Maksimov and Kolovsky, Equation (4)

Average Accuracy: 73.1% → 99.5%

Time: 15.4s

Precision: binary64

Cost: 20544

Math

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

↓

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

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
 (+
  (*
   (*
    J
    (+
     (* 0.3333333333333333 (pow l 3.0))
     (+ (* 0.016666666666666666 (pow l 5.0)) (* l 2.0))))
   (cos (/ K 2.0)))
  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) {
	return ((J * ((0.3333333333333333 * pow(l, 3.0)) + ((0.016666666666666666 * pow(l, 5.0)) + (l * 2.0)))) * cos((K / 2.0))) + 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
    code = ((j * ((0.3333333333333333d0 * (l ** 3.0d0)) + ((0.016666666666666666d0 * (l ** 5.0d0)) + (l * 2.0d0)))) * cos((k / 2.0d0))) + 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) {
	return ((J * ((0.3333333333333333 * Math.pow(l, 3.0)) + ((0.016666666666666666 * Math.pow(l, 5.0)) + (l * 2.0)))) * Math.cos((K / 2.0))) + 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):
	return ((J * ((0.3333333333333333 * math.pow(l, 3.0)) + ((0.016666666666666666 * math.pow(l, 5.0)) + (l * 2.0)))) * math.cos((K / 2.0))) + 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)
	return Float64(Float64(Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(l * 2.0)))) * cos(Float64(K / 2.0))) + 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)
	tmp = ((J * ((0.3333333333333333 * (l ^ 3.0)) + ((0.016666666666666666 * (l ^ 5.0)) + (l * 2.0)))) * cos((K / 2.0))) + 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_] := N[(N[(N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

2. Taylor expanded in l around 0 99.5%

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

3. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	13824

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

Alternative 2
Accuracy	99.2%
Cost	13376

\[\mathsf{fma}\left(J, 2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right), U\right) \]

Alternative 3
Accuracy	86.4%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;U \leq -7.5 \cdot 10^{-247} \lor \neg \left(U \leq 2.25 \cdot 10^{-257}\right):\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	7104

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

Alternative 5
Accuracy	99.1%
Cost	7104

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

Alternative 6
Accuracy	99.1%
Cost	7104

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

Alternative 7
Accuracy	86.0%
Cost	6720

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

Alternative 8
Accuracy	86.0%
Cost	448

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

Alternative 9
Accuracy	71.4%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023135 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))