Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.5% → 96.5%

Time: 7.2s

Alternatives: 10

Speedup: 1.1×

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: 75.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

3. Taylor expanded in K around 0

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

5. Applied rewrites 96.0%

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

6. Final simplification 96.0%

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

Alternative 2: 63.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 2.25 \cdot 10^{-215}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left(M \cdot M + \left(\ell - \left|m - n\right|\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(m \cdot K\right) \cdot 0.5\right) \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -55.0)
   (* 1.0 (exp (* -0.25 (* m m))))
   (if (<= m 2.25e-215)
     (*
      (cos (- (/ (* K (+ m n)) 2.0) M))
      (exp (- (+ (* M M) (- l (fabs (- m n)))))))
     (* (cos (* (* m K) 0.5)) (exp (* (* n n) -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -55.0) {
		tmp = 1.0 * exp((-0.25 * (m * m)));
	} else if (m <= 2.25e-215) {
		tmp = cos((((K * (m + n)) / 2.0) - M)) * exp(-((M * M) + (l - fabs((m - n)))));
	} else {
		tmp = cos(((m * K) * 0.5)) * exp(((n * n) * -0.25));
	}
	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 <= (-55.0d0)) then
        tmp = 1.0d0 * exp(((-0.25d0) * (m * m)))
    else if (m <= 2.25d-215) then
        tmp = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp(-((m_1 * m_1) + (l - abs((m - n)))))
    else
        tmp = cos(((m * k) * 0.5d0)) * exp(((n * n) * (-0.25d0)))
    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 <= -55.0) {
		tmp = 1.0 * Math.exp((-0.25 * (m * m)));
	} else if (m <= 2.25e-215) {
		tmp = Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp(-((M * M) + (l - Math.abs((m - n)))));
	} else {
		tmp = Math.cos(((m * K) * 0.5)) * Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -55.0:
		tmp = 1.0 * math.exp((-0.25 * (m * m)))
	elif m <= 2.25e-215:
		tmp = math.cos((((K * (m + n)) / 2.0) - M)) * math.exp(-((M * M) + (l - math.fabs((m - n)))))
	else:
		tmp = math.cos(((m * K) * 0.5)) * math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -55.0)
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))));
	elseif (m <= 2.25e-215)
		tmp = Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(-Float64(Float64(M * M) + Float64(l - abs(Float64(m - n)))))));
	else
		tmp = Float64(cos(Float64(Float64(m * K) * 0.5)) * exp(Float64(Float64(n * n) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -55.0)
		tmp = 1.0 * exp((-0.25 * (m * m)));
	elseif (m <= 2.25e-215)
		tmp = cos((((K * (m + n)) / 2.0) - M)) * exp(-((M * M) + (l - abs((m - n)))));
	else
		tmp = cos(((m * K) * 0.5)) * exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -55.0], N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.25e-215], N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[(-N[(N[(M * M), $MachinePrecision] + N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[(m * K), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -55

   1. Initial program 80.0%

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 23.5%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 23.5%

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

   12. Taylor expanded in m around inf

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

   14. Applied rewrites 97.0%

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

   if -55 < m < 2.25e-215

   1. Initial program 75.4%

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

   3. Taylor expanded in M around inf

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

   5. Applied rewrites 56.8%

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

   if 2.25e-215 < m

   1. Initial program 63.7%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 32.5%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 50.4%

      \[\leadsto \cos \color{blue}{\left(\left(m \cdot K\right) \cdot 0.5\right)} \cdot e^{\left(n \cdot n\right) \cdot -0.25} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 2.25 \cdot 10^{-215}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left(M \cdot M + \left(\ell - \left|m - n\right|\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(m \cdot K\right) \cdot 0.5\right) \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot n\right) \cdot -0.25\\ \mathbf{if}\;m \leq -2 \cdot 10^{+40}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -2.05 \cdot 10^{-91}:\\ \;\;\;\;\cos \left(\left(n \cdot K\right) \cdot 0.5 - M\right) \cdot e^{t\_0 - \left(\ell - \left|m - n\right|\right)}\\ \mathbf{elif}\;m \leq -3.2 \cdot 10^{-305}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(m \cdot K\right) \cdot 0.5\right) \cdot e^{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (* n n) -0.25)))
   (if (<= m -2e+40)
     (* 1.0 (exp (* -0.25 (* m m))))
     (if (<= m -2.05e-91)
       (* (cos (- (* (* n K) 0.5) M)) (exp (- t_0 (- l (fabs (- m n))))))
       (if (<= m -3.2e-305)
         (* (exp (* (- M) M)) 1.0)
         (* (cos (* (* m K) 0.5)) (exp t_0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (n * n) * -0.25;
	double tmp;
	if (m <= -2e+40) {
		tmp = 1.0 * exp((-0.25 * (m * m)));
	} else if (m <= -2.05e-91) {
		tmp = cos((((n * K) * 0.5) - M)) * exp((t_0 - (l - fabs((m - n)))));
	} else if (m <= -3.2e-305) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = cos(((m * K) * 0.5)) * exp(t_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) :: t_0
    real(8) :: tmp
    t_0 = (n * n) * (-0.25d0)
    if (m <= (-2d+40)) then
        tmp = 1.0d0 * exp(((-0.25d0) * (m * m)))
    else if (m <= (-2.05d-91)) then
        tmp = cos((((n * k) * 0.5d0) - m_1)) * exp((t_0 - (l - abs((m - n)))))
    else if (m <= (-3.2d-305)) then
        tmp = exp((-m_1 * m_1)) * 1.0d0
    else
        tmp = cos(((m * k) * 0.5d0)) * exp(t_0)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (n * n) * -0.25;
	double tmp;
	if (m <= -2e+40) {
		tmp = 1.0 * Math.exp((-0.25 * (m * m)));
	} else if (m <= -2.05e-91) {
		tmp = Math.cos((((n * K) * 0.5) - M)) * Math.exp((t_0 - (l - Math.abs((m - n)))));
	} else if (m <= -3.2e-305) {
		tmp = Math.exp((-M * M)) * 1.0;
	} else {
		tmp = Math.cos(((m * K) * 0.5)) * Math.exp(t_0);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (n * n) * -0.25
	tmp = 0
	if m <= -2e+40:
		tmp = 1.0 * math.exp((-0.25 * (m * m)))
	elif m <= -2.05e-91:
		tmp = math.cos((((n * K) * 0.5) - M)) * math.exp((t_0 - (l - math.fabs((m - n)))))
	elif m <= -3.2e-305:
		tmp = math.exp((-M * M)) * 1.0
	else:
		tmp = math.cos(((m * K) * 0.5)) * math.exp(t_0)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(n * n) * -0.25)
	tmp = 0.0
	if (m <= -2e+40)
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))));
	elseif (m <= -2.05e-91)
		tmp = Float64(cos(Float64(Float64(Float64(n * K) * 0.5) - M)) * exp(Float64(t_0 - Float64(l - abs(Float64(m - n))))));
	elseif (m <= -3.2e-305)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(cos(Float64(Float64(m * K) * 0.5)) * exp(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (n * n) * -0.25;
	tmp = 0.0;
	if (m <= -2e+40)
		tmp = 1.0 * exp((-0.25 * (m * m)));
	elseif (m <= -2.05e-91)
		tmp = cos((((n * K) * 0.5) - M)) * exp((t_0 - (l - abs((m - n)))));
	elseif (m <= -3.2e-305)
		tmp = exp((-M * M)) * 1.0;
	else
		tmp = cos(((m * K) * 0.5)) * exp(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]}, If[LessEqual[m, -2e+40], N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -2.05e-91], N[(N[Cos[N[(N[(N[(n * K), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -3.2e-305], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Cos[N[(N[(m * K), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -2.00000000000000006e40

   1. Initial program 78.3%

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

   3. Taylor expanded in 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

   5. Applied rewrites 19.8%

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

   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

   8. Applied rewrites 23.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 23.6%

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

   12. Taylor expanded in m around inf

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

   14. Applied rewrites 96.7%

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

   if -2.00000000000000006e40 < m < -2.05000000000000012e-91

   1. Initial program 90.1%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 81.8%

      \[\leadsto \cos \left(\color{blue}{\left(n \cdot K\right) \cdot 0.5} - M\right) \cdot e^{\left(n \cdot n\right) \cdot -0.25 - \left(\ell - \left|m - n\right|\right)} \]

   if -2.05000000000000012e-91 < m < -3.20000000000000009e-305

   1. Initial program 72.6%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 55.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 55.0%

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

   if -3.20000000000000009e-305 < m

   1. Initial program 65.2%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 34.9%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 53.3%

      \[\leadsto \cos \color{blue}{\left(\left(m \cdot K\right) \cdot 0.5\right)} \cdot e^{\left(n \cdot n\right) \cdot -0.25} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{+40}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -2.05 \cdot 10^{-91}:\\ \;\;\;\;\cos \left(\left(n \cdot K\right) \cdot 0.5 - M\right) \cdot e^{\left(n \cdot n\right) \cdot -0.25 - \left(\ell - \left|m - n\right|\right)}\\ \mathbf{elif}\;m \leq -3.2 \cdot 10^{-305}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(m \cdot K\right) \cdot 0.5\right) \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot n\right) \cdot -0.25\\ \mathbf{if}\;m \leq -2 \cdot 10^{+40}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.05 \cdot 10^{-90}:\\ \;\;\;\;\cos \left(\left(n \cdot K\right) \cdot 0.5\right) \cdot e^{t\_0 - \left(\ell - \left|m - n\right|\right)}\\ \mathbf{elif}\;m \leq -3.2 \cdot 10^{-305}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(m \cdot K\right) \cdot 0.5\right) \cdot e^{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (* n n) -0.25)))
   (if (<= m -2e+40)
     (* 1.0 (exp (* -0.25 (* m m))))
     (if (<= m -1.05e-90)
       (* (cos (* (* n K) 0.5)) (exp (- t_0 (- l (fabs (- m n))))))
       (if (<= m -3.2e-305)
         (* (exp (* (- M) M)) 1.0)
         (* (cos (* (* m K) 0.5)) (exp t_0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (n * n) * -0.25;
	double tmp;
	if (m <= -2e+40) {
		tmp = 1.0 * exp((-0.25 * (m * m)));
	} else if (m <= -1.05e-90) {
		tmp = cos(((n * K) * 0.5)) * exp((t_0 - (l - fabs((m - n)))));
	} else if (m <= -3.2e-305) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = cos(((m * K) * 0.5)) * exp(t_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) :: t_0
    real(8) :: tmp
    t_0 = (n * n) * (-0.25d0)
    if (m <= (-2d+40)) then
        tmp = 1.0d0 * exp(((-0.25d0) * (m * m)))
    else if (m <= (-1.05d-90)) then
        tmp = cos(((n * k) * 0.5d0)) * exp((t_0 - (l - abs((m - n)))))
    else if (m <= (-3.2d-305)) then
        tmp = exp((-m_1 * m_1)) * 1.0d0
    else
        tmp = cos(((m * k) * 0.5d0)) * exp(t_0)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (n * n) * -0.25;
	double tmp;
	if (m <= -2e+40) {
		tmp = 1.0 * Math.exp((-0.25 * (m * m)));
	} else if (m <= -1.05e-90) {
		tmp = Math.cos(((n * K) * 0.5)) * Math.exp((t_0 - (l - Math.abs((m - n)))));
	} else if (m <= -3.2e-305) {
		tmp = Math.exp((-M * M)) * 1.0;
	} else {
		tmp = Math.cos(((m * K) * 0.5)) * Math.exp(t_0);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (n * n) * -0.25
	tmp = 0
	if m <= -2e+40:
		tmp = 1.0 * math.exp((-0.25 * (m * m)))
	elif m <= -1.05e-90:
		tmp = math.cos(((n * K) * 0.5)) * math.exp((t_0 - (l - math.fabs((m - n)))))
	elif m <= -3.2e-305:
		tmp = math.exp((-M * M)) * 1.0
	else:
		tmp = math.cos(((m * K) * 0.5)) * math.exp(t_0)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(n * n) * -0.25)
	tmp = 0.0
	if (m <= -2e+40)
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))));
	elseif (m <= -1.05e-90)
		tmp = Float64(cos(Float64(Float64(n * K) * 0.5)) * exp(Float64(t_0 - Float64(l - abs(Float64(m - n))))));
	elseif (m <= -3.2e-305)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(cos(Float64(Float64(m * K) * 0.5)) * exp(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (n * n) * -0.25;
	tmp = 0.0;
	if (m <= -2e+40)
		tmp = 1.0 * exp((-0.25 * (m * m)));
	elseif (m <= -1.05e-90)
		tmp = cos(((n * K) * 0.5)) * exp((t_0 - (l - abs((m - n)))));
	elseif (m <= -3.2e-305)
		tmp = exp((-M * M)) * 1.0;
	else
		tmp = cos(((m * K) * 0.5)) * exp(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]}, If[LessEqual[m, -2e+40], N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -1.05e-90], N[(N[Cos[N[(N[(n * K), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -3.2e-305], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Cos[N[(N[(m * K), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -2.00000000000000006e40

   1. Initial program 78.3%

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

   3. Taylor expanded in 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

   5. Applied rewrites 19.8%

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

   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

   8. Applied rewrites 23.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 23.6%

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

   12. Taylor expanded in m around inf

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

   14. Applied rewrites 96.7%

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

   if -2.00000000000000006e40 < m < -1.05e-90

   1. Initial program 90.1%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 81.8%

      \[\leadsto \cos \left(\color{blue}{\left(n \cdot K\right) \cdot 0.5} - M\right) \cdot e^{\left(n \cdot n\right) \cdot -0.25 - \left(\ell - \left|m - n\right|\right)} \]

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 81.8%

       \[\leadsto \cos \color{blue}{\left(\left(n \cdot K\right) \cdot 0.5\right)} \cdot e^{\left(n \cdot n\right) \cdot -0.25 - \left(\ell - \left|m - n\right|\right)} \]

   if -1.05e-90 < m < -3.20000000000000009e-305

   1. Initial program 72.6%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 55.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 55.0%

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

   if -3.20000000000000009e-305 < m

   1. Initial program 65.2%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 34.9%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 53.3%

      \[\leadsto \cos \color{blue}{\left(\left(m \cdot K\right) \cdot 0.5\right)} \cdot e^{\left(n \cdot n\right) \cdot -0.25} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{+40}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.05 \cdot 10^{-90}:\\ \;\;\;\;\cos \left(\left(n \cdot K\right) \cdot 0.5\right) \cdot e^{\left(n \cdot n\right) \cdot -0.25 - \left(\ell - \left|m - n\right|\right)}\\ \mathbf{elif}\;m \leq -3.2 \cdot 10^{-305}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(m \cdot K\right) \cdot 0.5\right) \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.3 \cdot 10^{-251}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \cos M\\ \mathbf{elif}\;n \leq 72000:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.3e-251)
   (* (exp (* -0.25 (* m m))) (cos M))
   (if (<= n 72000.0)
     (* (exp (* (- M) M)) (cos M))
     (* 1.0 (exp (* (* n n) -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.3e-251) {
		tmp = exp((-0.25 * (m * m))) * cos(M);
	} else if (n <= 72000.0) {
		tmp = exp((-M * M)) * cos(M);
	} else {
		tmp = 1.0 * exp(((n * n) * -0.25));
	}
	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 <= 1.3d-251) then
        tmp = exp(((-0.25d0) * (m * m))) * cos(m_1)
    else if (n <= 72000.0d0) then
        tmp = exp((-m_1 * m_1)) * cos(m_1)
    else
        tmp = 1.0d0 * exp(((n * n) * (-0.25d0)))
    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 <= 1.3e-251) {
		tmp = Math.exp((-0.25 * (m * m))) * Math.cos(M);
	} else if (n <= 72000.0) {
		tmp = Math.exp((-M * M)) * Math.cos(M);
	} else {
		tmp = 1.0 * Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.3e-251:
		tmp = math.exp((-0.25 * (m * m))) * math.cos(M)
	elif n <= 72000.0:
		tmp = math.exp((-M * M)) * math.cos(M)
	else:
		tmp = 1.0 * math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.3e-251)
		tmp = Float64(exp(Float64(-0.25 * Float64(m * m))) * cos(M));
	elseif (n <= 72000.0)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * cos(M));
	else
		tmp = Float64(1.0 * exp(Float64(Float64(n * n) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1.3e-251)
		tmp = exp((-0.25 * (m * m))) * cos(M);
	elseif (n <= 72000.0)
		tmp = exp((-M * M)) * cos(M);
	else
		tmp = 1.0 * exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.3e-251], N[(N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 72000.0], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.3e-251

   1. Initial program 69.7%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 51.5%

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

   if 1.3e-251 < n < 72000

   1. Initial program 75.0%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 59.5%

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

   if 72000 < n

   1. Initial program 73.5%

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

   5. Applied rewrites 17.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 20.4%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 20.4%

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

   12. Taylor expanded in n around inf

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

   14. Applied rewrites 98.0%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.3 \cdot 10^{-251}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \cos M\\ \mathbf{elif}\;n \leq 72000:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{-251}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 72000:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 2e-251)
   (* 1.0 (exp (* -0.25 (* m m))))
   (if (<= n 72000.0)
     (* (exp (* (- M) M)) (cos M))
     (* 1.0 (exp (* (* n n) -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2e-251) {
		tmp = 1.0 * exp((-0.25 * (m * m)));
	} else if (n <= 72000.0) {
		tmp = exp((-M * M)) * cos(M);
	} else {
		tmp = 1.0 * exp(((n * n) * -0.25));
	}
	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 <= 2d-251) then
        tmp = 1.0d0 * exp(((-0.25d0) * (m * m)))
    else if (n <= 72000.0d0) then
        tmp = exp((-m_1 * m_1)) * cos(m_1)
    else
        tmp = 1.0d0 * exp(((n * n) * (-0.25d0)))
    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 <= 2e-251) {
		tmp = 1.0 * Math.exp((-0.25 * (m * m)));
	} else if (n <= 72000.0) {
		tmp = Math.exp((-M * M)) * Math.cos(M);
	} else {
		tmp = 1.0 * Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 2e-251:
		tmp = 1.0 * math.exp((-0.25 * (m * m)))
	elif n <= 72000.0:
		tmp = math.exp((-M * M)) * math.cos(M)
	else:
		tmp = 1.0 * math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 2e-251)
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 72000.0)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * cos(M));
	else
		tmp = Float64(1.0 * exp(Float64(Float64(n * n) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 2e-251)
		tmp = 1.0 * exp((-0.25 * (m * m)));
	elseif (n <= 72000.0)
		tmp = exp((-M * M)) * cos(M);
	else
		tmp = 1.0 * exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 2e-251], N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 72000.0], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 2.00000000000000003e-251

   1. Initial program 69.7%

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

   5. Applied rewrites 26.7%

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

   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

   8. Applied rewrites 33.2%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 33.2%

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

   12. Taylor expanded in m around inf

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

   14. Applied rewrites 51.5%

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

   if 2.00000000000000003e-251 < n < 72000

   1. Initial program 75.0%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 59.5%

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

   if 72000 < n

   1. Initial program 73.5%

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

   5. Applied rewrites 17.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 20.4%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 20.4%

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

   12. Taylor expanded in n around inf

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

   14. Applied rewrites 98.0%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{-251}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 72000:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{if}\;m \leq -1000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -6 \cdot 10^{-84}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{elif}\;m \leq 0.32:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 1.0 (exp (* -0.25 (* m m))))))
   (if (<= m -1000.0)
     t_0
     (if (<= m -6e-84)
       (* 1.0 (exp (- l)))
       (if (<= m 0.32) (* (exp (* (- M) M)) 1.0) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = 1.0 * exp((-0.25 * (m * m)));
	double tmp;
	if (m <= -1000.0) {
		tmp = t_0;
	} else if (m <= -6e-84) {
		tmp = 1.0 * exp(-l);
	} else if (m <= 0.32) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 * exp(((-0.25d0) * (m * m)))
    if (m <= (-1000.0d0)) then
        tmp = t_0
    else if (m <= (-6d-84)) then
        tmp = 1.0d0 * exp(-l)
    else if (m <= 0.32d0) then
        tmp = exp((-m_1 * m_1)) * 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = 1.0 * Math.exp((-0.25 * (m * m)));
	double tmp;
	if (m <= -1000.0) {
		tmp = t_0;
	} else if (m <= -6e-84) {
		tmp = 1.0 * Math.exp(-l);
	} else if (m <= 0.32) {
		tmp = Math.exp((-M * M)) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = 1.0 * math.exp((-0.25 * (m * m)))
	tmp = 0
	if m <= -1000.0:
		tmp = t_0
	elif m <= -6e-84:
		tmp = 1.0 * math.exp(-l)
	elif m <= 0.32:
		tmp = math.exp((-M * M)) * 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))))
	tmp = 0.0
	if (m <= -1000.0)
		tmp = t_0;
	elseif (m <= -6e-84)
		tmp = Float64(1.0 * exp(Float64(-l)));
	elseif (m <= 0.32)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = 1.0 * exp((-0.25 * (m * m)));
	tmp = 0.0;
	if (m <= -1000.0)
		tmp = t_0;
	elseif (m <= -6e-84)
		tmp = 1.0 * exp(-l);
	elseif (m <= 0.32)
		tmp = exp((-M * M)) * 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1000.0], t$95$0, If[LessEqual[m, -6e-84], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.32], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1e3 or 0.320000000000000007 < m

   1. Initial program 71.4%

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

   3. Taylor expanded in l around inf

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

   5. Applied rewrites 18.4%

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

   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

   8. Applied rewrites 26.8%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 26.8%

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

   12. Taylor expanded in m around inf

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

   14. Applied rewrites 96.9%

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

   if -1e3 < m < -6.0000000000000002e-84

   1. Initial program 86.8%

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

   3. Taylor expanded in 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

   5. Applied rewrites 47.8%

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

   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

   8. Applied rewrites 47.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 47.9%

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

   if -6.0000000000000002e-84 < m < 0.320000000000000007

   1. Initial program 70.0%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 61.2%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 61.2%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1000:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -6 \cdot 10^{-84}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{elif}\;m \leq 0.32:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{-251}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 72000:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 2e-251)
   (* 1.0 (exp (* -0.25 (* m m))))
   (if (<= n 72000.0)
     (* (exp (* (- M) M)) 1.0)
     (* 1.0 (exp (* (* n n) -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2e-251) {
		tmp = 1.0 * exp((-0.25 * (m * m)));
	} else if (n <= 72000.0) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = 1.0 * exp(((n * n) * -0.25));
	}
	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 <= 2d-251) then
        tmp = 1.0d0 * exp(((-0.25d0) * (m * m)))
    else if (n <= 72000.0d0) then
        tmp = exp((-m_1 * m_1)) * 1.0d0
    else
        tmp = 1.0d0 * exp(((n * n) * (-0.25d0)))
    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 <= 2e-251) {
		tmp = 1.0 * Math.exp((-0.25 * (m * m)));
	} else if (n <= 72000.0) {
		tmp = Math.exp((-M * M)) * 1.0;
	} else {
		tmp = 1.0 * Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 2e-251:
		tmp = 1.0 * math.exp((-0.25 * (m * m)))
	elif n <= 72000.0:
		tmp = math.exp((-M * M)) * 1.0
	else:
		tmp = 1.0 * math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 2e-251)
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 72000.0)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(1.0 * exp(Float64(Float64(n * n) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 2e-251)
		tmp = 1.0 * exp((-0.25 * (m * m)));
	elseif (n <= 72000.0)
		tmp = exp((-M * M)) * 1.0;
	else
		tmp = 1.0 * exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 2e-251], N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 72000.0], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 2.00000000000000003e-251

   1. Initial program 69.7%

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

   5. Applied rewrites 26.7%

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

   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

   8. Applied rewrites 33.2%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 33.2%

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

   12. Taylor expanded in m around inf

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

   14. Applied rewrites 51.5%

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

   if 2.00000000000000003e-251 < n < 72000

   1. Initial program 75.0%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 59.5%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 59.5%

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

   if 72000 < n

   1. Initial program 73.5%

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

   5. Applied rewrites 17.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 20.4%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 20.4%

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

   12. Taylor expanded in n around inf

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

   14. Applied rewrites 98.0%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{-251}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 72000:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -2.7 \cdot 10^{-18} \lor \neg \left(M \leq 26.5\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -2.7e-18) (not (<= M 26.5)))
   (* (exp (* (- M) M)) 1.0)
   (* 1.0 (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2.7e-18) || !(M <= 26.5)) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = 1.0 * exp(-l);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-2.7d-18)) .or. (.not. (m_1 <= 26.5d0))) then
        tmp = exp((-m_1 * m_1)) * 1.0d0
    else
        tmp = 1.0d0 * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2.7e-18) || !(M <= 26.5)) {
		tmp = Math.exp((-M * M)) * 1.0;
	} else {
		tmp = 1.0 * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -2.7e-18) or not (M <= 26.5):
		tmp = math.exp((-M * M)) * 1.0
	else:
		tmp = 1.0 * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -2.7e-18) || !(M <= 26.5))
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(1.0 * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -2.7e-18) || ~((M <= 26.5)))
		tmp = exp((-M * M)) * 1.0;
	else
		tmp = 1.0 * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -2.7e-18], N[Not[LessEqual[M, 26.5]], $MachinePrecision]], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -2.69999999999999989e-18 or 26.5 < M

   1. Initial program 76.1%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 95.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 95.9%

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

   if -2.69999999999999989e-18 < M < 26.5

   1. Initial program 66.3%

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

   3. Taylor expanded in 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

   5. Applied rewrites 34.6%

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

   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

   8. Applied rewrites 41.5%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 41.5%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.7 \cdot 10^{-18} \lor \neg \left(M \leq 26.5\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

5. Applied rewrites 26.8%

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

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

8. Applied rewrites 32.9%

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

9. Taylor expanded in M around 0

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

11. Applied rewrites 32.5%

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

Reproduce

herbie shell --seed 2025018 
(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)))))))