Maksimov and Kolovsky, Equation (32)

Average Error: 14.8 → 1.3

Time: 24.3s

Precision: binary64

Cost: 26624

Math

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

↓

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

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 (- (- (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((((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(((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((((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((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((((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.abs((m - n)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
}

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.fabs((m - n)) - l) - math.pow((((m + n) / 2.0) - M), 2.0)))

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(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((((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((m - n)) - l) - ((((m + n) / 2.0) - M) ^ 2.0)));
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[(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 14.8

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

2. Simplified 14.8

   \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   Proof
   (*.f64 (cos.f64 (-.f64 (*.f64 (+.f64 m n) (/.f64 K 2)) M)) (exp.f64 (-.f64 (-.f64 (fabs.f64 (-.f64 m n)) l) (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 (-.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 m n) K) 2)) M)) (exp.f64 (-.f64 (-.f64 (fabs.f64 (-.f64 m n)) l) (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 K (+.f64 m n))) 2) M)) (exp.f64 (-.f64 (-.f64 (fabs.f64 (-.f64 m n)) l) (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (Rewrite=> associate--l-_binary64 (-.f64 (fabs.f64 (-.f64 m n)) (+.f64 l (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (fabs.f64 (-.f64 m n)) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2) l))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (Rewrite=> associate--r+_binary64 (-.f64 (-.f64 (fabs.f64 (-.f64 m n)) (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) l)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (fabs.f64 (-.f64 m n)) (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)))) l))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (Rewrite<= associate-+r-_binary64 (+.f64 (fabs.f64 (-.f64 m n)) (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) l))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) l) (fabs.f64 (-.f64 m n)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (Rewrite<= associate--r-_binary64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) (-.f64 l (fabs.f64 (-.f64 m n))))))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in K around 0 1.3

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

4. Simplified 1.3

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   Proof
   (cos.f64 M): 0 points increase in error, 0 points decrease in error
   (Rewrite<= cos-neg_binary64 (cos.f64 (neg.f64 M))): 0 points increase in error, 0 points decrease in error

5. Final simplification 1.3

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

Alternatives

Alternative 1
Error	17.4
Cost	20428

\[\begin{array}{l} t_0 := \left|m - n\right| - \ell\\ t_1 := \cos M \cdot e^{t_0 + n \cdot \left(n \cdot -0.25\right)}\\ \mathbf{if}\;m \leq -580000:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq -2.95 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;m \leq 8.4 \cdot 10^{-308}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	18.5
Cost	20168

\[\begin{array}{l} \mathbf{if}\;n \leq -4.6 \cdot 10^{-222}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 3
Error	12.4
Cost	13776

\[\begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ t_1 := \cos M \cdot e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{if}\;m \leq -580000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;m \leq -2.9 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq -1.85 \cdot 10^{-193}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;m \leq 3.5 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	20.3
Cost	13776

\[\begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;m \leq -580000:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq -4.5 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq -2 \cdot 10^{-193}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;m \leq 6 \cdot 10^{-300}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 5
Error	14.4
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\ell \leq -380:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 6
Error	19.0
Cost	6788

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 7
Error	43.4
Cost	6528

\[e^{-\ell} \]

Alternative 8
Error	59.4
Cost	6464

\[\cos M \]

Alternative 9
Error	59.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022325 
(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)))))))