Maksimov and Kolovsky, Equation (32)

Average Accuracy: 71.8% → 92.0%

Time: 27.5s

Precision: binary64

Cost: 39424

Math

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

↓

\[\frac{\cos M}{e^{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)\right) + \left(\ell - \left|n - m\right|\right)}} \]

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
   (+ (log1p (expm1 (pow (- (* (+ m n) 0.5) M) 2.0))) (- l (fabs (- n 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((log1p(expm1(pow((((m + n) * 0.5) - M), 2.0))) + (l - fabs((n - m)))));
}

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.log1p(Math.expm1(Math.pow((((m + n) * 0.5) - M), 2.0))) + (l - Math.abs((n - 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.log1p(math.expm1(math.pow((((m + n) * 0.5) - M), 2.0))) + (l - math.fabs((n - 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(log1p(expm1((Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0))) + Float64(l - abs(Float64(n - 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[Log[1 + N[(Exp[N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] + N[(l - N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

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

2. Simplified 72.9%

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

   [Start] 72.9	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   sub-neg [=>] 72.9	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]
   associate--r+ [=>] 72.9	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]
   exp-diff [=>] 21.0	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]
   associate-*r/ [=>] 21.0	\[ \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]
   associate-/l* [=>] 21.0	\[ \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]
   associate-*r/ [<=] 21.0	\[ \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]
   exp-diff [=>] 19.0	\[ \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

3. Taylor expanded in K around 0 91.6%

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

4. Simplified 91.6%

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

   [Start] 91.6	\[ \frac{\cos \left(-M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}} \]
   cos-neg [=>] 91.6	\[ \frac{\color{blue}{\cos M}}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}} \]

5. Applied egg-rr 91.6%

   \[\leadsto \frac{\cos M}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)\right)} + \left(\ell - \left|n - m\right|\right)}} \]
   Step-by-step derivation

   [Start] 91.6	\[ \frac{\cos M}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}} \]
   log1p-expm1-u [=>] 91.6	\[ \frac{\cos M}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} + \left(\ell - \left|n - m\right|\right)}} \]
   div-inv [=>] 91.6	\[ \frac{\cos M}{e^{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}\right)\right) + \left(\ell - \left|n - m\right|\right)}} \]
   metadata-eval [=>] 91.6	\[ \frac{\cos M}{e^{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2}\right)\right) + \left(\ell - \left|n - m\right|\right)}} \]

6. Final simplification 91.6%

   \[\leadsto \frac{\cos M}{e^{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)\right) + \left(\ell - \left|n - m\right|\right)}} \]

Alternatives

Alternative 1
Accuracy	92.0%
Cost	26624

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

Alternative 2
Accuracy	65.5%
Cost	20428

\[\begin{array}{l} t_0 := \ell - \left|n - m\right|\\ t_1 := \frac{\cos M}{e^{t_0 + \left(m \cdot m\right) \cdot 0.25}}\\ \mathbf{if}\;n \leq 5.6 \cdot 10^{-179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-92}:\\ \;\;\;\;\frac{\cos M}{e^{t_0 + M \cdot M}}\\ \mathbf{elif}\;n \leq 140:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{t_0 + 0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]

Alternative 3
Accuracy	55.2%
Cost	20168

\[\begin{array}{l} t_0 := \ell - \left|n - m\right|\\ \mathbf{if}\;m \leq -1 \cdot 10^{+43}:\\ \;\;\;\;\frac{\cos M}{e^{t_0 + \left(m \cdot m\right) \cdot 0.25}}\\ \mathbf{elif}\;m \leq 4.5 \cdot 10^{-131}:\\ \;\;\;\;\frac{\cos M}{e^{t_0 + M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]

Alternative 4
Accuracy	58.5%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;\ell \leq 340:\\ \;\;\;\;\frac{\cos \left(\frac{m + n}{2} \cdot K - M\right)}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]

Alternative 5
Accuracy	30.3%
Cost	12992

\[\frac{\cos M}{e^{\ell}} \]

Alternative 6
Accuracy	7.2%
Cost	6464

\[\cos M \]

Reproduce

herbie shell --seed 2023157 
(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)))))))