Maksimov and Kolovsky, Equation (32)

Average Error: 23.35% → 2.5%

Time: 16.2s

Precision: binary64

Cost: 26436

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\left(m - n\right) - \left(\ell + M \cdot M\right)}\right)\right)\\ \end{array} \]

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)))))))

↓

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -1e+101)
   (* (cos M) (exp l))
   (expm1 (log1p (* (cos M) (exp (- (- m n) (+ l (* M 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) {
	double tmp;
	if (l <= -1e+101) {
		tmp = cos(M) * exp(l);
	} else {
		tmp = expm1(log1p((cos(M) * exp(((m - n) - (l + (M * M)))))));
	}
	return tmp;
}

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) {
	double tmp;
	if (l <= -1e+101) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.expm1(Math.log1p((Math.cos(M) * Math.exp(((m - n) - (l + (M * M)))))));
	}
	return tmp;
}

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):
	tmp = 0
	if l <= -1e+101:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.expm1(math.log1p((math.cos(M) * math.exp(((m - n) - (l + (M * M)))))))
	return tmp

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)
	tmp = 0.0
	if (l <= -1e+101)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = expm1(log1p(Float64(cos(M) * exp(Float64(Float64(m - n) - Float64(l + Float64(M * M)))))));
	end
	return tmp
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_] := If[LessEqual[l, -1e+101], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(Exp[N[Log[1 + N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] - N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.9999999999999998e100

   1. Initial program 28.3

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

   2. Simplified 28.3

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

      [Start] 28.3	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
      *-commutative [=>] 28.3	\[ \cos \left(\frac{\color{blue}{\left(m + n\right) \cdot K}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
      associate-*r/ [<=] 28.3	\[ \cos \left(\color{blue}{\left(m + n\right) \cdot \frac{K}{2}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
      associate--r- [=>] 28.3	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]
      +-commutative [=>] 28.3	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left|m - n\right| + \left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right)}} \]
      sub-neg [=>] 28.3	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left|m - n\right| + \color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) + \left(-\ell\right)\right)}} \]
      distribute-neg-out [=>] 28.3	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left|m - n\right| + \color{blue}{\left(-\left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)\right)}} \]
      sub-neg [<=] 28.3	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)}} \]
      +-commutative [=>] 28.3	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left|m - n\right| - \color{blue}{\left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}} \]
      associate--l- [<=] 28.3	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]

   3. Taylor expanded in K around 0 0

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

   4. Simplified 0

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

      [Start] 0	\[ \cos \left(-M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
      cos-neg [=>] 0	\[ \color{blue}{\cos M} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   5. Taylor expanded in l around inf 100

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]

   6. Simplified 100

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]
      Proof

      [Start] 100	\[ \cos M \cdot e^{-1 \cdot \ell} \]
      mul-1-neg [=>] 100	\[ \cos M \cdot e^{\color{blue}{-\ell}} \]

   7. Applied egg-rr 0

      \[\leadsto \color{blue}{\left(1 + \cos M \cdot e^{\ell}\right) - 1} \]

   8. Simplified 0

      \[\leadsto \color{blue}{\cos M \cdot e^{\ell}} \]
      Proof

      [Start] 0	\[ \left(1 + \cos M \cdot e^{\ell}\right) - 1 \]
      +-commutative [=>] 0	\[ \color{blue}{\left(\cos M \cdot e^{\ell} + 1\right)} - 1 \]
      associate--l+ [=>] 0	\[ \color{blue}{\cos M \cdot e^{\ell} + \left(1 - 1\right)} \]
      metadata-eval [=>] 0	\[ \cos M \cdot e^{\ell} + \color{blue}{0} \]
      +-rgt-identity [=>] 0	\[ \color{blue}{\cos M \cdot e^{\ell}} \]

   if -9.9999999999999998e100 < l

   1. Initial program 22.65

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

   2. Simplified 22.65

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

      [Start] 22.65	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
      *-commutative [=>] 22.65	\[ \cos \left(\frac{\color{blue}{\left(m + n\right) \cdot K}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
      associate-*r/ [<=] 22.65	\[ \cos \left(\color{blue}{\left(m + n\right) \cdot \frac{K}{2}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
      associate--r- [=>] 22.65	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]
      +-commutative [=>] 22.65	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left|m - n\right| + \left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right)}} \]
      sub-neg [=>] 22.65	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left|m - n\right| + \color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) + \left(-\ell\right)\right)}} \]
      distribute-neg-out [=>] 22.65	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left|m - n\right| + \color{blue}{\left(-\left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)\right)}} \]
      sub-neg [<=] 22.65	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)}} \]
      +-commutative [=>] 22.65	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left|m - n\right| - \color{blue}{\left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}} \]
      associate--l- [<=] 22.65	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]

   3. Taylor expanded in M around inf 52.83

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

   4. Simplified 52.83

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

      [Start] 52.83	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {M}^{2}} \]
      unpow2 [=>] 52.83	\[ \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{M \cdot M}} \]

   5. Applied egg-rr 23.17

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \cdot e^{\left(m - n\right) - \left(\ell + M \cdot M\right)}\right)\right)} \]

   6. Taylor expanded in K around 0 2.85

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(-M\right)} \cdot e^{\left(m - n\right) - \left(\ell + M \cdot M\right)}\right)\right) \]

   7. Simplified 2.85

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos M} \cdot e^{\left(m - n\right) - \left(\ell + M \cdot M\right)}\right)\right) \]
      Proof

      [Start] 2.85	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(-M\right) \cdot e^{\left(m - n\right) - \left(\ell + M \cdot M\right)}\right)\right) \]
      cos-neg [=>] 2.85	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos M} \cdot e^{\left(m - n\right) - \left(\ell + M \cdot M\right)}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\left(m - n\right) - \left(\ell + M \cdot M\right)}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.99%
Cost	26624

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

Alternative 2
Error	3.88%
Cost	20100

\[\begin{array}{l} \mathbf{if}\;m \leq -8200000:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-n\right) - \mathsf{fma}\left(M, M, \ell\right)} \cdot \left(\cos M \cdot \left(m + 1\right)\right)\\ \end{array} \]

Alternative 3
Error	5.34%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;M \leq -5.6 \cdot 10^{+28} \lor \neg \left(M \leq 28\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{m - \left(n + \ell\right)}\\ \end{array} \]

Alternative 4
Error	9.02%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{m - \left(n + \ell\right)}\\ \end{array} \]

Alternative 5
Error	17.62%
Cost	6720

\[e^{m - \left(n + \ell\right)} \]

Alternative 6
Error	67.58%
Cost	6528

\[e^{-\ell} \]

Alternative 7
Error	92.5%
Cost	6464

\[\cos M \]

Reproduce

herbie shell --seed 2023121 
(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)))))))