Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.4% → 96.1%

Time: 4.4s

Alternatives: 8

Speedup: 3.0×

Specification

Math

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

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: 75.4% accurate, 1.0× speedup

Math

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

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

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

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 75.4%

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

2. 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 96.8%

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in M around 0

   \[\leadsto 1 \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
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: 95.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := 1 \cdot e^{-1 \cdot {M}^{2}}\\ \mathbf{if}\;M \leq -2800000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 4 \cdot 10^{+42}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25 \cdot \left(n + m\right), n + m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 1.0 (exp (* -1.0 (pow M 2.0))))))
   (if (<= M -2800000000000.0)
     t_0
     (if (<= M 4e+42)
       (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 = 1.0 * exp((-1.0 * pow(M, 2.0)));
	double tmp;
	if (M <= -2800000000000.0) {
		tmp = t_0;
	} else if (M <= 4e+42) {
		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(1.0 * exp(Float64(-1.0 * (M ^ 2.0))))
	tmp = 0.0
	if (M <= -2800000000000.0)
		tmp = t_0;
	elseif (M <= 4e+42)
		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[(1.0 * N[Exp[N[(-1.0 * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -2800000000000.0], t$95$0, If[LessEqual[M, 4e+42], 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 < -2.8e12 or 4.0000000000000002e42 < M

   1. Initial program 75.4%

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

   2. 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 96.8%

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   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 54.0%

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

   10. Applied rewrites 54.0%

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

   if -2.8e12 < M < 4.0000000000000002e42

   1. Initial program 75.4%

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

   2. 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 96.8%

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

   4. Applied rewrites 96.8%

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

   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 86.7%

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

   7. Applied rewrites 86.7%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   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 86.7%

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

      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

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

      13. lower-*.f64 86.7%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 86.7%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 86.7%

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

   9. Applied rewrites 86.7%

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

Alternative 3: 86.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(m, n\right) \leq 54:\\ \;\;\;\;e^{\left|\mathsf{max}\left(m, n\right) - \mathsf{min}\left(m, n\right)\right| - \mathsf{fma}\left(0.25 \cdot \mathsf{min}\left(m, n\right), \mathsf{max}\left(m, n\right) + \mathsf{min}\left(m, n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {\left(\mathsf{max}\left(m, n\right)\right)}^{2}}\\ \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= (fmax m n) 54.0)
   (exp
    (-
     (fabs (- (fmax m n) (fmin m n)))
     (fma (* 0.25 (fmin m n)) (+ (fmax m n) (fmin m n)) l)))
   (exp (* -0.25 (pow (fmax m n) 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (fmax(m, n) <= 54.0) {
		tmp = exp((fabs((fmax(m, n) - fmin(m, n))) - fma((0.25 * fmin(m, n)), (fmax(m, n) + fmin(m, n)), l)));
	} else {
		tmp = exp((-0.25 * pow(fmax(m, n), 2.0)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (fmax(m, n) <= 54.0)
		tmp = exp(Float64(abs(Float64(fmax(m, n) - fmin(m, n))) - fma(Float64(0.25 * fmin(m, n)), Float64(fmax(m, n) + fmin(m, n)), l)));
	else
		tmp = exp(Float64(-0.25 * (fmax(m, n) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[N[Max[m, n], $MachinePrecision], 54.0], N[Exp[N[(N[Abs[N[(N[Max[m, n], $MachinePrecision] - N[Min[m, n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[(0.25 * N[Min[m, n], $MachinePrecision]), $MachinePrecision] * N[(N[Max[m, n], $MachinePrecision] + N[Min[m, n], $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.25 * N[Power[N[Max[m, n], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 54

   1. Initial program 75.4%

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

   2. 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 96.8%

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

   4. Applied rewrites 96.8%

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

   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 86.7%

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

   7. Applied rewrites 86.7%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   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 86.7%

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

      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

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

      13. lower-*.f64 86.7%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 86.7%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 86.7%

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

   9. Applied rewrites 86.7%

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

   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 61.1%

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

   12. Applied rewrites 61.1%

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

   if 54 < n

   1. Initial program 75.4%

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

   2. 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 96.8%

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

   4. Applied rewrites 96.8%

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

   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 86.7%

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

   7. Applied rewrites 86.7%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   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 86.7%

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

      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

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

      13. lower-*.f64 86.7%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 86.7%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 86.7%

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

   9. Applied rewrites 86.7%

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

   10. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 53.8%

          \[\leadsto e^{-0.25 \cdot {n}^{2}} \]

   12. Applied rewrites 53.8%

       \[\leadsto e^{-0.25 \cdot {n}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.7% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (fabs (- n m)) (fma (* 0.25 (+ n m)) (+ n m) l))))

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - fma((0.25 * (n + m)), (n + m), l)));
}

Julia

function code(K, m, n, M, l)
	return exp(Float64(abs(Float64(n - m)) - fma(Float64(0.25 * Float64(n + m)), Float64(n + m), l)))
end

Wolfram

code[K_, m_, n_, M_, l_] := 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]

Derivation

1. Initial program 75.4%

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

2. 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 96.8%

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

4. Applied rewrites 96.8%

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

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 86.7%

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

7. Applied rewrites 86.7%

   \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
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 86.7%

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

   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

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

   13. lower-*.f64 86.7%

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 86.7%

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 86.7%

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

9. Applied rewrites 86.7%

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

Alternative 5: 86.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(m, n\right) \leq -55:\\ \;\;\;\;e^{-0.25 \cdot {\left(\mathsf{min}\left(m, n\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|\mathsf{max}\left(m, n\right) - \mathsf{min}\left(m, n\right)\right| - \mathsf{fma}\left(0.25 \cdot \mathsf{max}\left(m, n\right), \mathsf{max}\left(m, n\right) + \mathsf{min}\left(m, n\right), \ell\right)}\\ \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= (fmin m n) -55.0)
   (exp (* -0.25 (pow (fmin m n) 2.0)))
   (exp
    (-
     (fabs (- (fmax m n) (fmin m n)))
     (fma (* 0.25 (fmax m n)) (+ (fmax m n) (fmin m n)) l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (fmin(m, n) <= -55.0) {
		tmp = exp((-0.25 * pow(fmin(m, n), 2.0)));
	} else {
		tmp = exp((fabs((fmax(m, n) - fmin(m, n))) - fma((0.25 * fmax(m, n)), (fmax(m, n) + fmin(m, n)), l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (fmin(m, n) <= -55.0)
		tmp = exp(Float64(-0.25 * (fmin(m, n) ^ 2.0)));
	else
		tmp = exp(Float64(abs(Float64(fmax(m, n) - fmin(m, n))) - fma(Float64(0.25 * fmax(m, n)), Float64(fmax(m, n) + fmin(m, n)), l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[N[Min[m, n], $MachinePrecision], -55.0], N[Exp[N[(-0.25 * N[Power[N[Min[m, n], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Abs[N[(N[Max[m, n], $MachinePrecision] - N[Min[m, n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[(0.25 * N[Max[m, n], $MachinePrecision]), $MachinePrecision] * N[(N[Max[m, n], $MachinePrecision] + N[Min[m, n], $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -55

   1. Initial program 75.4%

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

   2. 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 96.8%

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

   4. Applied rewrites 96.8%

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

   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 86.7%

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

   7. Applied rewrites 86.7%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   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 86.7%

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

      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

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

      13. lower-*.f64 86.7%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 86.7%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 86.7%

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

   9. Applied rewrites 86.7%

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

   10. Taylor expanded in m around inf

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 54.4%

          \[\leadsto e^{-0.25 \cdot {m}^{2}} \]

   12. Applied rewrites 54.4%

       \[\leadsto e^{-0.25 \cdot {m}^{2}} \]

   if -55 < m

   1. Initial program 75.4%

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

   2. 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 96.8%

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

   4. Applied rewrites 96.8%

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

   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 86.7%

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

   7. Applied rewrites 86.7%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   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 86.7%

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

      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

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

      13. lower-*.f64 86.7%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 86.7%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 86.7%

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

   9. Applied rewrites 86.7%

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

   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 60.2%

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

   12. Applied rewrites 60.2%

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

Alternative 6: 79.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(m, n\right) + \mathsf{min}\left(m, n\right)\\ t_1 := \left|\mathsf{max}\left(m, n\right) - \mathsf{min}\left(m, n\right)\right|\\ \mathbf{if}\;\mathsf{max}\left(m, n\right) \leq 2.2 \cdot 10^{+68}:\\ \;\;\;\;e^{t\_1 - \mathsf{fma}\left(0.25 \cdot \mathsf{min}\left(m, n\right), t\_0, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_1 - \mathsf{fma}\left(0.25 \cdot \mathsf{max}\left(m, n\right), t\_0, \ell\right)}\\ \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (+ (fmax m n) (fmin m n))) (t_1 (fabs (- (fmax m n) (fmin m n)))))
   (if (<= (fmax m n) 2.2e+68)
     (exp (- t_1 (fma (* 0.25 (fmin m n)) t_0 l)))
     (exp (- t_1 (fma (* 0.25 (fmax m n)) t_0 l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fmax(m, n) + fmin(m, n);
	double t_1 = fabs((fmax(m, n) - fmin(m, n)));
	double tmp;
	if (fmax(m, n) <= 2.2e+68) {
		tmp = exp((t_1 - fma((0.25 * fmin(m, n)), t_0, l)));
	} else {
		tmp = exp((t_1 - fma((0.25 * fmax(m, n)), t_0, l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(fmax(m, n) + fmin(m, n))
	t_1 = abs(Float64(fmax(m, n) - fmin(m, n)))
	tmp = 0.0
	if (fmax(m, n) <= 2.2e+68)
		tmp = exp(Float64(t_1 - fma(Float64(0.25 * fmin(m, n)), t_0, l)));
	else
		tmp = exp(Float64(t_1 - fma(Float64(0.25 * fmax(m, n)), t_0, l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Max[m, n], $MachinePrecision] + N[Min[m, n], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(N[Max[m, n], $MachinePrecision] - N[Min[m, n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[m, n], $MachinePrecision], 2.2e+68], N[Exp[N[(t$95$1 - N[(N[(0.25 * N[Min[m, n], $MachinePrecision]), $MachinePrecision] * t$95$0 + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$1 - N[(N[(0.25 * N[Max[m, n], $MachinePrecision]), $MachinePrecision] * t$95$0 + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if n < 2.1999999999999999e68

   1. Initial program 75.4%

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

   2. 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 96.8%

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

   4. Applied rewrites 96.8%

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

   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 86.7%

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

   7. Applied rewrites 86.7%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   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 86.7%

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

      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

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

      13. lower-*.f64 86.7%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 86.7%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 86.7%

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

   9. Applied rewrites 86.7%

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

   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 61.1%

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

   12. Applied rewrites 61.1%

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

   if 2.1999999999999999e68 < n

   1. Initial program 75.4%

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

   2. 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 96.8%

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

   4. Applied rewrites 96.8%

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

   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 86.7%

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

   7. Applied rewrites 86.7%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   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 86.7%

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

      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

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

      13. lower-*.f64 86.7%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 86.7%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 86.7%

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

   9. Applied rewrites 86.7%

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

   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 60.2%

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

   12. Applied rewrites 60.2%

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

Alternative 7: 65.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq 740:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25 \cdot m, n + m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell} \cdot 1\\ \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 740.0)
   (exp (- (fabs (- n m)) (fma (* 0.25 m) (+ n m) l)))
   (* (exp (- l)) 1.0)))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 740.0) {
		tmp = exp((fabs((n - m)) - fma((0.25 * m), (n + m), l)));
	} else {
		tmp = exp(-l) * 1.0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 740.0)
		tmp = exp(Float64(abs(Float64(n - m)) - fma(Float64(0.25 * m), Float64(n + m), l)));
	else
		tmp = Float64(exp(Float64(-l)) * 1.0);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 740.0], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[(0.25 * m), $MachinePrecision] * N[(n + m), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Exp[(-l)], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 740

   1. Initial program 75.4%

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

   2. 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 96.8%

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

   4. Applied rewrites 96.8%

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

   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 86.7%

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

   7. Applied rewrites 86.7%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
   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 86.7%

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

      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

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

      13. lower-*.f64 86.7%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 86.7%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 86.7%

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

   9. Applied rewrites 86.7%

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

   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 61.1%

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

   12. Applied rewrites 61.1%

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

   if 740 < l

   1. Initial program 75.4%

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

   2. 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 30.0%

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

   4. Applied rewrites 30.0%

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

   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 29.5%

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 34.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 34.9%

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

       4. lift-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 34.9%

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

   12. Applied rewrites 34.9%

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

Alternative 8: 34.9% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

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 = exp(-l) * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

2. 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 30.0%

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

4. Applied rewrites 30.0%

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

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 29.5%

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

7. Applied rewrites 29.5%

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

8. Taylor expanded in K around 0

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

10. Applied rewrites 34.9%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 34.9%

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

    4. lift-*.f64 N/A

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

    5. mul-1-neg N/A

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

    6. lower-neg.f64 34.9%

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

12. Applied rewrites 34.9%

    \[\leadsto \color{blue}{e^{-\ell} \cdot 1} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025187 
(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)))))))