Maksimov and Kolovsky, Equation (4)

Average Error: 17.3 → 0.5

Time: 12.5s

Precision: binary64

Cost: 20608

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ \left(2 \cdot \left(t_0 \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(t_0 \cdot \left(J \cdot {\ell}^{3}\right)\right)\right) + U \end{array} \]

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
 (let* ((t_0 (cos (* 0.5 K))))
   (+
    (+
     (* 2.0 (* t_0 (* l J)))
     (* 0.3333333333333333 (* t_0 (* J (pow l 3.0)))))
    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) {
	double t_0 = cos((0.5 * K));
	return ((2.0 * (t_0 * (l * J))) + (0.3333333333333333 * (t_0 * (J * pow(l, 3.0))))) + 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
    real(8) :: t_0
    t_0 = cos((0.5d0 * k))
    code = ((2.0d0 * (t_0 * (l * j))) + (0.3333333333333333d0 * (t_0 * (j * (l ** 3.0d0))))) + 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) {
	double t_0 = Math.cos((0.5 * K));
	return ((2.0 * (t_0 * (l * J))) + (0.3333333333333333 * (t_0 * (J * Math.pow(l, 3.0))))) + 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):
	t_0 = math.cos((0.5 * K))
	return ((2.0 * (t_0 * (l * J))) + (0.3333333333333333 * (t_0 * (J * math.pow(l, 3.0))))) + 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)
	t_0 = cos(Float64(0.5 * K))
	return Float64(Float64(Float64(2.0 * Float64(t_0 * Float64(l * J))) + Float64(0.3333333333333333 * Float64(t_0 * Float64(J * (l ^ 3.0))))) + 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)
	t_0 = cos((0.5 * K));
	tmp = ((2.0 * (t_0 * (l * J))) + (0.3333333333333333 * (t_0 * (J * (l ^ 3.0))))) + 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_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(2.0 * N[(t$95$0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(t$95$0 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $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

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

3. Final simplification 0.5

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

Alternatives

Alternative 1
Error	9.0
Cost	7240

\[\begin{array}{l} \mathbf{if}\;U \leq -2.1 \cdot 10^{-276}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \mathbf{elif}\;U \leq 3.3 \cdot 10^{-192}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \left(\ell \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, 2 \cdot \ell, U\right)\\ \end{array} \]

Alternative 2
Error	0.7
Cost	7104

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

Alternative 3
Error	0.7
Cost	7104

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

Alternative 4
Error	0.7
Cost	7104

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

Alternative 5
Error	8.8
Cost	6848

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

Alternative 6
Error	9.1
Cost	6720

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

Alternative 7
Error	18.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;U \leq -2.8 \cdot 10^{-236}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.4 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 8
Error	9.1
Cost	448

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

Alternative 9
Error	18.6
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022343 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))