Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.5% → 96.6%

Time: 9.7s

Alternatives: 8

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: 76.5% 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: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.9%

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

6. Final simplification 96.9%

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

Alternative 2: 75.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-66}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{+64}:\\ \;\;\;\;e^{-\left(M \cdot M + \left(\ell - \left(m - n\right)\right)\right)} \cdot \cos \left(M - 0.5 \cdot \left(K \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -1.1e-66)
   (* (cos M) (exp (* (* m m) -0.25)))
   (if (<= n 7.5e+64)
     (* (exp (- (+ (* M M) (- l (- m n))))) (cos (- M (* 0.5 (* K n)))))
     (* (exp (* (* n n) -0.25)) (cos M)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -1.1e-66) {
		tmp = cos(M) * exp(((m * m) * -0.25));
	} else if (n <= 7.5e+64) {
		tmp = exp(-((M * M) + (l - (m - n)))) * cos((M - (0.5 * (K * n))));
	} else {
		tmp = exp(((n * n) * -0.25)) * cos(M);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= (-1.1d-66)) then
        tmp = cos(m_1) * exp(((m * m) * (-0.25d0)))
    else if (n <= 7.5d+64) then
        tmp = exp(-((m_1 * m_1) + (l - (m - n)))) * cos((m_1 - (0.5d0 * (k * n))))
    else
        tmp = exp(((n * n) * (-0.25d0))) * cos(m_1)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -1.1e-66) {
		tmp = Math.cos(M) * Math.exp(((m * m) * -0.25));
	} else if (n <= 7.5e+64) {
		tmp = Math.exp(-((M * M) + (l - (m - n)))) * Math.cos((M - (0.5 * (K * n))));
	} else {
		tmp = Math.exp(((n * n) * -0.25)) * Math.cos(M);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -1.1e-66:
		tmp = math.cos(M) * math.exp(((m * m) * -0.25))
	elif n <= 7.5e+64:
		tmp = math.exp(-((M * M) + (l - (m - n)))) * math.cos((M - (0.5 * (K * n))))
	else:
		tmp = math.exp(((n * n) * -0.25)) * math.cos(M)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -1.1e-66)
		tmp = Float64(cos(M) * exp(Float64(Float64(m * m) * -0.25)));
	elseif (n <= 7.5e+64)
		tmp = Float64(exp(Float64(-Float64(Float64(M * M) + Float64(l - Float64(m - n))))) * cos(Float64(M - Float64(0.5 * Float64(K * n)))));
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * cos(M));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -1.1e-66)
		tmp = cos(M) * exp(((m * m) * -0.25));
	elseif (n <= 7.5e+64)
		tmp = exp(-((M * M) + (l - (m - n)))) * cos((M - (0.5 * (K * n))));
	else
		tmp = exp(((n * n) * -0.25)) * cos(M);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -1.1e-66], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.5e+64], N[(N[Exp[(-N[(N[(M * M), $MachinePrecision] + N[(l - N[(m - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Cos[N[(M - N[(0.5 * N[(K * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.1000000000000001e-66

   1. Initial program 68.7%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 54.1

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

   8. Applied rewrites 54.1%

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

   if -1.1000000000000001e-66 < n < 7.5000000000000005e64

   1. Initial program 86.5%

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

   3. Taylor expanded in M around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in m around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 73.8

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

   8. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 73.8

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

      4. lift-fabs.f64 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. sqrt-prod N/A

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

      7. rem-square-sqrt 84.6

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

      8. lift-cos.f64 N/A

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

      9. lift--.f64 N/A

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

   10. Applied rewrites 84.6%

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

   if 7.5000000000000005e64 < n

   1. Initial program 70.2%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-66}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{+64}:\\ \;\;\;\;e^{-\left(M \cdot M + \left(\ell - \left(m - n\right)\right)\right)} \cdot \cos \left(M - 0.5 \cdot \left(K \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{if}\;M \leq -4.9 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{-206}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) (cos M))))
   (if (<= M -4.9e-5)
     t_0
     (if (<= M 1.9e-206)
       (* (exp (* (* n n) -0.25)) (cos M))
       (if (<= M 26.5) (* 1.0 (exp (- l))) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * cos(M);
	double tmp;
	if (M <= -4.9e-5) {
		tmp = t_0;
	} else if (M <= 1.9e-206) {
		tmp = exp(((n * n) * -0.25)) * cos(M);
	} else if (M <= 26.5) {
		tmp = 1.0 * exp(-l);
	} 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 = exp((-m_1 * m_1)) * cos(m_1)
    if (m_1 <= (-4.9d-5)) then
        tmp = t_0
    else if (m_1 <= 1.9d-206) then
        tmp = exp(((n * n) * (-0.25d0))) * cos(m_1)
    else if (m_1 <= 26.5d0) then
        tmp = 1.0d0 * exp(-l)
    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 = Math.exp((-M * M)) * Math.cos(M);
	double tmp;
	if (M <= -4.9e-5) {
		tmp = t_0;
	} else if (M <= 1.9e-206) {
		tmp = Math.exp(((n * n) * -0.25)) * Math.cos(M);
	} else if (M <= 26.5) {
		tmp = 1.0 * Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-M * M)) * math.cos(M)
	tmp = 0
	if M <= -4.9e-5:
		tmp = t_0
	elif M <= 1.9e-206:
		tmp = math.exp(((n * n) * -0.25)) * math.cos(M)
	elif M <= 26.5:
		tmp = 1.0 * math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * cos(M))
	tmp = 0.0
	if (M <= -4.9e-5)
		tmp = t_0;
	elseif (M <= 1.9e-206)
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * cos(M));
	elseif (M <= 26.5)
		tmp = Float64(1.0 * exp(Float64(-l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-M * M)) * cos(M);
	tmp = 0.0;
	if (M <= -4.9e-5)
		tmp = t_0;
	elseif (M <= 1.9e-206)
		tmp = exp(((n * n) * -0.25)) * cos(M);
	elseif (M <= 26.5)
		tmp = 1.0 * exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -4.9e-5], t$95$0, If[LessEqual[M, 1.9e-206], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 26.5], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if M < -4.9e-5 or 26.5 < M

   1. Initial program 85.7%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   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%

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

   if -4.9e-5 < M < 1.90000000000000001e-206

   1. Initial program 68.1%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 61.5%

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

   if 1.90000000000000001e-206 < M < 26.5

   1. Initial program 79.4%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 60.6

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

   5. Applied rewrites 60.6%

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

   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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 66.4

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

   8. Applied rewrites 66.4%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 66.4%

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

Alternative 4: 73.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{if}\;M \leq -4.9 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 3.7 \cdot 10^{-211}:\\ \;\;\;\;1 \cdot e^{\left(\frac{0.5 \cdot m - M}{n} + 0.25\right) \cdot \left(\left(-n\right) \cdot n\right)}\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) (cos M))))
   (if (<= M -4.9e-5)
     t_0
     (if (<= M 3.7e-211)
       (* 1.0 (exp (* (+ (/ (- (* 0.5 m) M) n) 0.25) (* (- n) n))))
       (if (<= M 26.5) (* 1.0 (exp (- l))) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * cos(M);
	double tmp;
	if (M <= -4.9e-5) {
		tmp = t_0;
	} else if (M <= 3.7e-211) {
		tmp = 1.0 * exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)));
	} else if (M <= 26.5) {
		tmp = 1.0 * exp(-l);
	} 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 = exp((-m_1 * m_1)) * cos(m_1)
    if (m_1 <= (-4.9d-5)) then
        tmp = t_0
    else if (m_1 <= 3.7d-211) then
        tmp = 1.0d0 * exp((((((0.5d0 * m) - m_1) / n) + 0.25d0) * (-n * n)))
    else if (m_1 <= 26.5d0) then
        tmp = 1.0d0 * exp(-l)
    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 = Math.exp((-M * M)) * Math.cos(M);
	double tmp;
	if (M <= -4.9e-5) {
		tmp = t_0;
	} else if (M <= 3.7e-211) {
		tmp = 1.0 * Math.exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)));
	} else if (M <= 26.5) {
		tmp = 1.0 * Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-M * M)) * math.cos(M)
	tmp = 0
	if M <= -4.9e-5:
		tmp = t_0
	elif M <= 3.7e-211:
		tmp = 1.0 * math.exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)))
	elif M <= 26.5:
		tmp = 1.0 * math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * cos(M))
	tmp = 0.0
	if (M <= -4.9e-5)
		tmp = t_0;
	elseif (M <= 3.7e-211)
		tmp = Float64(1.0 * exp(Float64(Float64(Float64(Float64(Float64(0.5 * m) - M) / n) + 0.25) * Float64(Float64(-n) * n))));
	elseif (M <= 26.5)
		tmp = Float64(1.0 * exp(Float64(-l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-M * M)) * cos(M);
	tmp = 0.0;
	if (M <= -4.9e-5)
		tmp = t_0;
	elseif (M <= 3.7e-211)
		tmp = 1.0 * exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)));
	elseif (M <= 26.5)
		tmp = 1.0 * exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -4.9e-5], t$95$0, If[LessEqual[M, 3.7e-211], N[(1.0 * N[Exp[N[(N[(N[(N[(N[(0.5 * m), $MachinePrecision] - M), $MachinePrecision] / n), $MachinePrecision] + 0.25), $MachinePrecision] * N[((-n) * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 26.5], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if M < -4.9e-5 or 26.5 < M

   1. Initial program 85.7%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   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%

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

   if -4.9e-5 < M < 3.6999999999999998e-211

   1. Initial program 68.1%

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

   3. Taylor expanded in 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

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

      2. lower-neg.f64 36.1

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

   5. Applied rewrites 36.1%

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

   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

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

      2. lower-cos.f64 43.8

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

   8. Applied rewrites 43.8%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 43.8%

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

   12. Taylor expanded in n around -inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower--.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 61.2

           \[\leadsto 1 \cdot e^{\left(\frac{-\left(0.5 \cdot m - M\right)}{n} - 0.25\right) \cdot \color{blue}{\left(n \cdot n\right)}} \]

   14. Applied rewrites 61.2%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(\frac{-\left(0.5 \cdot m - M\right)}{n} - 0.25\right) \cdot \left(n \cdot n\right)}} \]

   if 3.6999999999999998e-211 < M < 26.5

   1. Initial program 79.4%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 60.6

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

   5. Applied rewrites 60.6%

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

   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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 66.4

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

   8. Applied rewrites 66.4%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 66.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4.9 \cdot 10^{-5}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{elif}\;M \leq 3.7 \cdot 10^{-211}:\\ \;\;\;\;1 \cdot e^{\left(\frac{0.5 \cdot m - M}{n} + 0.25\right) \cdot \left(\left(-n\right) \cdot n\right)}\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.05 \cdot 10^{-297}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.05e-297)
   (* (cos M) (exp (* (* m m) -0.25)))
   (if (<= n 55.0)
     (* (exp (* (- M) M)) (cos M))
     (* (exp (* (* n n) -0.25)) (cos M)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.05e-297) {
		tmp = cos(M) * exp(((m * m) * -0.25));
	} else if (n <= 55.0) {
		tmp = exp((-M * M)) * cos(M);
	} else {
		tmp = exp(((n * n) * -0.25)) * cos(M);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 1.05d-297) then
        tmp = cos(m_1) * exp(((m * m) * (-0.25d0)))
    else if (n <= 55.0d0) then
        tmp = exp((-m_1 * m_1)) * cos(m_1)
    else
        tmp = exp(((n * n) * (-0.25d0))) * cos(m_1)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.05e-297) {
		tmp = Math.cos(M) * Math.exp(((m * m) * -0.25));
	} else if (n <= 55.0) {
		tmp = Math.exp((-M * M)) * Math.cos(M);
	} else {
		tmp = Math.exp(((n * n) * -0.25)) * Math.cos(M);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.05e-297:
		tmp = math.cos(M) * math.exp(((m * m) * -0.25))
	elif n <= 55.0:
		tmp = math.exp((-M * M)) * math.cos(M)
	else:
		tmp = math.exp(((n * n) * -0.25)) * math.cos(M)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.05e-297)
		tmp = Float64(cos(M) * exp(Float64(Float64(m * m) * -0.25)));
	elseif (n <= 55.0)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * cos(M));
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * cos(M));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1.05e-297)
		tmp = cos(M) * exp(((m * m) * -0.25));
	elseif (n <= 55.0)
		tmp = exp((-M * M)) * cos(M);
	else
		tmp = exp(((n * n) * -0.25)) * cos(M);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.05e-297], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 55.0], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.05000000000000007e-297

   1. Initial program 75.5%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 53.8

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

   8. Applied rewrites 53.8%

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

   if 1.05000000000000007e-297 < n < 55

   1. Initial program 86.4%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 55.7%

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

   if 55 < 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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 92.8%

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

Alternative 6: 63.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot e^{\left(\frac{0.5 \cdot m - M}{n} + 0.25\right) \cdot \left(\left(-n\right) \cdot n\right)}\\ \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -7.5 \cdot 10^{-112}:\\ \;\;\;\;1 \cdot e^{\left(\frac{0.5 \cdot n - M}{m} + 0.25\right) \cdot \left(\left(-m\right) \cdot m\right)}\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 1.0 (exp (* (+ (/ (- (* 0.5 m) M) n) 0.25) (* (- n) n))))))
   (if (<= l -4.3e+192)
     t_0
     (if (<= l -7.5e-112)
       (* 1.0 (exp (* (+ (/ (- (* 0.5 n) M) m) 0.25) (* (- m) m))))
       (if (<= l 720.0) t_0 (* 1.0 (exp (- l))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = 1.0 * exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)));
	double tmp;
	if (l <= -4.3e+192) {
		tmp = t_0;
	} else if (l <= -7.5e-112) {
		tmp = 1.0 * exp((((((0.5 * n) - M) / m) + 0.25) * (-m * m)));
	} else if (l <= 720.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 * exp((((((0.5d0 * m) - m_1) / n) + 0.25d0) * (-n * n)))
    if (l <= (-4.3d+192)) then
        tmp = t_0
    else if (l <= (-7.5d-112)) then
        tmp = 1.0d0 * exp((((((0.5d0 * n) - m_1) / m) + 0.25d0) * (-m * m)))
    else if (l <= 720.0d0) then
        tmp = t_0
    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 t_0 = 1.0 * Math.exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)));
	double tmp;
	if (l <= -4.3e+192) {
		tmp = t_0;
	} else if (l <= -7.5e-112) {
		tmp = 1.0 * Math.exp((((((0.5 * n) - M) / m) + 0.25) * (-m * m)));
	} else if (l <= 720.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = 1.0 * math.exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)))
	tmp = 0
	if l <= -4.3e+192:
		tmp = t_0
	elif l <= -7.5e-112:
		tmp = 1.0 * math.exp((((((0.5 * n) - M) / m) + 0.25) * (-m * m)))
	elif l <= 720.0:
		tmp = t_0
	else:
		tmp = 1.0 * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(1.0 * exp(Float64(Float64(Float64(Float64(Float64(0.5 * m) - M) / n) + 0.25) * Float64(Float64(-n) * n))))
	tmp = 0.0
	if (l <= -4.3e+192)
		tmp = t_0;
	elseif (l <= -7.5e-112)
		tmp = Float64(1.0 * exp(Float64(Float64(Float64(Float64(Float64(0.5 * n) - M) / m) + 0.25) * Float64(Float64(-m) * m))));
	elseif (l <= 720.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = 1.0 * exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)));
	tmp = 0.0;
	if (l <= -4.3e+192)
		tmp = t_0;
	elseif (l <= -7.5e-112)
		tmp = 1.0 * exp((((((0.5 * n) - M) / m) + 0.25) * (-m * m)));
	elseif (l <= 720.0)
		tmp = t_0;
	else
		tmp = 1.0 * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(1.0 * N[Exp[N[(N[(N[(N[(N[(0.5 * m), $MachinePrecision] - M), $MachinePrecision] / n), $MachinePrecision] + 0.25), $MachinePrecision] * N[((-n) * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.3e+192], t$95$0, If[LessEqual[l, -7.5e-112], N[(1.0 * N[Exp[N[(N[(N[(N[(N[(0.5 * n), $MachinePrecision] - M), $MachinePrecision] / m), $MachinePrecision] + 0.25), $MachinePrecision] * N[((-m) * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 720.0], t$95$0, N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.29999999999999976e192 or -7.5000000000000002e-112 < l < 720

   1. Initial program 77.6%

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

   3. Taylor expanded in 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

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

      2. lower-neg.f64 17.5

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

   5. Applied rewrites 17.5%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 18.1

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

   8. Applied rewrites 18.1%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 16.5%

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

   12. Taylor expanded in n around -inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower--.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 53.3

           \[\leadsto 1 \cdot e^{\left(\frac{-\left(0.5 \cdot m - M\right)}{n} - 0.25\right) \cdot \color{blue}{\left(n \cdot n\right)}} \]

   14. Applied rewrites 53.3%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(\frac{-\left(0.5 \cdot m - M\right)}{n} - 0.25\right) \cdot \left(n \cdot n\right)}} \]

   if -4.29999999999999976e192 < l < -7.5000000000000002e-112

   1. Initial program 77.2%

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

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

      2. lower-neg.f64 18.6

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

   5. Applied rewrites 18.6%

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

   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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 20.6

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

   8. Applied rewrites 20.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 17.6%

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

   12. Taylor expanded in m around -inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. mul-1-neg N/A

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

       5. distribute-neg-frac2 N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-neg.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 45.9

           \[\leadsto 1 \cdot e^{\left(\frac{0.5 \cdot n - M}{-m} - 0.25\right) \cdot \color{blue}{\left(m \cdot m\right)}} \]

   14. Applied rewrites 45.9%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(\frac{0.5 \cdot n - M}{-m} - 0.25\right) \cdot \left(m \cdot m\right)}} \]

   if 720 < l

   1. Initial program 80.3%

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

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

      2. lower-neg.f64 80.3

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

   5. Applied rewrites 80.3%

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

   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

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

      2. lower-cos.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+192}:\\ \;\;\;\;1 \cdot e^{\left(\frac{0.5 \cdot m - M}{n} + 0.25\right) \cdot \left(\left(-n\right) \cdot n\right)}\\ \mathbf{elif}\;\ell \leq -7.5 \cdot 10^{-112}:\\ \;\;\;\;1 \cdot e^{\left(\frac{0.5 \cdot n - M}{m} + 0.25\right) \cdot \left(\left(-m\right) \cdot m\right)}\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;1 \cdot e^{\left(\frac{0.5 \cdot m - M}{n} + 0.25\right) \cdot \left(\left(-n\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;1 \cdot e^{\left(\frac{0.5 \cdot m - M}{n} + 0.25\right) \cdot \left(\left(-n\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 720.0)
   (* 1.0 (exp (* (+ (/ (- (* 0.5 m) M) n) 0.25) (* (- n) n))))
   (* 1.0 (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = 1.0 * exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)));
	} else {
		tmp = 1.0 * exp(-l);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 720.0d0) then
        tmp = 1.0d0 * exp((((((0.5d0 * m) - m_1) / n) + 0.25d0) * (-n * n)))
    else
        tmp = 1.0d0 * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = 1.0 * Math.exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)));
	} else {
		tmp = 1.0 * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 720.0:
		tmp = 1.0 * math.exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)))
	else:
		tmp = 1.0 * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 720.0)
		tmp = Float64(1.0 * exp(Float64(Float64(Float64(Float64(Float64(0.5 * m) - M) / n) + 0.25) * Float64(Float64(-n) * n))));
	else
		tmp = Float64(1.0 * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 720.0)
		tmp = 1.0 * exp((((((0.5 * m) - M) / n) + 0.25) * (-n * n)));
	else
		tmp = 1.0 * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 720.0], N[(1.0 * N[Exp[N[(N[(N[(N[(N[(0.5 * m), $MachinePrecision] - M), $MachinePrecision] / n), $MachinePrecision] + 0.25), $MachinePrecision] * N[((-n) * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 720

   1. Initial program 77.5%

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 17.9

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

   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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 19.0

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

   8. Applied rewrites 19.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 16.9%

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

   12. Taylor expanded in n around -inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower--.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 50.2

           \[\leadsto 1 \cdot e^{\left(\frac{-\left(0.5 \cdot m - M\right)}{n} - 0.25\right) \cdot \color{blue}{\left(n \cdot n\right)}} \]

   14. Applied rewrites 50.2%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(\frac{-\left(0.5 \cdot m - M\right)}{n} - 0.25\right) \cdot \left(n \cdot n\right)}} \]

   if 720 < l

   1. Initial program 80.3%

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

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

      2. lower-neg.f64 80.3

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

   5. Applied rewrites 80.3%

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

   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

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

      2. lower-cos.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;1 \cdot e^{\left(\frac{0.5 \cdot m - M}{n} + 0.25\right) \cdot \left(\left(-n\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.8% 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 78.2%

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

3. Taylor expanded in l around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 34.0

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

5. Applied rewrites 34.0%

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

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

   1. cos-neg-rev N/A

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

   2. lower-cos.f64 39.9

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

8. Applied rewrites 39.9%

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

9. Taylor expanded in M around 0

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

11. Applied rewrites 38.3%

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

Reproduce

herbie shell --seed 2025006 
(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)))))))