Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.5% → 96.1%

Time: 4.8s

Alternatives: 7

Speedup: 2.6×

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: 96.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 1 \cdot e^{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 * exp((fabs((m - n)) - (l + pow(((0.5 * (m + n)) - M), 2.0))));
}

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((abs((m - n)) - (l + (((0.5d0 * (m + n)) - m_1) ** 2.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 * Math.exp((Math.abs((m - n)) - (l + Math.pow(((0.5 * (m + n)) - M), 2.0))));
}

Python

def code(K, m, n, M, l):
	return 1.0 * math.exp((math.fabs((m - n)) - (l + math.pow(((0.5 * (m + n)) - M), 2.0))))

Julia

function code(K, m, n, M, l)
	return Float64(1.0 * exp(Float64(abs(Float64(m - n)) - Float64(l + (Float64(Float64(0.5 * Float64(m + n)) - M) ^ 2.0)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 * exp((abs((m - n)) - (l + (((0.5 * (m + n)) - M) ^ 2.0))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 * N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[Power[N[(N[(0.5 * N[(m + n), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

2. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 96.6

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

4. Applied rewrites 96.6%

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

5. Taylor expanded in M around 0

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

7. Applied rewrites 96.1%

   \[\leadsto 1 \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
8. Add Preprocessing

Alternative 2: 95.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -940000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 240:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25 \cdot \left(n + m\right), n + m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -940000000000.0)
     t_0
     (if (<= M 240.0)
       (exp (- (fabs (- n m)) (fma (* 0.25 (+ n m)) (+ n m) l)))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -940000000000.0) {
		tmp = t_0;
	} else if (M <= 240.0) {
		tmp = exp((fabs((n - m)) - fma((0.25 * (n + m)), (n + m), l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -940000000000.0)
		tmp = t_0;
	elseif (M <= 240.0)
		tmp = exp(Float64(abs(Float64(n - m)) - fma(Float64(0.25 * Float64(n + m)), Float64(n + m), l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -940000000000.0], t$95$0, If[LessEqual[M, 240.0], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[(0.25 * N[(n + m), $MachinePrecision]), $MachinePrecision] * N[(n + m), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -9.4e11 or 240 < M

   1. Initial program 76.5%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

   7. Applied rewrites 96.1%

      \[\leadsto 1 \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   8. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 53.3

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

   10. Applied rewrites 53.3%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 53.3

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

       4. lift-*.f64 N/A

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

       5. mul-1-neg N/A

          \[\leadsto e^{\mathsf{neg}\left({M}^{2}\right)} \cdot 1 \]

       6. lift-pow.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left({M}^{2}\right)} \cdot 1 \]

       7. unpow2 N/A

          \[\leadsto e^{\mathsf{neg}\left(M \cdot M\right)} \cdot 1 \]

       8. distribute-lft-neg-in N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(M\right)\right) \cdot M} \cdot 1 \]

       9. lift-neg.f64 N/A

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

       10. lower-*.f64 53.3

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

   12. Applied rewrites 53.3%

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

   if -9.4e11 < M < 240

   1. Initial program 76.5%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 87.1

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

   7. Applied rewrites 87.1%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower--.f64 87.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   9. Applied rewrites 87.1%

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

Alternative 3: 95.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -940000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 240:\\ \;\;\;\;e^{\left|n\right| - \mathsf{fma}\left(0.25 \cdot \left(n + m\right), n + m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -940000000000.0)
     t_0
     (if (<= M 240.0)
       (exp (- (fabs n) (fma (* 0.25 (+ n m)) (+ n m) l)))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -940000000000.0) {
		tmp = t_0;
	} else if (M <= 240.0) {
		tmp = exp((fabs(n) - fma((0.25 * (n + m)), (n + m), l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -940000000000.0)
		tmp = t_0;
	elseif (M <= 240.0)
		tmp = exp(Float64(abs(n) - fma(Float64(0.25 * Float64(n + m)), Float64(n + m), l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -940000000000.0], t$95$0, If[LessEqual[M, 240.0], N[Exp[N[(N[Abs[n], $MachinePrecision] - N[(N[(0.25 * N[(n + m), $MachinePrecision]), $MachinePrecision] * N[(n + m), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -9.4e11 or 240 < M

   1. Initial program 76.5%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

   7. Applied rewrites 96.1%

      \[\leadsto 1 \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   8. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 53.3

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

   10. Applied rewrites 53.3%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 53.3

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

       4. lift-*.f64 N/A

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

       5. mul-1-neg N/A

          \[\leadsto e^{\mathsf{neg}\left({M}^{2}\right)} \cdot 1 \]

       6. lift-pow.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left({M}^{2}\right)} \cdot 1 \]

       7. unpow2 N/A

          \[\leadsto e^{\mathsf{neg}\left(M \cdot M\right)} \cdot 1 \]

       8. distribute-lft-neg-in N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(M\right)\right) \cdot M} \cdot 1 \]

       9. lift-neg.f64 N/A

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

       10. lower-*.f64 53.3

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

   12. Applied rewrites 53.3%

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

   if -9.4e11 < M < 240

   1. Initial program 76.5%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 87.1

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

   7. Applied rewrites 87.1%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower--.f64 87.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   9. Applied rewrites 87.1%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 87.0%

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

Alternative 4: 82.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ t_1 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -49000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq 10^{-112}:\\ \;\;\;\;e^{t\_0 - \mathsf{fma}\left(0.25 \cdot n, n + m, \ell\right)}\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;e^{t\_0 - \mathsf{fma}\left(0.25 \cdot m, n + m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))) (t_1 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -49000000000.0)
     t_1
     (if (<= M 1e-112)
       (exp (- t_0 (fma (* 0.25 n) (+ n m) l)))
       (if (<= M 26.5) (exp (- t_0 (fma (* 0.25 m) (+ n m) l))) t_1)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double t_1 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -49000000000.0) {
		tmp = t_1;
	} else if (M <= 1e-112) {
		tmp = exp((t_0 - fma((0.25 * n), (n + m), l)));
	} else if (M <= 26.5) {
		tmp = exp((t_0 - fma((0.25 * m), (n + m), l)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	t_1 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -49000000000.0)
		tmp = t_1;
	elseif (M <= 1e-112)
		tmp = exp(Float64(t_0 - fma(Float64(0.25 * n), Float64(n + m), l)));
	elseif (M <= 26.5)
		tmp = exp(Float64(t_0 - fma(Float64(0.25 * m), Float64(n + m), l)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -49000000000.0], t$95$1, If[LessEqual[M, 1e-112], N[Exp[N[(t$95$0 - N[(N[(0.25 * n), $MachinePrecision] * N[(n + m), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[M, 26.5], N[Exp[N[(t$95$0 - N[(N[(0.25 * m), $MachinePrecision] * N[(n + m), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -4.9e10 or 26.5 < M

   1. Initial program 76.5%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

   7. Applied rewrites 96.1%

      \[\leadsto 1 \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   8. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 53.3

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

   10. Applied rewrites 53.3%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 53.3

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

       4. lift-*.f64 N/A

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

       5. mul-1-neg N/A

          \[\leadsto e^{\mathsf{neg}\left({M}^{2}\right)} \cdot 1 \]

       6. lift-pow.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left({M}^{2}\right)} \cdot 1 \]

       7. unpow2 N/A

          \[\leadsto e^{\mathsf{neg}\left(M \cdot M\right)} \cdot 1 \]

       8. distribute-lft-neg-in N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(M\right)\right) \cdot M} \cdot 1 \]

       9. lift-neg.f64 N/A

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

       10. lower-*.f64 53.3

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

   12. Applied rewrites 53.3%

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

   if -4.9e10 < M < 9.9999999999999995e-113

   1. Initial program 76.5%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 87.1

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

   7. Applied rewrites 87.1%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower--.f64 87.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   9. Applied rewrites 87.1%

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

   10. Taylor expanded in m around 0

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

       1. lower-*.f64 61.1

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

   12. Applied rewrites 61.1%

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

   if 9.9999999999999995e-113 < M < 26.5

   1. Initial program 76.5%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 87.1

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

   7. Applied rewrites 87.1%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower--.f64 87.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   9. Applied rewrites 87.1%

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

   10. Taylor expanded in m around inf

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

       1. lower-*.f64 61.3

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

   12. Applied rewrites 61.3%

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

Alternative 5: 82.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -6.9 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25 \cdot m, n + m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -6.9e-6)
     t_0
     (if (<= M 26.5)
       (exp (- (fabs (- n m)) (fma (* 0.25 m) (+ n m) l)))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -6.9e-6) {
		tmp = t_0;
	} else if (M <= 26.5) {
		tmp = exp((fabs((n - m)) - fma((0.25 * m), (n + m), l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -6.9e-6)
		tmp = t_0;
	elseif (M <= 26.5)
		tmp = exp(Float64(abs(Float64(n - m)) - fma(Float64(0.25 * m), Float64(n + m), l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -6.9e-6], t$95$0, If[LessEqual[M, 26.5], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[(0.25 * m), $MachinePrecision] * N[(n + m), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -6.9e-6 or 26.5 < M

   1. Initial program 76.5%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

   7. Applied rewrites 96.1%

      \[\leadsto 1 \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   8. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 53.3

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

   10. Applied rewrites 53.3%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 53.3

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

       4. lift-*.f64 N/A

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

       5. mul-1-neg N/A

          \[\leadsto e^{\mathsf{neg}\left({M}^{2}\right)} \cdot 1 \]

       6. lift-pow.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left({M}^{2}\right)} \cdot 1 \]

       7. unpow2 N/A

          \[\leadsto e^{\mathsf{neg}\left(M \cdot M\right)} \cdot 1 \]

       8. distribute-lft-neg-in N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(M\right)\right) \cdot M} \cdot 1 \]

       9. lift-neg.f64 N/A

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

       10. lower-*.f64 53.3

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

   12. Applied rewrites 53.3%

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

   if -6.9e-6 < M < 26.5

   1. Initial program 76.5%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 87.1

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

   7. Applied rewrites 87.1%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower--.f64 87.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   9. Applied rewrites 87.1%

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

   10. Taylor expanded in m around inf

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

       1. lower-*.f64 61.3

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

   12. Applied rewrites 61.3%

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

Alternative 6: 69.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -6.9 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;1 \cdot e^{-1 \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -6.9e-6) t_0 (if (<= M 26.5) (* 1.0 (exp (* -1.0 l))) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -6.9e-6) {
		tmp = t_0;
	} else if (M <= 26.5) {
		tmp = 1.0 * exp((-1.0 * 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)) * 1.0d0
    if (m_1 <= (-6.9d-6)) then
        tmp = t_0
    else if (m_1 <= 26.5d0) then
        tmp = 1.0d0 * exp(((-1.0d0) * 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)) * 1.0;
	double tmp;
	if (M <= -6.9e-6) {
		tmp = t_0;
	} else if (M <= 26.5) {
		tmp = 1.0 * Math.exp((-1.0 * l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-M * M)) * 1.0
	tmp = 0
	if M <= -6.9e-6:
		tmp = t_0
	elif M <= 26.5:
		tmp = 1.0 * math.exp((-1.0 * l))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -6.9e-6)
		tmp = t_0;
	elseif (M <= 26.5)
		tmp = Float64(1.0 * exp(Float64(-1.0 * l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-M * M)) * 1.0;
	tmp = 0.0;
	if (M <= -6.9e-6)
		tmp = t_0;
	elseif (M <= 26.5)
		tmp = 1.0 * exp((-1.0 * 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] * 1.0), $MachinePrecision]}, If[LessEqual[M, -6.9e-6], t$95$0, If[LessEqual[M, 26.5], N[(1.0 * N[Exp[N[(-1.0 * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -6.9e-6 or 26.5 < M

   1. Initial program 76.5%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

   7. Applied rewrites 96.1%

      \[\leadsto 1 \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   8. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 53.3

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

   10. Applied rewrites 53.3%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 53.3

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

       4. lift-*.f64 N/A

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

       5. mul-1-neg N/A

          \[\leadsto e^{\mathsf{neg}\left({M}^{2}\right)} \cdot 1 \]

       6. lift-pow.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left({M}^{2}\right)} \cdot 1 \]

       7. unpow2 N/A

          \[\leadsto e^{\mathsf{neg}\left(M \cdot M\right)} \cdot 1 \]

       8. distribute-lft-neg-in N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(M\right)\right) \cdot M} \cdot 1 \]

       9. lift-neg.f64 N/A

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

       10. lower-*.f64 53.3

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

   12. Applied rewrites 53.3%

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

   if -6.9e-6 < M < 26.5

   1. Initial program 76.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 30.7

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

   4. Applied rewrites 30.7%

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

   5. Taylor expanded in M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 30.3

         \[\leadsto \cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right)\right) \cdot e^{-1 \cdot \ell} \]

   7. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right)\right)} \cdot e^{-1 \cdot \ell} \]

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 35.7%

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

Alternative 7: 35.7% accurate, 5.4× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 * exp((-1.0 * 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 = 1.0d0 * exp(((-1.0d0) * l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

2. Taylor expanded in l around inf

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

   1. lower-*.f64 30.7

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

4. Applied rewrites 30.7%

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

5. Taylor expanded in M around 0

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

   1. lower-cos.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 30.3

      \[\leadsto \cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right)\right) \cdot e^{-1 \cdot \ell} \]

7. Applied rewrites 30.3%

   \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right)\right)} \cdot e^{-1 \cdot \ell} \]

8. Taylor expanded in K around 0

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

10. Applied rewrites 35.7%

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

Reproduce

herbie shell --seed 2025142 
(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)))))))