elif n <= 53.0:
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.25e-213)
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(0.25 * Float64(m * m))));
	elseif (n <= 53.0)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = 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 <= 1.25e-213)
		tmp = exp((abs((m - n)) - (0.25 * (m * m))));
	elseif (n <= 53.0)
		tmp = cos(M) * exp(-l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.25e-213], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 53.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.24999999999999994e-213

   1. Initial program 82.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 93.8%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 81.1

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

   8. Applied rewrites 81.1%

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

   9. Taylor expanded in m around inf

      \[\leadsto e^{\left|n - m\right| - \frac{1}{4} \cdot {m}^{2}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \frac{1}{4} \cdot {m}^{2}} \]

       2. pow2 N/A

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

       3. lift-*.f64 49.4

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

   11. Applied rewrites 49.4%

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

   if 1.24999999999999994e-213 < n < 53

   1. Initial program 76.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 33.4

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

   5. Applied rewrites 33.4%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. lift-cos.f64 38.9

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

   8. Applied rewrites 38.9%

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

   if 53 < n

   1. Initial program 69.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 98.4%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 92.2

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

   8. Applied rewrites 92.2%

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

   9. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 98.4

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

   11. Applied rewrites 98.4%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;e^{\left|m - n\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 53:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;e^{\left|m - n\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 53:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.25e-213)
   (exp (- (fabs (- m n)) (* 0.25 (* m m))))
   (if (<= n 53.0) (exp (- l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.25e-213) {
		tmp = exp((fabs((m - n)) - (0.25 * (m * m))));
	} else if (n <= 53.0) {
		tmp = exp(-l);
	} else {
		tmp = 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 <= 1.25d-213) then
        tmp = exp((abs((m - n)) - (0.25d0 * (m * m))))
    else if (n <= 53.0d0) then
        tmp = exp(-l)
    else
        tmp = 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 <= 1.25e-213) {
		tmp = Math.exp((Math.abs((m - n)) - (0.25 * (m * m))));
	} else if (n <= 53.0) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.25e-213:
		tmp = math.exp((math.fabs((m - n)) - (0.25 * (m * m))))
	elif n <= 53.0:
		tmp = math.exp(-l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.25e-213)
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(0.25 * Float64(m * m))));
	elseif (n <= 53.0)
		tmp = exp(Float64(-l));
	else
		tmp = 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 <= 1.25e-213)
		tmp = exp((abs((m - n)) - (0.25 * (m * m))));
	elseif (n <= 53.0)
		tmp = exp(-l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.25e-213], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 53.0], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.24999999999999994e-213

   1. Initial program 82.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 93.8%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 81.1

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

   8. Applied rewrites 81.1%

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

   9. Taylor expanded in m around inf

      \[\leadsto e^{\left|n - m\right| - \frac{1}{4} \cdot {m}^{2}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \frac{1}{4} \cdot {m}^{2}} \]

       2. pow2 N/A

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

       3. lift-*.f64 49.4

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

   11. Applied rewrites 49.4%

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

   if 1.24999999999999994e-213 < n < 53

   1. Initial program 76.7%

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

   3. Taylor expanded in 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 93.0%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 67.7

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in l around inf

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

       1. lower-*.f64 38.9

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

   11. Applied rewrites 38.9%

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

   if 53 < n

   1. Initial program 69.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 98.4%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 92.2

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

   8. Applied rewrites 92.2%

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

   9. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 98.4

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

   11. Applied rewrites 98.4%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;e^{\left|m - n\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 53:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8.2 \cdot 10^{-5} \lor \neg \left(m \leq 4.5 \cdot 10^{-11}\right):\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -8.2e-5) (not (<= m 4.5e-11)))
   (exp (* -0.25 (* m m)))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -8.2e-5) || !(m <= 4.5e-11)) {
		tmp = exp((-0.25 * (m * m)));
	} else {
		tmp = 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 ((m <= (-8.2d-5)) .or. (.not. (m <= 4.5d-11))) then
        tmp = exp(((-0.25d0) * (m * m)))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -8.2e-5) || !(m <= 4.5e-11)) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -8.2e-5) or not (m <= 4.5e-11):
		tmp = math.exp((-0.25 * (m * m)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -8.2e-5) || !(m <= 4.5e-11))
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -8.2e-5) || ~((m <= 4.5e-11)))
		tmp = exp((-0.25 * (m * m)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -8.2e-5], N[Not[LessEqual[m, 4.5e-11]], $MachinePrecision]], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -8.20000000000000009e-5 or 4.5e-11 < m

   1. Initial program 75.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 99.2%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 97.0

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in m around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 96.3

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

   11. Applied rewrites 96.3%

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

   if -8.20000000000000009e-5 < m < 4.5e-11

   1. Initial program 81.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 90.0%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 64.0

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

   8. Applied rewrites 64.0%

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

   9. Taylor expanded in l around inf

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

       1. lower-*.f64 38.6

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

   11. Applied rewrites 38.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.2 \cdot 10^{-5} \lor \neg \left(m \leq 4.5 \cdot 10^{-11}\right):\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 53:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.25e-213)
   (exp (* -0.25 (* m m)))
   (if (<= n 53.0) (exp (- l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.25e-213) {
		tmp = exp((-0.25 * (m * m)));
	} else if (n <= 53.0) {
		tmp = exp(-l);
	} else {
		tmp = 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 <= 1.25d-213) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (n <= 53.0d0) then
        tmp = exp(-l)
    else
        tmp = 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 <= 1.25e-213) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (n <= 53.0) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.25e-213:
		tmp = math.exp((-0.25 * (m * m)))
	elif n <= 53.0:
		tmp = math.exp(-l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.25e-213)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (n <= 53.0)
		tmp = exp(Float64(-l));
	else
		tmp = 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 <= 1.25e-213)
		tmp = exp((-0.25 * (m * m)));
	elseif (n <= 53.0)
		tmp = exp(-l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.25e-213], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 53.0], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.24999999999999994e-213

   1. Initial program 82.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 93.8%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 81.1

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

   8. Applied rewrites 81.1%

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

   9. Taylor expanded in m around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 55.9

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

   11. Applied rewrites 55.9%

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

   if 1.24999999999999994e-213 < n < 53

   1. Initial program 76.7%

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

   3. Taylor expanded in 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 93.0%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 67.7

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in l around inf

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

       1. lower-*.f64 38.9

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

   11. Applied rewrites 38.9%

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

   if 53 < n

   1. Initial program 69.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 98.4%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 92.2

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

   8. Applied rewrites 92.2%

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

   9. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 98.4

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

   11. Applied rewrites 98.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 53:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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 = exp(-l)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 0

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-+.f64 81.0

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

8. Applied rewrites 81.0%

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

9. Taylor expanded in l around inf

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

    1. lower-*.f64 34.5

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

11. Applied rewrites 34.5%

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

12. Final simplification 34.5%

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

Reproduce

herbie shell --seed 2025073 
(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)))))))