Maksimov and Kolovsky, Equation (32)

Average Error: 15.3 → 1.3

Time: 11.0s

Precision: binary64

Cost: 20096

Math

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

↓

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

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
 (exp (- (- (fabs (- n m)) (pow (- (* 0.5 (+ n m)) M) 2.0)) l)))

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

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

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

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

Derivation

1. Initial program 15.3

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

2. Taylor expanded in n around inf 9.5

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

3. Simplified 9.5

   \[\leadsto \cos \color{blue}{\left(K \cdot \left(n \cdot 0.5\right)\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   Proof
   (*.f64 K (*.f64 n 1/2)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 K n) 1/2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 n K)) 1/2): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 1/2 (*.f64 n K))): 0 points increase in error, 0 points decrease in error

4. Taylor expanded in K around 0 1.3

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

5. Simplified 1.3

   \[\leadsto \color{blue}{e^{\left(\left|n - m\right| - {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}\right) - \ell}} \]
   Proof
   (exp.f64 (-.f64 (-.f64 (fabs.f64 (-.f64 n m)) (pow.f64 (-.f64 (*.f64 1/2 (+.f64 n m)) M) 2)) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (-.f64 (Rewrite<= fabs-sub_binary64 (fabs.f64 (-.f64 m n))) (pow.f64 (-.f64 (*.f64 1/2 (+.f64 n m)) M) 2)) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (-.f64 (fabs.f64 (-.f64 m n)) (Rewrite=> unpow2_binary64 (*.f64 (-.f64 (*.f64 1/2 (+.f64 n m)) M) (-.f64 (*.f64 1/2 (+.f64 n m)) M)))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (fabs.f64 (-.f64 m n)) (*.f64 (neg.f64 (-.f64 (*.f64 1/2 (+.f64 n m)) M)) (-.f64 (*.f64 1/2 (+.f64 n m)) M)))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (Rewrite=> fabs-sub_binary64 (fabs.f64 (-.f64 n m))) (*.f64 (neg.f64 (-.f64 (*.f64 1/2 (+.f64 n m)) M)) (-.f64 (*.f64 1/2 (+.f64 n m)) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (Rewrite<= unsub-neg_binary64 (+.f64 n (neg.f64 m)))) (*.f64 (neg.f64 (-.f64 (*.f64 1/2 (+.f64 n m)) M)) (-.f64 (*.f64 1/2 (+.f64 n m)) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (+.f64 n (Rewrite<= mul-1-neg_binary64 (*.f64 -1 m)))) (*.f64 (neg.f64 (-.f64 (*.f64 1/2 (+.f64 n m)) M)) (-.f64 (*.f64 1/2 (+.f64 n m)) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (Rewrite<= fabs-neg_binary64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m))))) (*.f64 (neg.f64 (-.f64 (*.f64 1/2 (+.f64 n m)) M)) (-.f64 (*.f64 1/2 (+.f64 n m)) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (*.f64 (neg.f64 (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 n m) 1/2)) M)) (-.f64 (*.f64 1/2 (+.f64 n m)) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (*.f64 (neg.f64 (-.f64 (*.f64 (+.f64 n (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 m)))) 1/2) M)) (-.f64 (*.f64 1/2 (+.f64 n m)) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (*.f64 (neg.f64 (-.f64 (*.f64 (+.f64 n (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 m)))) 1/2) M)) (-.f64 (*.f64 1/2 (+.f64 n m)) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (*.f64 (neg.f64 (-.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 n (*.f64 -1 m))) 1/2) M)) (-.f64 (*.f64 1/2 (+.f64 n m)) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (*.f64 (neg.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (-.f64 n (*.f64 -1 m)))) M)) (-.f64 (*.f64 1/2 (+.f64 n m)) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (*.f64 (neg.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M)) (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 n m) 1/2)) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (*.f64 (neg.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M)) (-.f64 (*.f64 (+.f64 n (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 m)))) 1/2) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (*.f64 (neg.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M)) (-.f64 (*.f64 (+.f64 n (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 m)))) 1/2) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (*.f64 (neg.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M)) (-.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 n (*.f64 -1 m))) 1/2) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (+.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (*.f64 (neg.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M)) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (-.f64 n (*.f64 -1 m)))) M))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (*.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M) (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M)))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (-.f64 (-.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M) 2))) l)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (Rewrite<= associate--r+_binary64 (-.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m)))) (+.f64 (pow.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M) 2) l)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> exp-diff_binary64 (/.f64 (exp.f64 (fabs.f64 (neg.f64 (+.f64 n (*.f64 -1 m))))) (exp.f64 (+.f64 (pow.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M) 2) l)))): 200 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (Rewrite=> fabs-neg_binary64 (fabs.f64 (+.f64 n (*.f64 -1 m))))) (exp.f64 (+.f64 (pow.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M) 2) l))): 0 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (fabs.f64 (+.f64 n (Rewrite=> mul-1-neg_binary64 (neg.f64 m))))) (exp.f64 (+.f64 (pow.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M) 2) l))): 0 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (fabs.f64 (Rewrite=> unsub-neg_binary64 (-.f64 n m)))) (exp.f64 (+.f64 (pow.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M) 2) l))): 0 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (Rewrite<= fabs-sub_binary64 (fabs.f64 (-.f64 m n)))) (exp.f64 (+.f64 (pow.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M) 2) l))): 0 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (fabs.f64 (-.f64 m n))) (exp.f64 (Rewrite=> +-commutative_binary64 (+.f64 l (pow.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (fabs.f64 (-.f64 m n))) (Rewrite=> exp-sum_binary64 (*.f64 (exp.f64 l) (exp.f64 (pow.f64 (-.f64 (*.f64 1/2 (-.f64 n (*.f64 -1 m))) M) 2))))): 5 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (fabs.f64 (-.f64 m n))) (*.f64 (exp.f64 l) (exp.f64 (pow.f64 (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 n (*.f64 -1 m)) 1/2)) M) 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (fabs.f64 (-.f64 m n))) (*.f64 (exp.f64 l) (exp.f64 (pow.f64 (-.f64 (*.f64 (Rewrite=> sub-neg_binary64 (+.f64 n (neg.f64 (*.f64 -1 m)))) 1/2) M) 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (fabs.f64 (-.f64 m n))) (*.f64 (exp.f64 l) (exp.f64 (pow.f64 (-.f64 (*.f64 (+.f64 n (neg.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 m)))) 1/2) M) 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (fabs.f64 (-.f64 m n))) (*.f64 (exp.f64 l) (exp.f64 (pow.f64 (-.f64 (*.f64 (+.f64 n (Rewrite=> remove-double-neg_binary64 m)) 1/2) M) 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (fabs.f64 (-.f64 m n))) (*.f64 (exp.f64 l) (exp.f64 (pow.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (+.f64 n m))) M) 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (exp.f64 (fabs.f64 (-.f64 m n))) (Rewrite<= exp-sum_binary64 (exp.f64 (+.f64 l (pow.f64 (-.f64 (*.f64 1/2 (+.f64 n m)) M) 2))))): 0 points increase in error, 5 points decrease in error
   (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 (fabs.f64 (-.f64 m n)) (+.f64 l (pow.f64 (-.f64 (*.f64 1/2 (+.f64 n m)) M) 2))))): 0 points increase in error, 200 points decrease in error

6. Final simplification 1.3

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

Alternatives

Alternative 1
Error	10.5
Cost	20232

\[\begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -0.0004583516628081433:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 0.41504941807140094:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - {\left(m \cdot 0.5 - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	11.1
Cost	13640

\[\begin{array}{l} t_0 := e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{if}\;m \leq -424.7482284755198:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.1758322959806462 \cdot 10^{-26}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.5
Cost	7052

\[\begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;m \leq -0.18669673730072953:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq -4.409450604229991 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq -1.0292247521690727 \cdot 10^{-193}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	19.3
Cost	6788

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.7278069601340592 \cdot 10^{-10}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 5
Error	43.6
Cost	6528

\[e^{-\ell} \]

Reproduce

herbie shell --seed 2022298 
(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)))))))