Maksimov and Kolovsky, Equation (4)

Average Error: 17.4 → 0.4

Time: 14.8s

Precision: binary64

Cost: 13824

Math

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

↓

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

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
 (+
  (* (+ (* 2.0 l) (* 0.3333333333333333 (pow l 3.0))) (* (cos (* 0.5 K)) J))
  U))

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 (((2.0 * l) + (0.3333333333333333 * pow(l, 3.0))) * (cos((0.5 * K)) * J)) + U;
}

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 = (((2.0d0 * l) + (0.3333333333333333d0 * (l ** 3.0d0))) * (cos((0.5d0 * k)) * j)) + u
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 (((2.0 * l) + (0.3333333333333333 * Math.pow(l, 3.0))) * (Math.cos((0.5 * K)) * J)) + U;
}

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 (((2.0 * l) + (0.3333333333333333 * math.pow(l, 3.0))) * (math.cos((0.5 * K)) * J)) + U

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(Float64(Float64(Float64(2.0 * l) + Float64(0.3333333333333333 * (l ^ 3.0))) * Float64(cos(Float64(0.5 * K)) * J)) + U)
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 = (((2.0 * l) + (0.3333333333333333 * (l ^ 3.0))) * (cos((0.5 * K)) * J)) + U;
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[(N[(N[(N[(2.0 * l), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 17.4

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

2. Taylor expanded in l around 0 0.5

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

3. Simplified 0.4

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

   [Start] 0.5	\[ \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]
   rational_best-simplify-44 [=>] 0.4	\[ \left(2 \cdot \color{blue}{\left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]
   rational_best-simplify-2 [=>] 0.4	\[ \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \ell\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]
   rational_best-simplify-44 [=>] 0.4	\[ \left(\color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \ell\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]
   rational_best-simplify-44 [=>] 0.4	\[ \left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \ell\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)}\right) + U \]
   rational_best-simplify-2 [=>] 0.4	\[ \left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \ell\right) + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot {\ell}^{3}\right)}\right) + U \]
   rational_best-simplify-44 [=>] 0.4	\[ \left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \ell\right) + \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)}\right) + U \]
   rational_best-simplify-2 [=>] 0.4	\[ \left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \ell\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)}\right) + U \]
   rational_best-simplify-51 [=>] 0.4	\[ \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)} + U \]
   rational_best-simplify-2 [=>] 0.4	\[ \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} + U \]
   rational_best-simplify-1 [=>] 0.4	\[ \color{blue}{\left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right) + U \]

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	10.5
Cost	7504

\[\begin{array}{l} t_0 := 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ t_1 := \left(2 \cdot \ell\right) \cdot J + U\\ \mathbf{if}\;U \leq -7.2 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq -2.7 \cdot 10^{-215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 9 \cdot 10^{-273}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{-146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	10.6
Cost	7504

\[\begin{array}{l} t_0 := \left(2 \cdot \ell\right) \cdot J + U\\ \mathbf{if}\;U \leq -4 \cdot 10^{-148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq -1.9 \cdot 10^{-215}:\\ \;\;\;\;\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J + J\right)\right)\\ \mathbf{elif}\;U \leq 9.5 \cdot 10^{-273}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 9.2 \cdot 10^{-146}:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.6
Cost	7104

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

Alternative 4
Error	19.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;U \leq -1.8 \cdot 10^{-133}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.25 \cdot 10^{-131}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 5
Error	8.8
Cost	448

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

Alternative 6
Error	18.6
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023096 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))