Maksimov and Kolovsky, Equation (4)

Average Accuracy: 72.1% → 99.3%

Time: 15.9s

Precision: binary64

Cost: 20544

Math

\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

↓

\[U + \left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + \ell \cdot 2\right)\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right) \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

↓

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (*
   (+
    (* 0.3333333333333333 (pow l 3.0))
    (+ (* 0.016666666666666666 (pow l 5.0)) (* l 2.0)))
   (* (cos (* 0.5 K)) J))))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

↓

double code(double J, double l, double K, double U) {
	return U + (((0.3333333333333333 * pow(l, 3.0)) + ((0.016666666666666666 * pow(l, 5.0)) + (l * 2.0))) * (cos((0.5 * K)) * J));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

↓

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (((0.3333333333333333d0 * (l ** 3.0d0)) + ((0.016666666666666666d0 * (l ** 5.0d0)) + (l * 2.0d0))) * (cos((0.5d0 * k)) * j))
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

↓

public static double code(double J, double l, double K, double U) {
	return U + (((0.3333333333333333 * Math.pow(l, 3.0)) + ((0.016666666666666666 * Math.pow(l, 5.0)) + (l * 2.0))) * (Math.cos((0.5 * K)) * J));
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

↓

def code(J, l, K, U):
	return U + (((0.3333333333333333 * math.pow(l, 3.0)) + ((0.016666666666666666 * math.pow(l, 5.0)) + (l * 2.0))) * (math.cos((0.5 * K)) * J))

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

↓

function code(J, l, K, U)
	return Float64(U + Float64(Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(l * 2.0))) * Float64(cos(Float64(0.5 * K)) * J)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

↓

function tmp = code(J, l, K, U)
	tmp = U + (((0.3333333333333333 * (l ^ 3.0)) + ((0.016666666666666666 * (l ^ 5.0)) + (l * 2.0))) * (cos((0.5 * K)) * J));
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

↓

code[J_, l_, K_, U_] := N[(U + N[(N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

2. Simplified 72.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
   Proof

   [Start] 72.1	\[ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   *-commutative [=>] 72.1	\[ \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} + U \]
   associate-*r* [=>] 72.1	\[ \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]
   *-commutative [<=] 72.1	\[ \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} + U \]
   fma-def [=>] 72.1	\[ \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, \cos \left(\frac{K}{2}\right) \cdot J, U\right)} \]
   *-commutative [=>] 72.1	\[ \mathsf{fma}\left(e^{\ell} - e^{-\ell}, \color{blue}{J \cdot \cos \left(\frac{K}{2}\right)}, U\right) \]

3. Taylor expanded in l around 0 99.3%

   \[\leadsto \color{blue}{2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + \left(0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right) + \left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right) + U\right)\right)} \]

4. Simplified 99.3%

   \[\leadsto \color{blue}{U + \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333\right) + J \cdot \left(\ell \cdot 2 + {\ell}^{5} \cdot 0.016666666666666666\right)\right)} \]
   Proof

   [Start] 99.3	\[ 2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + \left(0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right) + \left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right) + U\right)\right) \]
   associate-+r+ [=>] 99.3	\[ \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)\right) + \left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right) + U\right)} \]
   associate-+r+ [=>] 99.3	\[ \color{blue}{\left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U} \]
   +-commutative [=>] 99.3	\[ \color{blue}{U + \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} \]
   +-commutative [=>] 99.3	\[ U + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right) + \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)\right)\right)} \]
   *-commutative [=>] 99.3	\[ U + \left(\color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right) \cdot 0.3333333333333333} + \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)\right)\right) \]
   associate-*l* [=>] 99.3	\[ U + \left(\color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left({\ell}^{3} \cdot J\right) \cdot 0.3333333333333333\right)} + \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)\right)\right) \]

5. Taylor expanded in J around inf 99.3%

   \[\leadsto U + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]

6. Final simplification 99.3%

   \[\leadsto U + \left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + \ell \cdot 2\right)\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right) \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	13824

\[U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \cdot \cos \left(\frac{K}{2}\right) \]

Alternative 2
Accuracy	99.0%
Cost	13376

\[\mathsf{fma}\left(\ell \cdot 2, J \cdot \cos \left(\frac{K}{2}\right), U\right) \]

Alternative 3
Accuracy	85.6%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;U \leq -8.5 \cdot 10^{-256} \lor \neg \left(U \leq 6.4 \cdot 10^{-263}\right):\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(J \cdot \left(2 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	99.0%
Cost	7104

\[U + 2 \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right) \]

Alternative 5
Accuracy	85.5%
Cost	6720

\[\mathsf{fma}\left(\ell \cdot 2, J, U\right) \]

Alternative 6
Accuracy	70.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;J \leq -1.35 \cdot 10^{+173} \lor \neg \left(J \leq 1.75 \cdot 10^{+263}\right):\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 7
Accuracy	85.5%
Cost	448

\[U + \left(\ell \cdot 2\right) \cdot J \]

Alternative 8
Accuracy	3.6%
Cost	64

\[1 \]

Alternative 9
Accuracy	70.0%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023137 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))