Maksimov and Kolovsky, Equation (32)

Average Error: 15.8 → 1.4

Time: 16.9s

Precision: binary64

Cost: 20224

\[ \begin{array}{c}[m, n] = \mathsf{sort}([m, n])\\ \end{array} \]

Math

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

↓

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

FPCore

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

↓

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

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) / exp((pow(((0.5 * (n + m)) - M), 2.0) + ((n + l) - m)));
}

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(((((0.5d0 * (n + m)) - m_1) ** 2.0d0) + ((n + l) - m)))
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.exp((Math.pow(((0.5 * (n + m)) - M), 2.0) + ((n + l) - m)));
}

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.exp((math.pow(((0.5 * (n + m)) - M), 2.0) + ((n + l) - m)))

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(Float64(0.5 * Float64(n + m)) - M) ^ 2.0) + Float64(Float64(n + l) - m))))
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(((((0.5 * (n + m)) - M) ^ 2.0) + ((n + l) - m)));
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[Exp[N[(N[Power[N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(n + l), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.8

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

2. Simplified 15.7

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

   [Start] 15.8	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   *-commutative [=>] 15.8	\[ \cos \left(\frac{\color{blue}{\left(m + n\right) \cdot K}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   associate-*r/ [<=] 15.7	\[ \cos \left(\color{blue}{\left(m + n\right) \cdot \frac{K}{2}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   associate--r- [=>] 15.7	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]
   +-commutative [=>] 15.7	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left|m - n\right| + \left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right)}} \]
   sub-neg [=>] 15.7	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left|m - n\right| + \color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) + \left(-\ell\right)\right)}} \]
   distribute-neg-out [=>] 15.7	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left|m - n\right| + \color{blue}{\left(-\left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)\right)}} \]
   sub-neg [<=] 15.7	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)}} \]
   +-commutative [=>] 15.7	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left|m - n\right| - \color{blue}{\left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}} \]
   associate--l- [<=] 15.7	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]

3. Applied egg-rr 15.8

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \cdot e^{\left(\left(m - n\right) - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\right)} - 1} \]

4. Simplified 15.9

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

   [Start] 15.8	\[ e^{\mathsf{log1p}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \cdot e^{\left(\left(m - n\right) - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\right)} - 1 \]
   expm1-def [=>] 15.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \cdot e^{\left(\left(m - n\right) - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}\right)\right)} \]
   expm1-log1p [=>] 15.8	\[ \color{blue}{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \cdot e^{\left(\left(m - n\right) - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}} \]
   exp-diff [=>] 20.6	\[ \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \cdot \color{blue}{\frac{e^{\left(m - n\right) - \ell}}{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}}} \]
   associate-*r/ [=>] 20.6	\[ \color{blue}{\frac{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \cdot e^{\left(m - n\right) - \ell}}{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}}} \]
   associate-/l* [=>] 20.6	\[ \color{blue}{\frac{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}{\frac{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}}{e^{\left(m - n\right) - \ell}}}} \]
   +-commutative [=>] 20.6	\[ \frac{\cos \left(\color{blue}{\left(n + m\right)} \cdot \left(K \cdot 0.5\right) - M\right)}{\frac{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}}{e^{\left(m - n\right) - \ell}}} \]
   *-commutative [=>] 20.6	\[ \frac{\cos \left(\color{blue}{\left(K \cdot 0.5\right) \cdot \left(n + m\right)} - M\right)}{\frac{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}}{e^{\left(m - n\right) - \ell}}} \]
   *-commutative [=>] 20.6	\[ \frac{\cos \left(\color{blue}{\left(0.5 \cdot K\right)} \cdot \left(n + m\right) - M\right)}{\frac{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}}{e^{\left(m - n\right) - \ell}}} \]
   associate-*r* [<=] 20.6	\[ \frac{\cos \left(\color{blue}{0.5 \cdot \left(K \cdot \left(n + m\right)\right)} - M\right)}{\frac{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}}{e^{\left(m - n\right) - \ell}}} \]
   div-exp [=>] 15.9	\[ \frac{\cos \left(0.5 \cdot \left(K \cdot \left(n + m\right)\right) - M\right)}{\color{blue}{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \left(\left(m - n\right) - \ell\right)}}} \]

5. Taylor expanded in K around 0 1.4

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

6. Simplified 1.4

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

   [Start] 1.4	\[ \frac{\cos \left(-M\right)}{e^{{\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} - \left(m - \left(n + \ell\right)\right)}} \]
   cos-neg [=>] 1.4	\[ \frac{\color{blue}{\cos M}}{e^{{\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} - \left(m - \left(n + \ell\right)\right)}} \]

7. Final simplification 1.4

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

Alternatives

Alternative 1
Error	2.6
Cost	13896

\[\begin{array}{l} \mathbf{if}\;M \leq -1050000000:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{elif}\;M \leq 10^{-65}:\\ \;\;\;\;\frac{\cos M}{e^{m \cdot \left(m \cdot 0.25\right) + \left(\left(n + \ell\right) - m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) - \left(\ell + M \cdot M\right)}\\ \end{array} \]

Alternative 2
Error	2.6
Cost	13636

\[\begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{+120}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) - \left(\ell + M \cdot M\right)}\\ \end{array} \]

Alternative 3
Error	10.0
Cost	13384

\[\begin{array}{l} \mathbf{if}\;m \leq -4000:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{elif}\;m \leq -1.35 \cdot 10^{-211}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \]

Alternative 4
Error	11.9
Cost	13188

\[\begin{array}{l} \mathbf{if}\;m \leq -1.8 \cdot 10^{-9}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \]

Alternative 5
Error	21.8
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\ell \leq 115:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]

Alternative 6
Error	31.2
Cost	12992

\[\cos M \cdot e^{m} \]

Reproduce

herbie shell --seed 2023046 
(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)))))))