Maksimov and Kolovsky, Equation (4)

Average Accuracy: 71.7% → 99.4%

Time: 12.6s

Precision: binary64

Cost: 34048

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ \left(t_0 \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot J\right) + t_0 \cdot \left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7} + 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right) + U \end{array} \]

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))))
   (+
    (+
     (* t_0 (* (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)) J))
     (*
      t_0
      (*
       J
       (+
        (* 0.0003968253968253968 (pow l 7.0))
        (* 0.016666666666666666 (pow l 5.0))))))
    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 ((t_0 * (((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)) * J)) + (t_0 * (J * ((0.0003968253968253968 * pow(l, 7.0)) + (0.016666666666666666 * pow(l, 5.0)))))) + 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 = ((t_0 * (((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)) * j)) + (t_0 * (j * ((0.0003968253968253968d0 * (l ** 7.0d0)) + (0.016666666666666666d0 * (l ** 5.0d0)))))) + 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 ((t_0 * (((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)) * J)) + (t_0 * (J * ((0.0003968253968253968 * Math.pow(l, 7.0)) + (0.016666666666666666 * Math.pow(l, 5.0)))))) + 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 ((t_0 * (((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)) * J)) + (t_0 * (J * ((0.0003968253968253968 * math.pow(l, 7.0)) + (0.016666666666666666 * math.pow(l, 5.0)))))) + 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(t_0 * Float64(Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)) * J)) + Float64(t_0 * Float64(J * Float64(Float64(0.0003968253968253968 * (l ^ 7.0)) + Float64(0.016666666666666666 * (l ^ 5.0)))))) + 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 = ((t_0 * (((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)) * J)) + (t_0 * (J * ((0.0003968253968253968 * (l ^ 7.0)) + (0.016666666666666666 * (l ^ 5.0)))))) + 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[(t$95$0 * N[(N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(J * N[(N[(0.0003968253968253968 * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Initial program 71.7%

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

2. Taylor expanded in l around 0 99.4%

   \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + \left(0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right) + \left(0.0003968253968253968 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{7} \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)\right)\right)} + U \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(\mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right) \cdot J\right) + \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(0.016666666666666666, {\ell}^{5}, 0.0003968253968253968 \cdot {\ell}^{7}\right)\right)\right)} + U \]
   Proof

   [Start] 99.4	\[ \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + \left(0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right) + \left(0.0003968253968253968 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{7} \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)\right)\right) + U \]
   associate-+r+ [=>] 99.4	\[ \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + \color{blue}{\left(\left(0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right) + 0.0003968253968253968 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{7} \cdot J\right)\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)}\right) + U \]
   +-commutative [=>] 99.4	\[ \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right) + \left(0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right) + 0.0003968253968253968 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{7} \cdot J\right)\right)\right)\right)}\right) + U \]
   associate-+r+ [=>] 99.4	\[ \color{blue}{\left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + \left(0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right) + 0.0003968253968253968 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{7} \cdot J\right)\right)\right)\right)} + U \]

4. Taylor expanded in J around 0 99.4%

   \[\leadsto \left(\cos \left(0.5 \cdot K\right) \cdot \left(\mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right) \cdot J\right) + \cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(\left(0.0003968253968253968 \cdot {\ell}^{7} + 0.016666666666666666 \cdot {\ell}^{5}\right) \cdot J\right)}\right) + U \]

5. Taylor expanded in K around inf 99.4%

   \[\leadsto \left(\color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + \cos \left(0.5 \cdot K\right) \cdot \left(\left(0.0003968253968253968 \cdot {\ell}^{7} + 0.016666666666666666 \cdot {\ell}^{5}\right) \cdot J\right)\right) + U \]

6. Final simplification 99.4%

   \[\leadsto \left(\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot J\right) + \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7} + 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right) + U \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	27264

\[U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(\ell \cdot 2 + 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) \]

Alternative 2
Accuracy	99.3%
Cost	20608

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

Alternative 3
Accuracy	99.3%
Cost	13952

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

Alternative 4
Accuracy	99.3%
Cost	13824

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

Alternative 5
Accuracy	99.1%
Cost	7104

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

Alternative 6
Accuracy	99.1%
Cost	7104

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

Alternative 7
Accuracy	69.0%
Cost	584

\[\begin{array}{l} \mathbf{if}\;J \leq 2.6 \cdot 10^{+154}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.1 \cdot 10^{+220}:\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + U\\ \end{array} \]

Alternative 8
Accuracy	85.6%
Cost	448

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

Alternative 9
Accuracy	3.6%
Cost	64

\[1 \]

Alternative 10
Accuracy	69.8%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023151 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))