Maksimov and Kolovsky, Equation (4)

Average Error: 17.3 → 0.1

Time: 16.5s

Precision: binary64

Cost: 13504

Math

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

↓

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

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
 (+ (* J (* (* 2.0 (sinh l)) (cos (* K 0.5)))) 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 * ((2.0 * sinh(l)) * cos((K * 0.5)))) + 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 * ((2.0d0 * sinh(l)) * cos((k * 0.5d0)))) + 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 * ((2.0 * Math.sinh(l)) * Math.cos((K * 0.5)))) + 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 * ((2.0 * math.sinh(l)) * math.cos((K * 0.5)))) + 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(J * Float64(Float64(2.0 * sinh(l)) * cos(Float64(K * 0.5)))) + 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 * ((2.0 * sinh(l)) * cos((K * 0.5)))) + 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[(J * N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $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. Applied egg-rr 0.1

   \[\leadsto \color{blue}{{\left(J \cdot \left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right)\right)\right)}^{1}} + U \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13504

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

Alternative 2
Error	0.6
Cost	7488

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

Alternative 3
Error	9.1
Cost	7240

\[\begin{array}{l} \mathbf{if}\;U \leq -5.351806381557179 \cdot 10^{-192}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;U \leq 1.1382380770620905 \cdot 10^{-284}:\\ \;\;\;\;\ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \end{array} \]

Alternative 4
Error	0.8
Cost	7104

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

Alternative 5
Error	0.8
Cost	7104

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

Alternative 6
Error	9.2
Cost	6720

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

Alternative 7
Error	19.3
Cost	980

\[\begin{array}{l} t_0 := 2 \cdot \left(J \cdot \ell\right)\\ \mathbf{if}\;J \leq -3.381316968543226 \cdot 10^{+230}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4.389827766998613 \cdot 10^{+171}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.201148231643713 \cdot 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 6.220690576779446 \cdot 10^{+205}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 7.949603653824938 \cdot 10^{+247}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 8
Error	9.2
Cost	448

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

Alternative 9
Error	18.8
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022311 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))