Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.0% → 96.8%

Time: 5.2s

Alternatives: 8

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(n + m\right) \cdot K}{2} - M\\ t_1 := \left|n - m\right|\\ t_2 := e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - t\_1\right)}\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t\_2 \leq 0.499:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(-t\_0\right), \cos \left(\frac{\pi}{2}\right), \cos t\_0 \cdot \sin \left(\frac{\pi}{2}\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t\_1 - \left({\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (/ (* (+ n m) K) 2.0) M))
        (t_1 (fabs (- n m)))
        (t_2 (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l t_1)))))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_2) 0.499)
     (*
      (fma (sin (- t_0)) (cos (/ PI 2.0)) (* (cos t_0) (sin (/ PI 2.0))))
      t_2)
     (* (cos M) (exp (- t_1 (+ (pow (- (* 0.5 (+ n m)) M) 2.0) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (((n + m) * K) / 2.0) - M;
	double t_1 = fabs((n - m));
	double t_2 = exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)));
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_2) <= 0.499) {
		tmp = fma(sin(-t_0), cos((((double) M_PI) / 2.0)), (cos(t_0) * sin((((double) M_PI) / 2.0)))) * t_2;
	} else {
		tmp = cos(M) * exp((t_1 - (pow(((0.5 * (n + m)) - M), 2.0) + l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(Float64(Float64(n + m) * K) / 2.0) - M)
	t_1 = abs(Float64(n - m))
	t_2 = exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - t_1)))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_2) <= 0.499)
		tmp = Float64(fma(sin(Float64(-t_0)), cos(Float64(pi / 2.0)), Float64(cos(t_0) * sin(Float64(pi / 2.0)))) * t_2);
	else
		tmp = Float64(cos(M) * exp(Float64(t_1 - Float64((Float64(Float64(0.5 * Float64(n + m)) - M) ^ 2.0) + l))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], 0.499], N[(N[(N[Sin[(-t$95$0)], $MachinePrecision] * N[Cos[N[(Pi / 2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[N[(Pi / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$1 - N[(N[Power[N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < 0.499

   1. Initial program 97.4%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. sin-+PI/2-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 96.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sin-sum N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 97.4%

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

   if 0.499 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 24.5%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\cos M \cdot e^{\left|n - m\right| - \left({\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} + \ell\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \leq 0.499:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(-\left(\frac{\left(n + m\right) \cdot K}{2} - M\right)\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot \sin \left(\frac{\pi}{2}\right)\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left({\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ t_1 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - t\_0\right)}\\ \mathbf{if}\;t\_1 \leq 0.499:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t\_0 - \left({\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m)))
        (t_1
         (*
          (cos (- (/ (* K (+ m n)) 2.0) M))
          (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l t_0))))))
   (if (<= t_1 0.499)
     t_1
     (* (cos M) (exp (- t_0 (+ (pow (- (* 0.5 (+ n m)) M) 2.0) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double t_1 = cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - t_0)));
	double tmp;
	if (t_1 <= 0.499) {
		tmp = t_1;
	} else {
		tmp = cos(M) * exp((t_0 - (pow(((0.5 * (n + m)) - M), 2.0) + l)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((n - m))
    t_1 = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - t_0)))
    if (t_1 <= 0.499d0) then
        tmp = t_1
    else
        tmp = cos(m_1) * exp((t_0 - ((((0.5d0 * (n + m)) - m_1) ** 2.0d0) + l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double t_1 = Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - t_0)));
	double tmp;
	if (t_1 <= 0.499) {
		tmp = t_1;
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - (Math.pow(((0.5 * (n + m)) - M), 2.0) + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	t_1 = math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - t_0)))
	tmp = 0
	if t_1 <= 0.499:
		tmp = t_1
	else:
		tmp = math.cos(M) * math.exp((t_0 - (math.pow(((0.5 * (n + m)) - M), 2.0) + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	t_1 = 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 - t_0))))
	tmp = 0.0
	if (t_1 <= 0.499)
		tmp = t_1;
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64((Float64(Float64(0.5 * Float64(n + m)) - M) ^ 2.0) + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	t_1 = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - t_0)));
	tmp = 0.0;
	if (t_1 <= 0.499)
		tmp = t_1;
	else
		tmp = cos(M) * exp((t_0 - ((((0.5 * (n + m)) - M) ^ 2.0) + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.499], t$95$1, N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[Power[N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < 0.499

   1. Initial program 97.4%

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

   if 0.499 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 24.5%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\cos M \cdot e^{\left|n - m\right| - \left({\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} + \ell\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \leq 0.499:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left({\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)\\ t_1 := \left|n - m\right|\\ \mathbf{if}\;t\_0 \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - t\_1\right)} \leq -0.5:\\ \;\;\;\;t\_0 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t\_1 - \left({\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cos (- (/ (* K (+ m n)) 2.0) M))) (t_1 (fabs (- n m))))
   (if (<=
        (* t_0 (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l t_1))))
        -0.5)
     (* t_0 (exp (- l)))
     (* (cos M) (exp (- t_1 (+ (pow (- (* 0.5 (+ n m)) M) 2.0) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos((((K * (m + n)) / 2.0) - M));
	double t_1 = fabs((n - m));
	double tmp;
	if ((t_0 * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)))) <= -0.5) {
		tmp = t_0 * exp(-l);
	} else {
		tmp = cos(M) * exp((t_1 - (pow(((0.5 * (n + m)) - M), 2.0) + l)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((((k * (m + n)) / 2.0d0) - m_1))
    t_1 = abs((n - m))
    if ((t_0 * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - t_1)))) <= (-0.5d0)) then
        tmp = t_0 * exp(-l)
    else
        tmp = cos(m_1) * exp((t_1 - ((((0.5d0 * (n + m)) - m_1) ** 2.0d0) + l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos((((K * (m + n)) / 2.0) - M));
	double t_1 = Math.abs((n - m));
	double tmp;
	if ((t_0 * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)))) <= -0.5) {
		tmp = t_0 * Math.exp(-l);
	} else {
		tmp = Math.cos(M) * Math.exp((t_1 - (Math.pow(((0.5 * (n + m)) - M), 2.0) + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos((((K * (m + n)) / 2.0) - M))
	t_1 = math.fabs((n - m))
	tmp = 0
	if (t_0 * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)))) <= -0.5:
		tmp = t_0 * math.exp(-l)
	else:
		tmp = math.cos(M) * math.exp((t_1 - (math.pow(((0.5 * (n + m)) - M), 2.0) + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M))
	t_1 = abs(Float64(n - m))
	tmp = 0.0
	if (Float64(t_0 * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - t_1)))) <= -0.5)
		tmp = Float64(t_0 * exp(Float64(-l)));
	else
		tmp = Float64(cos(M) * exp(Float64(t_1 - Float64((Float64(Float64(0.5 * Float64(n + m)) - M) ^ 2.0) + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos((((K * (m + n)) / 2.0) - M));
	t_1 = abs((n - m));
	tmp = 0.0;
	if ((t_0 * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - t_1)))) <= -0.5)
		tmp = t_0 * exp(-l);
	else
		tmp = cos(M) * exp((t_1 - ((((0.5 * (n + m)) - M) ^ 2.0) + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], N[(t$95$0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$1 - N[(N[Power[N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < -0.5

   1. Initial program 70.7%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.3

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

   5. Applied rewrites 70.3%

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

   if -0.5 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 78.6%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{\cos M \cdot e^{\left|n - m\right| - \left({\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} + \ell\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \leq -0.5:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left({\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ t_1 := \cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{if}\;M \leq -1.05 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq -3.3 \cdot 10^{-139}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 1.3 \cdot 10^{-177}:\\ \;\;\;\;\left(-0.5 \cdot \left(M \cdot M\right)\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (* -0.25 (* n n)))))
        (t_1 (* (cos M) (exp (* (- M) M)))))
   (if (<= M -1.05e+18)
     t_1
     (if (<= M -3.3e-139)
       t_0
       (if (<= M 1.3e-177)
         (* (* -0.5 (* M M)) (exp (* (* m m) -0.25)))
         (if (<= M 27.0) t_0 t_1))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp((-0.25 * (n * n)));
	double t_1 = cos(M) * exp((-M * M));
	double tmp;
	if (M <= -1.05e+18) {
		tmp = t_1;
	} else if (M <= -3.3e-139) {
		tmp = t_0;
	} else if (M <= 1.3e-177) {
		tmp = (-0.5 * (M * M)) * exp(((m * m) * -0.25));
	} else if (M <= 27.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(m_1) * exp(((-0.25d0) * (n * n)))
    t_1 = cos(m_1) * exp((-m_1 * m_1))
    if (m_1 <= (-1.05d+18)) then
        tmp = t_1
    else if (m_1 <= (-3.3d-139)) then
        tmp = t_0
    else if (m_1 <= 1.3d-177) then
        tmp = ((-0.5d0) * (m_1 * m_1)) * exp(((m * m) * (-0.25d0)))
    else if (m_1 <= 27.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	double t_1 = Math.cos(M) * Math.exp((-M * M));
	double tmp;
	if (M <= -1.05e+18) {
		tmp = t_1;
	} else if (M <= -3.3e-139) {
		tmp = t_0;
	} else if (M <= 1.3e-177) {
		tmp = (-0.5 * (M * M)) * Math.exp(((m * m) * -0.25));
	} else if (M <= 27.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp((-0.25 * (n * n)))
	t_1 = math.cos(M) * math.exp((-M * M))
	tmp = 0
	if M <= -1.05e+18:
		tmp = t_1
	elif M <= -3.3e-139:
		tmp = t_0
	elif M <= 1.3e-177:
		tmp = (-0.5 * (M * M)) * math.exp(((m * m) * -0.25))
	elif M <= 27.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))))
	t_1 = Float64(cos(M) * exp(Float64(Float64(-M) * M)))
	tmp = 0.0
	if (M <= -1.05e+18)
		tmp = t_1;
	elseif (M <= -3.3e-139)
		tmp = t_0;
	elseif (M <= 1.3e-177)
		tmp = Float64(Float64(-0.5 * Float64(M * M)) * exp(Float64(Float64(m * m) * -0.25)));
	elseif (M <= 27.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp((-0.25 * (n * n)));
	t_1 = cos(M) * exp((-M * M));
	tmp = 0.0;
	if (M <= -1.05e+18)
		tmp = t_1;
	elseif (M <= -3.3e-139)
		tmp = t_0;
	elseif (M <= 1.3e-177)
		tmp = (-0.5 * (M * M)) * exp(((m * m) * -0.25));
	elseif (M <= 27.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -1.05e+18], t$95$1, If[LessEqual[M, -3.3e-139], t$95$0, If[LessEqual[M, 1.3e-177], N[(N[(-0.5 * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 27.0], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -1.05e18 or 27 < M

   1. Initial program 83.0%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.4

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

   8. Applied rewrites 96.4%

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

   if -1.05e18 < M < -3.3e-139 or 1.3e-177 < M < 27

   1. Initial program 70.4%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 50.0

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

   8. Applied rewrites 50.0%

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

   if -3.3e-139 < M < 1.3e-177

   1. Initial program 77.1%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. lift-cos.f64 60.0

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

   8. Applied rewrites 60.0%

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

   9. Taylor expanded in M around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 60.0

          \[\leadsto \left(1 + -0.5 \cdot \left(M \cdot M\right)\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   11. Applied rewrites 60.0%

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

   12. Taylor expanded in M around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 81.3

          \[\leadsto \left(-0.5 \cdot \left(M \cdot M\right)\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   14. Applied rewrites 81.3%

       \[\leadsto \left(-0.5 \cdot \left(M \cdot \color{blue}{M}\right)\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.05 \cdot 10^{+18}:\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{elif}\;M \leq -3.3 \cdot 10^{-139}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;M \leq 1.3 \cdot 10^{-177}:\\ \;\;\;\;\left(-0.5 \cdot \left(M \cdot M\right)\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(M \cdot M\right)\\ t_1 := e^{\left(m \cdot m\right) \cdot -0.25}\\ t_2 := \cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{if}\;M \leq -27:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;M \leq -2.5 \cdot 10^{-111}:\\ \;\;\;\;\left(1 + t\_0\right) \cdot t\_1\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* -0.5 (* M M)))
        (t_1 (exp (* (* m m) -0.25)))
        (t_2 (* (cos M) (exp (* (- M) M)))))
   (if (<= M -27.0)
     t_2
     (if (<= M -2.5e-111)
       (* (+ 1.0 t_0) t_1)
       (if (<= M 27.0) (* t_0 t_1) t_2)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = -0.5 * (M * M);
	double t_1 = exp(((m * m) * -0.25));
	double t_2 = cos(M) * exp((-M * M));
	double tmp;
	if (M <= -27.0) {
		tmp = t_2;
	} else if (M <= -2.5e-111) {
		tmp = (1.0 + t_0) * t_1;
	} else if (M <= 27.0) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (-0.5d0) * (m_1 * m_1)
    t_1 = exp(((m * m) * (-0.25d0)))
    t_2 = cos(m_1) * exp((-m_1 * m_1))
    if (m_1 <= (-27.0d0)) then
        tmp = t_2
    else if (m_1 <= (-2.5d-111)) then
        tmp = (1.0d0 + t_0) * t_1
    else if (m_1 <= 27.0d0) then
        tmp = t_0 * t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = -0.5 * (M * M);
	double t_1 = Math.exp(((m * m) * -0.25));
	double t_2 = Math.cos(M) * Math.exp((-M * M));
	double tmp;
	if (M <= -27.0) {
		tmp = t_2;
	} else if (M <= -2.5e-111) {
		tmp = (1.0 + t_0) * t_1;
	} else if (M <= 27.0) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = -0.5 * (M * M)
	t_1 = math.exp(((m * m) * -0.25))
	t_2 = math.cos(M) * math.exp((-M * M))
	tmp = 0
	if M <= -27.0:
		tmp = t_2
	elif M <= -2.5e-111:
		tmp = (1.0 + t_0) * t_1
	elif M <= 27.0:
		tmp = t_0 * t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(-0.5 * Float64(M * M))
	t_1 = exp(Float64(Float64(m * m) * -0.25))
	t_2 = Float64(cos(M) * exp(Float64(Float64(-M) * M)))
	tmp = 0.0
	if (M <= -27.0)
		tmp = t_2;
	elseif (M <= -2.5e-111)
		tmp = Float64(Float64(1.0 + t_0) * t_1);
	elseif (M <= 27.0)
		tmp = Float64(t_0 * t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = -0.5 * (M * M);
	t_1 = exp(((m * m) * -0.25));
	t_2 = cos(M) * exp((-M * M));
	tmp = 0.0;
	if (M <= -27.0)
		tmp = t_2;
	elseif (M <= -2.5e-111)
		tmp = (1.0 + t_0) * t_1;
	elseif (M <= 27.0)
		tmp = t_0 * t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(-0.5 * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -27.0], t$95$2, If[LessEqual[M, -2.5e-111], N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[M, 27.0], N[(t$95$0 * t$95$1), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -27 or 27 < M

   1. Initial program 82.5%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.4

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

   8. Applied rewrites 96.4%

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

   if -27 < M < -2.5000000000000001e-111

   1. Initial program 76.3%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 48.5

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

   5. Applied rewrites 48.5%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. lift-cos.f64 63.5

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

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in M around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 63.5

          \[\leadsto \left(1 + -0.5 \cdot \left(M \cdot M\right)\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   11. Applied rewrites 63.5%

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

   if -2.5000000000000001e-111 < M < 27

   1. Initial program 73.0%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. lift-cos.f64 66.0

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

   8. Applied rewrites 66.0%

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

   9. Taylor expanded in M around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 66.0

          \[\leadsto \left(1 + -0.5 \cdot \left(M \cdot M\right)\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   11. Applied rewrites 66.0%

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

   12. Taylor expanded in M around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 73.9

          \[\leadsto \left(-0.5 \cdot \left(M \cdot M\right)\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   14. Applied rewrites 73.9%

       \[\leadsto \left(-0.5 \cdot \left(M \cdot \color{blue}{M}\right)\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -27:\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \mathbf{elif}\;M \leq -2.5 \cdot 10^{-111}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(M \cdot M\right)\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;\left(-0.5 \cdot \left(M \cdot M\right)\right) \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(-M\right) \cdot M}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{-12} \lor \neg \left(m \leq 53\right):\\ \;\;\;\;1 \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -2e-12) (not (<= m 53.0)))
   (* 1.0 (exp (* (* m m) -0.25)))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -2e-12) || !(m <= 53.0)) {
		tmp = 1.0 * exp(((m * m) * -0.25));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m <= (-2d-12)) .or. (.not. (m <= 53.0d0))) then
        tmp = 1.0d0 * exp(((m * m) * (-0.25d0)))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -2e-12) || !(m <= 53.0)) {
		tmp = 1.0 * Math.exp(((m * m) * -0.25));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -2e-12) or not (m <= 53.0):
		tmp = 1.0 * math.exp(((m * m) * -0.25))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -2e-12) || !(m <= 53.0))
		tmp = Float64(1.0 * exp(Float64(Float64(m * m) * -0.25)));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -2e-12) || ~((m <= 53.0)))
		tmp = 1.0 * exp(((m * m) * -0.25));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -2e-12], N[Not[LessEqual[m, 53.0]], $MachinePrecision]], N[(1.0 * N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.99999999999999996e-12 or 53 < m

   1. Initial program 74.6%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. lift-cos.f64 96.2

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

   8. Applied rewrites 96.2%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 96.2%

       \[\leadsto 1 \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   if -1.99999999999999996e-12 < m < 53

   1. Initial program 82.2%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. lift-cos.f64 38.8

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

   8. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{-12} \lor \neg \left(m \leq 53\right):\\ \;\;\;\;1 \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3600 \lor \neg \left(m \leq 2.45 \cdot 10^{-6}\right):\\ \;\;\;\;1 \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -3600.0) (not (<= m 2.45e-6)))
   (* 1.0 (exp (* (* m m) -0.25)))
   (* 1.0 (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -3600.0) || !(m <= 2.45e-6)) {
		tmp = 1.0 * exp(((m * m) * -0.25));
	} else {
		tmp = 1.0 * exp(-l);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m <= (-3600.0d0)) .or. (.not. (m <= 2.45d-6))) then
        tmp = 1.0d0 * exp(((m * m) * (-0.25d0)))
    else
        tmp = 1.0d0 * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -3600.0) || !(m <= 2.45e-6)) {
		tmp = 1.0 * Math.exp(((m * m) * -0.25));
	} else {
		tmp = 1.0 * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -3600.0) or not (m <= 2.45e-6):
		tmp = 1.0 * math.exp(((m * m) * -0.25))
	else:
		tmp = 1.0 * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -3600.0) || !(m <= 2.45e-6))
		tmp = Float64(1.0 * exp(Float64(Float64(m * m) * -0.25)));
	else
		tmp = Float64(1.0 * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -3600.0) || ~((m <= 2.45e-6)))
		tmp = 1.0 * exp(((m * m) * -0.25));
	else
		tmp = 1.0 * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -3600.0], N[Not[LessEqual[m, 2.45e-6]], $MachinePrecision]], N[(1.0 * N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3600 or 2.44999999999999984e-6 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. lift-cos.f64 96.9

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

   8. Applied rewrites 96.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 96.9%

       \[\leadsto 1 \cdot e^{\left(m \cdot m\right) \cdot -0.25} \]

   if -3600 < m < 2.44999999999999984e-6

   1. Initial program 82.4%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.6

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg-rev N/A

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

      2. lift-cos.f64 38.9

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

   8. Applied rewrites 38.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 38.9%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3600 \lor \neg \left(m \leq 2.45 \cdot 10^{-6}\right):\\ \;\;\;\;1 \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(k, m, n, m_1, l)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0 * exp(-l)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in l around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 33.1

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

5. Applied rewrites 33.1%

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

6. Taylor expanded in K around 0

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

   1. cos-neg-rev N/A

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

   2. lift-cos.f64 34.9

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

8. Applied rewrites 34.9%

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

9. Taylor expanded in M around 0

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

11. Applied rewrites 34.9%

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

Reproduce

herbie shell --seed 2025084 
(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)))))))