Maksimov and Kolovsky, Equation (32)

Average Error: 15.2 → 1.2

Time: 10.0s

Precision: binary64

Math

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

↓

\[\cos M \cdot {\left({\left(e^{\left(\left|n - m\right| - {\left(\left(n + m\right) \cdot 0.5 - M\right)}^{2}\right) - \ell}\right)}^{3}\right)}^{0.3333333333333333} \]

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)))))))

↓

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos M)
  (pow
   (pow (exp (- (- (fabs (- n m)) (pow (- (* (+ n m) 0.5) M) 2.0)) l)) 3.0)
   0.3333333333333333)))

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)))));
}

↓

double code(double K, double m, double n, double M, double l) {
	return cos(M) * pow(pow(exp(((fabs((n - m)) - pow((((n + m) * 0.5) - M), 2.0)) - l)), 3.0), 0.3333333333333333);
}

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

↓

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((n - m)) - ((((n + m) * 0.5d0) - m_1) ** 2.0d0)) - l)) ** 3.0d0) ** 0.3333333333333333d0)
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)))));
}

↓

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.pow(Math.pow(Math.exp(((Math.abs((n - m)) - Math.pow((((n + m) * 0.5) - M), 2.0)) - l)), 3.0), 0.3333333333333333);
}

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)))))

↓

def code(K, m, n, M, l):
	return math.cos(M) * math.pow(math.pow(math.exp(((math.fabs((n - m)) - math.pow((((n + m) * 0.5) - M), 2.0)) - l)), 3.0), 0.3333333333333333)

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

↓

function code(K, m, n, M, l)
	return Float64(cos(M) * ((exp(Float64(Float64(abs(Float64(n - m)) - (Float64(Float64(Float64(n + m) * 0.5) - M) ^ 2.0)) - l)) ^ 3.0) ^ 0.3333333333333333))
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

↓

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * ((exp(((abs((n - m)) - ((((n + m) * 0.5) - M) ^ 2.0)) - l)) ^ 3.0) ^ 0.3333333333333333);
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]

↓

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Power[N[Power[N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[Power[N[(N[(N[(n + m), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.2

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

2. Simplified 15.2

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

3. Taylor expanded in K around 0 1.2

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

4. Simplified 1.2

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

5. Applied egg-rr 1.2

   \[\leadsto \cos M \cdot \color{blue}{{\left({\left(e^{\left(\left|n - m\right| - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right) - \ell}\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Final simplification 1.2

   \[\leadsto \cos M \cdot {\left({\left(e^{\left(\left|n - m\right| - {\left(\left(n + m\right) \cdot 0.5 - M\right)}^{2}\right) - \ell}\right)}^{3}\right)}^{0.3333333333333333} \]

Reproduce

herbie shell --seed 2022207 
(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)))))))