Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.8% → 96.6%

Time: 5.2s

Alternatives: 7

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: 76.8% 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: 96.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := M - \left(n + m\right) \cdot 0.5\\ \cos \left(-M\right) \cdot \frac{1}{e^{\mathsf{fma}\left(t\_0, t\_0, \ell\right) - \left|n - m\right|}} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* (+ n m) 0.5))))
   (* (cos (- M)) (/ 1.0 (exp (- (fma t_0 t_0 l) (fabs (- n m))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - ((n + m) * 0.5);
	return cos(-M) * (1.0 / exp((fma(t_0, t_0, l) - fabs((n - m)))));
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M - Float64(Float64(n + m) * 0.5))
	return Float64(cos(Float64(-M)) * Float64(1.0 / exp(Float64(fma(t_0, t_0, l) - abs(Float64(n - m))))))
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(N[(n + m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[(-M)], $MachinePrecision] * N[(1.0 / N[Exp[N[(N[(t$95$0 * t$95$0 + l), $MachinePrecision] - N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.8%

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

2. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 96.6

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

4. Applied rewrites 96.6%

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

6. Applied rewrites 96.6%

   \[\leadsto \cos \left(-M\right) \cdot \frac{1}{\color{blue}{e^{\mathsf{fma}\left(M - \left(n + m\right) \cdot 0.5, M - \left(n + m\right) \cdot 0.5, \ell\right) - \left|n - m\right|}}} \]
7. Add Preprocessing

Alternative 2: 95.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, n + m, M\right)\\ 1 \cdot \frac{1}{\frac{1}{e^{\left|n - m\right| - \mathsf{fma}\left(t\_0, t\_0, \ell\right)}}} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma -0.5 (+ n m) M)))
   (* 1.0 (/ 1.0 (/ 1.0 (exp (- (fabs (- n m)) (fma t_0 t_0 l))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma(-0.5, (n + m), M);
	return 1.0 * (1.0 / (1.0 / exp((fabs((n - m)) - fma(t_0, t_0, l)))));
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(-0.5, Float64(n + m), M)
	return Float64(1.0 * Float64(1.0 / Float64(1.0 / exp(Float64(abs(Float64(n - m)) - fma(t_0, t_0, l))))))
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(-0.5 * N[(n + m), $MachinePrecision] + M), $MachinePrecision]}, N[(1.0 * N[(1.0 / N[(1.0 / N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(t$95$0 * t$95$0 + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.8%

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

2. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 96.6

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

4. Applied rewrites 96.6%

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

6. Applied rewrites 96.6%

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

7. Taylor expanded in M around 0

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

9. Applied rewrites 95.9%

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

11. Applied rewrites 95.9%

    \[\leadsto 1 \cdot \frac{1}{\frac{1}{\color{blue}{e^{\left|n - m\right| - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, n + m, M\right), \mathsf{fma}\left(-0.5, n + m, M\right), \ell\right)}}}} \]
12. Add Preprocessing

Alternative 3: 95.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, n + m, M\right)\\ e^{\left|n - m\right| - \mathsf{fma}\left(t\_0, t\_0, \ell\right)} \cdot 1 \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma -0.5 (+ n m) M)))
   (* (exp (- (fabs (- n m)) (fma t_0 t_0 l))) 1.0)))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma(-0.5, (n + m), M);
	return exp((fabs((n - m)) - fma(t_0, t_0, l))) * 1.0;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(-0.5, Float64(n + m), M)
	return Float64(exp(Float64(abs(Float64(n - m)) - fma(t_0, t_0, l))) * 1.0)
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(-0.5 * N[(n + m), $MachinePrecision] + M), $MachinePrecision]}, N[(N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(t$95$0 * t$95$0 + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Initial program 76.8%

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

2. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 96.6

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

4. Applied rewrites 96.6%

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

5. Taylor expanded in M around 0

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

7. Applied rewrites 95.9%

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

9. Applied rewrites 95.9%

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

Alternative 4: 95.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(m, n\right) + \mathsf{min}\left(m, n\right)\\ t_1 := M - 0.5 \cdot \mathsf{max}\left(m, n\right)\\ \mathbf{if}\;\mathsf{min}\left(m, n\right) \leq -1920000000:\\ \;\;\;\;e^{\left|\mathsf{min}\left(m, n\right) - \mathsf{max}\left(m, n\right)\right| - \mathsf{fma}\left(t\_0, t\_0 \cdot 0.25, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{1}{e^{\mathsf{fma}\left(t\_1, t\_1, \ell\right) - \left|\mathsf{max}\left(m, n\right) - \mathsf{min}\left(m, n\right)\right|}}\\ \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (+ (fmax m n) (fmin m n))) (t_1 (- M (* 0.5 (fmax m n)))))
   (if (<= (fmin m n) -1920000000.0)
     (exp (- (fabs (- (fmin m n) (fmax m n))) (fma t_0 (* t_0 0.25) l)))
     (*
      1.0
      (/ 1.0 (exp (- (fma t_1 t_1 l) (fabs (- (fmax m n) (fmin m n))))))))))

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 = M - (0.5 * fmax(m, n));
	double tmp;
	if (fmin(m, n) <= -1920000000.0) {
		tmp = exp((fabs((fmin(m, n) - fmax(m, n))) - fma(t_0, (t_0 * 0.25), l)));
	} else {
		tmp = 1.0 * (1.0 / exp((fma(t_1, t_1, l) - fabs((fmax(m, n) - fmin(m, n))))));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(fmax(m, n) + fmin(m, n))
	t_1 = Float64(M - Float64(0.5 * fmax(m, n)))
	tmp = 0.0
	if (fmin(m, n) <= -1920000000.0)
		tmp = exp(Float64(abs(Float64(fmin(m, n) - fmax(m, n))) - fma(t_0, Float64(t_0 * 0.25), l)));
	else
		tmp = Float64(1.0 * Float64(1.0 / exp(Float64(fma(t_1, t_1, l) - abs(Float64(fmax(m, n) - fmin(m, n)))))));
	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[(M - N[(0.5 * N[Max[m, n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[m, n], $MachinePrecision], -1920000000.0], N[Exp[N[(N[Abs[N[(N[Min[m, n], $MachinePrecision] - N[Max[m, n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(t$95$0 * N[(t$95$0 * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 * N[(1.0 / N[Exp[N[(N[(t$95$1 * t$95$1 + l), $MachinePrecision] - N[Abs[N[(N[Max[m, n], $MachinePrecision] - N[Min[m, n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.92e9

   1. Initial program 76.8%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 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-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 86.7

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

   9. Applied rewrites 86.7%

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

   if -1.92e9 < m

   1. Initial program 76.8%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   6. Applied rewrites 96.6%

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

   7. Taylor expanded in M around 0

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

   9. Applied rewrites 95.9%

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

   10. Taylor expanded in m around 0

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 77.5

          \[\leadsto 1 \cdot \frac{1}{e^{\mathsf{fma}\left(M - 0.5 \cdot n, M - \left(n + m\right) \cdot 0.5, \ell\right) - \left|n - m\right|}} \]

   12. Applied rewrites 77.5%

       \[\leadsto 1 \cdot \frac{1}{e^{\mathsf{fma}\left(M - 0.5 \cdot n, M - \left(n + m\right) \cdot 0.5, \ell\right) - \left|n - m\right|}} \]

   13. Taylor expanded in m around 0

       \[\leadsto 1 \cdot \frac{1}{e^{\mathsf{fma}\left(M - \frac{1}{2} \cdot n, M - \frac{1}{2} \cdot n, \ell\right) - \left|n - m\right|}} \]
   14. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. lower-*.f64 78.4

          \[\leadsto 1 \cdot \frac{1}{e^{\mathsf{fma}\left(M - 0.5 \cdot n, M - 0.5 \cdot n, \ell\right) - \left|n - m\right|}} \]

   15. Applied rewrites 78.4%

       \[\leadsto 1 \cdot \frac{1}{e^{\mathsf{fma}\left(M - 0.5 \cdot n, M - 0.5 \cdot n, \ell\right) - \left|n - m\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

2. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 96.6

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

4. Applied rewrites 96.6%

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

5. Taylor expanded in M around 0

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-+.f64 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-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 86.7

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

9. Applied rewrites 86.7%

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

Alternative 6: 77.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(m, n\right) \leq -5400:\\ \;\;\;\;e^{\left(\mathsf{min}\left(m, n\right) \cdot \mathsf{min}\left(m, n\right)\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {\left(\mathsf{max}\left(m, n\right)\right)}^{2}}\\ \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= (fmin m n) -5400.0)
   (exp (* (* (fmin m n) (fmin m n)) -0.25))
   (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 (fmin(m, n) <= -5400.0) {
		tmp = exp(((fmin(m, n) * fmin(m, n)) * -0.25));
	} else {
		tmp = exp((-0.25 * pow(fmax(m, n), 2.0)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (fmin(m, n) <= (-5400.0d0)) then
        tmp = exp(((fmin(m, n) * fmin(m, n)) * (-0.25d0)))
    else
        tmp = exp(((-0.25d0) * (fmax(m, n) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (fmin(m, n) <= -5400.0) {
		tmp = Math.exp(((fmin(m, n) * fmin(m, n)) * -0.25));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(fmax(m, n), 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if fmin(m, n) <= -5400.0:
		tmp = math.exp(((fmin(m, n) * fmin(m, n)) * -0.25))
	else:
		tmp = math.exp((-0.25 * math.pow(fmax(m, n), 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (fmin(m, n) <= -5400.0)
		tmp = exp(Float64(Float64(fmin(m, n) * fmin(m, n)) * -0.25));
	else
		tmp = exp(Float64(-0.25 * (fmax(m, n) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (min(m, n) <= -5400.0)
		tmp = exp(((min(m, n) * min(m, n)) * -0.25));
	else
		tmp = exp((-0.25 * (max(m, n) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[N[Min[m, n], $MachinePrecision], -5400.0], N[Exp[N[(N[(N[Min[m, n], $MachinePrecision] * N[Min[m, n], $MachinePrecision]), $MachinePrecision] * -0.25), $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 m < -5400

   1. Initial program 76.8%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 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. Taylor expanded in m around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 54.0

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

   10. Applied rewrites 54.0%

       \[\leadsto e^{-0.25 \cdot {m}^{2}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 54.0

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

       4. lift-pow.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 54.0

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

   12. Applied rewrites 54.0%

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

   if -5400 < m

   1. Initial program 76.8%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 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. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 53.7

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

   10. Applied rewrites 53.7%

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

Alternative 7: 54.0% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

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(((m * m) * (-0.25d0)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(((m * m) * -0.25));
}

Python

def code(K, m, n, M, l):
	return math.exp(((m * m) * -0.25))

Julia

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

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((m * m) * -0.25));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.8%

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

2. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 96.6

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

4. Applied rewrites 96.6%

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

5. Taylor expanded in M around 0

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-+.f64 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. Taylor expanded in m around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 54.0

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

10. Applied rewrites 54.0%

    \[\leadsto e^{-0.25 \cdot {m}^{2}} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 54.0

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

    4. lift-pow.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 54.0

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

12. Applied rewrites 54.0%

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

Reproduce

herbie shell --seed 2025175 
(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)))))))