Maksimov and Kolovsky, Equation (4)

Average Accuracy: 72.7% → 99.4%

Time: 10.2s

Precision: binary64

Cost: 7488

Math

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

↓

\[U + \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right) + 2\right)\right)\right) \]

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 (* (cos (* 0.5 K)) (* J (* l (+ (* 0.3333333333333333 (* l l)) 2.0))))))

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 + (cos((0.5 * K)) * (J * (l * ((0.3333333333333333 * (l * l)) + 2.0))));
}

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 + (cos((0.5d0 * k)) * (j * (l * ((0.3333333333333333d0 * (l * l)) + 2.0d0))))
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 + (Math.cos((0.5 * K)) * (J * (l * ((0.3333333333333333 * (l * l)) + 2.0))));
}

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 + (math.cos((0.5 * K)) * (J * (l * ((0.3333333333333333 * (l * l)) + 2.0))))

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(cos(Float64(0.5 * K)) * Float64(J * Float64(l * Float64(Float64(0.3333333333333333 * Float64(l * l)) + 2.0)))))
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 + (cos((0.5 * K)) * (J * (l * ((0.3333333333333333 * (l * l)) + 2.0))));
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[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

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

2. Simplified 72.7%

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

   [Start] 72.7	\[ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   associate-*l* [=>] 72.7	\[ \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]
   fma-def [=>] 72.7	\[ \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

3. Taylor expanded in l around 0 99.4%

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

4. Simplified 99.4%

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

   [Start] 99.4	\[ 2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + \left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right) + U\right) \]
   associate-+r+ [=>] 99.4	\[ \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U} \]
   +-commutative [=>] 99.4	\[ \color{blue}{U + \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} \]
   *-commutative [=>] 99.4	\[ U + \left(\color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) \cdot 2} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) \]
   associate-*l* [=>] 99.4	\[ U + \left(\color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(\ell \cdot J\right) \cdot 2\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) \]
   *-commutative [=>] 99.4	\[ U + \left(\cos \left(0.5 \cdot K\right) \cdot \left(\left(\ell \cdot J\right) \cdot 2\right) + \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right) \cdot 0.3333333333333333}\right) \]
   associate-*l* [=>] 99.4	\[ U + \left(\cos \left(0.5 \cdot K\right) \cdot \left(\left(\ell \cdot J\right) \cdot 2\right) + \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left({\ell}^{3} \cdot J\right) \cdot 0.3333333333333333\right)}\right) \]
   distribute-lft-out [=>] 99.4	\[ U + \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(\ell \cdot J\right) \cdot 2 + \left({\ell}^{3} \cdot J\right) \cdot 0.3333333333333333\right)} \]
   *-commutative [<=] 99.4	\[ U + \cos \left(0.5 \cdot K\right) \cdot \left(\color{blue}{2 \cdot \left(\ell \cdot J\right)} + \left({\ell}^{3} \cdot J\right) \cdot 0.3333333333333333\right) \]
   associate-*r* [=>] 99.4	\[ U + \cos \left(0.5 \cdot K\right) \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot J} + \left({\ell}^{3} \cdot J\right) \cdot 0.3333333333333333\right) \]
   *-commutative [<=] 99.4	\[ U + \cos \left(0.5 \cdot K\right) \cdot \left(\left(2 \cdot \ell\right) \cdot J + \color{blue}{0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)}\right) \]
   associate-*r* [=>] 99.4	\[ U + \cos \left(0.5 \cdot K\right) \cdot \left(\left(2 \cdot \ell\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot J}\right) \]
   distribute-rgt-out [=>] 99.4	\[ U + \cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} \]
   +-commutative [<=] 99.4	\[ U + \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \]
   unpow3 [=>] 99.4	\[ U + \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \ell\right)} + 2 \cdot \ell\right)\right) \]
   associate-*r* [=>] 99.4	\[ U + \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \cdot \ell} + 2 \cdot \ell\right)\right) \]
   distribute-rgt-out [=>] 99.4	\[ U + \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right) + 2\right)\right)}\right) \]

5. Final simplification 99.4%

   \[\leadsto U + \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right) + 2\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	7104

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

Alternative 2
Accuracy	99.1%
Cost	7104

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

Alternative 3
Accuracy	99.1%
Cost	7104

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

Alternative 4
Accuracy	86.6%
Cost	960

\[U + J \cdot \left(\ell \cdot 2 + \ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right) \]

Alternative 5
Accuracy	86.5%
Cost	448

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

Alternative 6
Accuracy	3.6%
Cost	64

\[1 \]

Alternative 7
Accuracy	71.0%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023152 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))