Maksimov and Kolovsky, Equation (4)

Average Error: 17.3 → 0.5

Time: 24.2s

Precision: binary64

Cost: 13824

Math

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

↓

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

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
 (+
  (* (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)) (* J (cos (* 0.5 K))))
  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 (((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)) * (J * cos((0.5 * K)))) + 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 = (((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)) * (j * cos((0.5d0 * k)))) + 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 (((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)) * (J * Math.cos((0.5 * K)))) + 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 (((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)) * (J * math.cos((0.5 * K)))) + 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(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)) * Float64(J * cos(Float64(0.5 * K)))) + 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 = (((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)) * (J * cos((0.5 * K)))) + 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[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 17.3

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

2. Taylor expanded in l around 0 0.5

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

3. Simplified 0.5

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

   [Start] 0.5	\[ \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   rational.json-simplify-1 [=>] 0.5	\[ \left(J \cdot \color{blue}{\left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

4. Taylor expanded in J around 0 0.5

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

5. Simplified 0.5

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

   [Start] 0.5	\[ \cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right) + U \]
   rational.json-simplify-43 [=>] 0.5	\[ \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]
   rational.json-simplify-2 [=>] 0.5	\[ \left(0.3333333333333333 \cdot {\ell}^{3} + \color{blue}{\ell \cdot 2}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	13824

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

Alternative 2
Error	0.7
Cost	7104

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

Alternative 3
Error	0.7
Cost	7104

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

Alternative 4
Error	18.4
Cost	848

\[\begin{array}{l} t_0 := 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{if}\;U \leq -1.15 \cdot 10^{-97}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -3.4 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq -1.45 \cdot 10^{-244}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 8 \cdot 10^{-233}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 5
Error	9.0
Cost	448

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

Alternative 6
Error	18.7
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023073 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))