Maksimov and Kolovsky, Equation (32)

Average Accuracy: 71.0% → 92.2%

Time: 17.4s

Precision: binary64

Cost: 26564

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq 740:\\ \;\;\;\;\frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, m + n, -M\right)\right)}^{2} + \left(m - n\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

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 740.0)
   (/ (cos M) (exp (+ (pow (fma 0.5 (+ m n) (- M)) 2.0) (- m n))))
   (exp (- 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) {
	double tmp;
	if (l <= 740.0) {
		tmp = cos(M) / exp((pow(fma(0.5, (m + n), -M), 2.0) + (m - n)));
	} else {
		tmp = exp(-l);
	}
	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 <= 740.0)
		tmp = Float64(cos(M) / exp(Float64((fma(0.5, Float64(m + n), Float64(-M)) ^ 2.0) + Float64(m - n))));
	else
		tmp = exp(Float64(-l));
	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, 740.0], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[Power[N[(0.5 * N[(m + n), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision] + N[(m - n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 740

   1. Initial program 67.9%

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

   2. Simplified 68.0%

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

      [Start] 67.9	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
      sub-neg [=>] 67.9	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]
      associate--r+ [=>] 67.9	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]
      exp-diff [=>] 16.9	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]
      associate-*r/ [=>] 16.9	\[ \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]
      associate-/l* [=>] 16.9	\[ \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]
      associate-*r/ [<=] 16.9	\[ \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]
      exp-diff [=>] 13.8	\[ \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

   3. Taylor expanded in K around 0 89.7%

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

   4. Simplified 89.7%

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

      [Start] 89.7	\[ \frac{\cos \left(-M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}} \]
      cos-neg [=>] 89.7	\[ \frac{\color{blue}{\cos M}}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}} \]

   5. Taylor expanded in l around 0 89.7%

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

   6. Simplified 89.7%

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

      [Start] 89.7	\[ \frac{\cos M}{e^{{\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} - \left|n - m\right|}} \]
      sub-neg [=>] 89.7	\[ \frac{\cos M}{e^{\color{blue}{{\left(0.5 \cdot \left(n + m\right) - M\right)}^{2} + \left(-\left|n - m\right|\right)}}} \]
      fma-neg [=>] 89.7	\[ \frac{\cos M}{e^{{\color{blue}{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}}^{2} + \left(-\left|n - m\right|\right)}} \]
      neg-sub0 [=>] 89.7	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \color{blue}{\left(0 - \left|n - m\right|\right)}}} \]
      unpow1 [<=] 89.7	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \left(0 - \left|\color{blue}{{\left(n - m\right)}^{1}}\right|\right)}} \]
      sqr-pow [=>] 45.0	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \left(0 - \left|\color{blue}{{\left(n - m\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(n - m\right)}^{\left(\frac{1}{2}\right)}}\right|\right)}} \]
      fabs-sqr [=>] 45.0	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \left(0 - \color{blue}{{\left(n - m\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(n - m\right)}^{\left(\frac{1}{2}\right)}}\right)}} \]
      sqr-pow [<=] 89.7	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \left(0 - \color{blue}{{\left(n - m\right)}^{1}}\right)}} \]
      unpow1 [=>] 89.7	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \left(0 - \color{blue}{\left(n - m\right)}\right)}} \]
      associate--r- [=>] 89.7	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \color{blue}{\left(\left(0 - n\right) + m\right)}}} \]
      neg-sub0 [<=] 89.7	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \left(\color{blue}{\left(-n\right)} + m\right)}} \]
      mul-1-neg [<=] 89.7	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \left(\color{blue}{-1 \cdot n} + m\right)}} \]
      +-commutative [=>] 89.7	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \color{blue}{\left(m + -1 \cdot n\right)}}} \]
      mul-1-neg [=>] 89.7	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \left(m + \color{blue}{\left(-n\right)}\right)}} \]
      sub-neg [<=] 89.7	\[ \frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \color{blue}{\left(m - n\right)}}} \]

   if 740 < l

   1. Initial program 80.7%

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

   2. Simplified 80.7%

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

      [Start] 80.7	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
      sub-neg [=>] 80.7	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]
      associate--r+ [=>] 80.7	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]
      exp-diff [=>] 26.0	\[ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]
      associate-*r/ [=>] 26.0	\[ \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]
      associate-/l* [=>] 26.0	\[ \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]
      associate-*r/ [<=] 26.0	\[ \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]
      exp-diff [=>] 26.0	\[ \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

   3. Taylor expanded in K around 0 100.0%

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

   4. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{\cos \left(-M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}} \]
      cos-neg [=>] 100.0	\[ \frac{\color{blue}{\cos M}}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}} \]

   5. Taylor expanded in l around inf 100.0%

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

   6. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]

   7. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{1}{e^{\ell}} \]
      rec-exp [=>] 100.0	\[ \color{blue}{e^{-\ell}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 740:\\ \;\;\;\;\frac{\cos M}{e^{{\left(\mathsf{fma}\left(0.5, m + n, -M\right)\right)}^{2} + \left(m - n\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.1%
Cost	33088

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

Alternative 2
Accuracy	92.2%
Cost	26624

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

Alternative 3
Accuracy	64.0%
Cost	20689

\[\begin{array}{l} \mathbf{if}\;m \leq -11600:\\ \;\;\;\;\frac{\cos M}{e^{m \cdot \left(m \cdot 0.25\right)}}\\ \mathbf{elif}\;m \leq -2.35 \cdot 10^{-249}:\\ \;\;\;\;\frac{\cos M}{e^{\sqrt{\ell \cdot \ell}}}\\ \mathbf{elif}\;m \leq 4 \cdot 10^{-302} \lor \neg \left(m \leq 9.5 \cdot 10^{-254}\right):\\ \;\;\;\;\frac{\cos M}{e^{n \cdot \left(n \cdot 0.25\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(K \cdot \left(m \cdot 0.5\right)\right)}{e^{\left(\ell - \left|n - m\right|\right) + M \cdot M}}\\ \end{array} \]

Alternative 4
Accuracy	63.9%
Cost	19784

\[\begin{array}{l} \mathbf{if}\;n \leq 3.1 \cdot 10^{-270}:\\ \;\;\;\;\frac{\cos M}{e^{m \cdot \left(m \cdot 0.25\right)}}\\ \mathbf{elif}\;n \leq 0.0027:\\ \;\;\;\;\frac{\cos M}{e^{\sqrt{\ell \cdot \ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{n \cdot \left(n \cdot 0.25\right)}}\\ \end{array} \]

Alternative 5
Accuracy	65.3%
Cost	13380

\[\begin{array}{l} t_0 := \frac{1}{e^{M \cdot M}}\\ \mathbf{if}\;n \leq 2.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{\cos M}{e^{m \cdot \left(m \cdot 0.25\right)}}\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.08 \cdot 10^{-58}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;n \leq 52:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{n \cdot \left(n \cdot 0.25\right)}}\\ \end{array} \]

Alternative 6
Accuracy	64.0%
Cost	13256

\[\begin{array}{l} t_0 := \frac{1}{e^{M \cdot M}}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -2.7 \cdot 10^{-216}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.56 \cdot 10^{-58}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;n \leq 52:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{n \cdot \left(n \cdot 0.25\right)}}\\ \end{array} \]

Alternative 7
Accuracy	64.0%
Cost	13256

\[\begin{array}{l} t_0 := e^{M \cdot M}\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;n \leq -8.2 \cdot 10^{-157}:\\ \;\;\;\;\frac{\cos M}{t_0}\\ \mathbf{elif}\;n \leq -4.3 \cdot 10^{-216}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-58}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;n \leq 52:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{n \cdot \left(n \cdot 0.25\right)}}\\ \end{array} \]

Alternative 8
Accuracy	64.0%
Cost	7508

\[\begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \frac{1}{e^{M \cdot M}}\\ \mathbf{if}\;n \leq -8.2 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -2.7 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 52:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{n \cdot \left(n \cdot 0.25\right)}}\\ \end{array} \]

Alternative 9
Accuracy	65.8%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;\ell \leq 700:\\ \;\;\;\;\frac{1}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 10
Accuracy	29.6%
Cost	6528

\[e^{-\ell} \]

Alternative 11
Accuracy	6.9%
Cost	6464

\[\cos M \]

Alternative 12
Accuracy	6.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023153 
(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)))))))