Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.3% → 96.2%

Time: 13.1s

Alternatives: 12

Speedup: 2.8×

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.3% 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

real(8) function code(k, m, n, m_1, l)
    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: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ t_1 := t\_0 - \ell\\ t_2 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)\\ t_3 := \left(m + n\right) \cdot \left(m + n\right)\\ \mathbf{if}\;e^{t\_1 - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot t\_2 \leq 0:\\ \;\;\;\;t\_2 \cdot e^{t\_1 - M \cdot \mathsf{fma}\left(M, \frac{n + \mathsf{fma}\left(-0.25, \frac{t\_3}{M}, m\right)}{-M}, M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - \mathsf{fma}\left(0.25, t\_3, \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n)))
        (t_1 (- t_0 l))
        (t_2 (cos (- (/ (* K (+ m n)) 2.0) M)))
        (t_3 (* (+ m n) (+ m n))))
   (if (<= (* (exp (- t_1 (pow (- (/ (+ m n) 2.0) M) 2.0))) t_2) 0.0)
     (*
      t_2
      (exp (- t_1 (* M (fma M (/ (+ n (fma -0.25 (/ t_3 M) m)) (- M)) M)))))
     (exp (- t_0 (fma 0.25 t_3 l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double t_1 = t_0 - l;
	double t_2 = cos((((K * (m + n)) / 2.0) - M));
	double t_3 = (m + n) * (m + n);
	double tmp;
	if ((exp((t_1 - pow((((m + n) / 2.0) - M), 2.0))) * t_2) <= 0.0) {
		tmp = t_2 * exp((t_1 - (M * fma(M, ((n + fma(-0.25, (t_3 / M), m)) / -M), M))));
	} else {
		tmp = exp((t_0 - fma(0.25, t_3, l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	t_1 = Float64(t_0 - l)
	t_2 = cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M))
	t_3 = Float64(Float64(m + n) * Float64(m + n))
	tmp = 0.0
	if (Float64(exp(Float64(t_1 - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0))) * t_2) <= 0.0)
		tmp = Float64(t_2 * exp(Float64(t_1 - Float64(M * fma(M, Float64(Float64(n + fma(-0.25, Float64(t_3 / M), m)) / Float64(-M)), M)))));
	else
		tmp = exp(Float64(t_0 - fma(0.25, t_3, l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - l), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[N[(t$95$1 - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], 0.0], N[(t$95$2 * N[Exp[N[(t$95$1 - N[(M * N[(M * N[(N[(n + N[(-0.25 * N[(t$95$3 / M), $MachinePrecision] + m), $MachinePrecision]), $MachinePrecision] / (-M)), $MachinePrecision] + M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$0 - N[(0.25 * t$95$3 + l), $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)))))) < 0.0

   1. Initial program 98.6%

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

   3. Taylor expanded in M around -inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 98.6%

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

   if 0.0 < (*.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 21.8%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 96.3

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

   5. Simplified 96.3%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 95.2

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

   8. Simplified 95.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \leq 0:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot \mathsf{fma}\left(M, \frac{n + \mathsf{fma}\left(-0.25, \frac{\left(m + n\right) \cdot \left(m + n\right)}{M}, m\right)}{-M}, M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

3. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. lower-cos.f64 96.6

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

5. Simplified 96.6%

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

6. Final simplification 96.6%

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

Reproduce

herbie shell --seed 2024218 
(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)))))))