Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.6% → 95.2%

Time: 5.9s

Alternatives: 8

Speedup: 2.9×

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: 75.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+49} \lor \neg \left(M \leq 27.5\right):\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1e+49) (not (<= M 27.5)))
   (* (cos M) (exp (* (- M) M)))
   (exp (- (fabs (- m n)) (+ l (* 0.25 (pow (+ m n) 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1e+49) || !(M <= 27.5)) {
		tmp = cos(M) * exp((-M * M));
	} else {
		tmp = exp((fabs((m - n)) - (l + (0.25 * pow((m + n), 2.0)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-1d+49)) .or. (.not. (m_1 <= 27.5d0))) then
        tmp = cos(m_1) * exp((-m_1 * m_1))
    else
        tmp = exp((abs((m - n)) - (l + (0.25d0 * ((m + n) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1e+49) || !(M <= 27.5)) {
		tmp = Math.cos(M) * Math.exp((-M * M));
	} else {
		tmp = Math.exp((Math.abs((m - n)) - (l + (0.25 * Math.pow((m + n), 2.0)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1e+49) or not (M <= 27.5):
		tmp = math.cos(M) * math.exp((-M * M))
	else:
		tmp = math.exp((math.fabs((m - n)) - (l + (0.25 * math.pow((m + n), 2.0)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1e+49) || !(M <= 27.5))
		tmp = Float64(cos(M) * exp(Float64(Float64(-M) * M)));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(0.25 * (Float64(m + n) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1e+49) || ~((M <= 27.5)))
		tmp = cos(M) * exp((-M * M));
	else
		tmp = exp((abs((m - n)) - (l + (0.25 * ((m + n) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1e+49], N[Not[LessEqual[M, 27.5]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -9.99999999999999946e48 or 27.5 < M

   1. Initial program 73.8%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if -9.99999999999999946e48 < M < 27.5

   1. Initial program 71.3%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 70.5%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 94.3

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

   8. Applied rewrites 94.3%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+49} \lor \neg \left(M \leq 27.5\right):\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. fabs-sub N/A

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

   7. lower-fabs.f64 N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

5. Applied rewrites 96.7%

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

6. Final simplification 96.7%

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

Alternative 3: 85.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.65 \cdot 10^{+41} \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1.65e+41) (not (<= M 27.0)))
   (* (cos M) (exp (* (- M) M)))
   (exp (- (fabs n) (+ l (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.65e+41) || !(M <= 27.0)) {
		tmp = cos(M) * exp((-M * M));
	} else {
		tmp = exp((fabs(n) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-1.65d+41)) .or. (.not. (m_1 <= 27.0d0))) then
        tmp = cos(m_1) * exp((-m_1 * m_1))
    else
        tmp = exp((abs(n) - (l + (0.25d0 * (n * n)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.65e+41) || !(M <= 27.0)) {
		tmp = Math.cos(M) * Math.exp((-M * M));
	} else {
		tmp = Math.exp((Math.abs(n) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1.65e+41) or not (M <= 27.0):
		tmp = math.cos(M) * math.exp((-M * M))
	else:
		tmp = math.exp((math.fabs(n) - (l + (0.25 * (n * n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1.65e+41) || !(M <= 27.0))
		tmp = Float64(cos(M) * exp(Float64(Float64(-M) * M)));
	else
		tmp = exp(Float64(abs(n) - Float64(l + Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1.65e+41) || ~((M <= 27.0)))
		tmp = cos(M) * exp((-M * M));
	else
		tmp = exp((abs(n) - (l + (0.25 * (n * n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1.65e+41], N[Not[LessEqual[M, 27.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Abs[n], $MachinePrecision] - N[(l + N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1.65e41 or 27 < M

   1. Initial program 73.4%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if -1.65e41 < M < 27

   1. Initial program 71.6%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 70.9%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 94.2

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

   8. Applied rewrites 94.2%

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

   9. Taylor expanded in m around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 68.8

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

   11. Applied rewrites 68.8%

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

   12. Taylor expanded in m around 0

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

   14. Applied rewrites 79.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.65 \cdot 10^{+41} \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -10000000000000:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -10000000000000.0)
   (exp (* -0.25 (* m m)))
   (exp (- (fabs (- m n)) (+ l (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -10000000000000.0) {
		tmp = exp((-0.25 * (m * m)));
	} else {
		tmp = exp((fabs((m - n)) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-10000000000000.0d0)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else
        tmp = exp((abs((m - n)) - (l + (0.25d0 * (n * n)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -10000000000000.0) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.exp((Math.abs((m - n)) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -10000000000000.0:
		tmp = math.exp((-0.25 * (m * m)))
	else:
		tmp = math.exp((math.fabs((m - n)) - (l + (0.25 * (n * n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -10000000000000.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -10000000000000.0)
		tmp = exp((-0.25 * (m * m)));
	else
		tmp = exp((abs((m - n)) - (l + (0.25 * (n * n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -10000000000000.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1e13

   1. Initial program 51.1%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 51.1%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in m around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -1e13 < m

   1. Initial program 77.4%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 84.4

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

   8. Applied rewrites 84.4%

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

   9. Taylor expanded in m around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 67.4

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

   11. Applied rewrites 67.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -10000000000000:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3700000000000:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -3700000000000.0)
   (exp (* -0.25 (* m m)))
   (exp (- (fabs n) (+ l (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -3700000000000.0) {
		tmp = exp((-0.25 * (m * m)));
	} else {
		tmp = exp((fabs(n) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-3700000000000.0d0)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else
        tmp = exp((abs(n) - (l + (0.25d0 * (n * n)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -3700000000000.0) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.exp((Math.abs(n) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -3700000000000.0:
		tmp = math.exp((-0.25 * (m * m)))
	else:
		tmp = math.exp((math.fabs(n) - (l + (0.25 * (n * n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -3700000000000.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	else
		tmp = exp(Float64(abs(n) - Float64(l + Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -3700000000000.0)
		tmp = exp((-0.25 * (m * m)));
	else
		tmp = exp((abs(n) - (l + (0.25 * (n * n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -3700000000000.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Abs[n], $MachinePrecision] - N[(l + N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.7e12

   1. Initial program 51.1%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 51.1%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in m around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -3.7e12 < m

   1. Initial program 77.4%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 84.4

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

   8. Applied rewrites 84.4%

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

   9. Taylor expanded in m around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 67.4

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

   11. Applied rewrites 67.4%

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

   12. Taylor expanded in m around 0

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

   14. Applied rewrites 72.1%

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

Alternative 6: 68.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1050000000000 \lor \neg \left(m \leq 2.55 \cdot 10^{-48}\right):\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -1050000000000.0) (not (<= m 2.55e-48)))
   (exp (* -0.25 (* m m)))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -1050000000000.0) || !(m <= 2.55e-48)) {
		tmp = exp((-0.25 * (m * m)));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m <= (-1050000000000.0d0)) .or. (.not. (m <= 2.55d-48))) then
        tmp = exp(((-0.25d0) * (m * m)))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -1050000000000.0) || !(m <= 2.55e-48)) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -1050000000000.0) or not (m <= 2.55e-48):
		tmp = math.exp((-0.25 * (m * m)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -1050000000000.0) || !(m <= 2.55e-48))
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -1050000000000.0) || ~((m <= 2.55e-48)))
		tmp = exp((-0.25 * (m * m)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -1050000000000.0], N[Not[LessEqual[m, 2.55e-48]], $MachinePrecision]], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.05e12 or 2.55000000000000006e-48 < m

   1. Initial program 63.1%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 62.3%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 97.9

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in m around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 92.2

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

   11. Applied rewrites 92.2%

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

   if -1.05e12 < m < 2.55000000000000006e-48

   1. Initial program 81.1%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 63.0%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 77.6

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

   8. Applied rewrites 77.6%

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

   9. Taylor expanded in l around inf

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

       1. lower-*.f64 44.7

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

   11. Applied rewrites 44.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1050000000000 \lor \neg \left(m \leq 2.55 \cdot 10^{-48}\right):\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.35 \cdot 10^{-271}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 1.08 \cdot 10^{-10}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.35e-271)
   (exp (* -0.25 (* m m)))
   (if (<= n 1.08e-10) (exp (- l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.35e-271) {
		tmp = exp((-0.25 * (m * m)));
	} else if (n <= 1.08e-10) {
		tmp = exp(-l);
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 1.35d-271) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (n <= 1.08d-10) then
        tmp = exp(-l)
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.35e-271) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (n <= 1.08e-10) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.35e-271:
		tmp = math.exp((-0.25 * (m * m)))
	elif n <= 1.08e-10:
		tmp = math.exp(-l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.35e-271)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (n <= 1.08e-10)
		tmp = exp(Float64(-l));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1.35e-271)
		tmp = exp((-0.25 * (m * m)));
	elseif (n <= 1.08e-10)
		tmp = exp(-l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.35e-271], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.08e-10], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.3499999999999999e-271

   1. Initial program 69.4%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 59.6%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 86.8

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

   8. Applied rewrites 86.8%

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

   9. Taylor expanded in m around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 56.8

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

   11. Applied rewrites 56.8%

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

   if 1.3499999999999999e-271 < n < 1.08000000000000002e-10

   1. Initial program 92.3%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.5%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 72.7

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

   8. Applied rewrites 72.7%

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

   9. Taylor expanded in l around inf

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

       1. lower-*.f64 46.5

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

   11. Applied rewrites 46.5%

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

   if 1.08000000000000002e-10 < n

   1. Initial program 64.7%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.8%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in n around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 94.3

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

   11. Applied rewrites 94.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.35 \cdot 10^{-271}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 1.08 \cdot 10^{-10}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in M around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 62.7%

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

6. Taylor expanded in K around 0

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-+.f64 87.2

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

8. Applied rewrites 87.2%

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

9. Taylor expanded in l around inf

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

    1. lower-*.f64 36.5

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

11. Applied rewrites 36.5%

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

12. Final simplification 36.5%

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

Reproduce

herbie shell --seed 2025085 
(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)))))))