Maksimov and Kolovsky, Equation (4)

Average Error: 16.9 → 0.5

Time: 21.5s

Precision: binary64

Cost: 7104

Math

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

↓

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

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

Derivation

1. Initial program 16.9

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

2. Simplified 16.9

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

   [Start] 16.9	\[ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   rational.json-simplify-2 [=>] 16.9	\[ \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} + U \]
   rational.json-simplify-2 [=>] 16.9	\[ \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]
   rational.json-simplify-43 [=>] 16.9	\[ \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

3. Taylor expanded in l around 0 0.5

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

4. Final simplification 0.5

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

Alternatives

Alternative 1
Error	9.6
Cost	7108

\[\begin{array}{l} \mathbf{if}\;J \leq -5 \cdot 10^{+169}:\\ \;\;\;\;\ell \cdot \left(\left(J \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot J + U\\ \end{array} \]

Alternative 2
Error	9.6
Cost	7108

\[\begin{array}{l} \mathbf{if}\;J \leq -4.8 \cdot 10^{+169}:\\ \;\;\;\;\left(\ell \cdot \left(J + J\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot J + U\\ \end{array} \]

Alternative 3
Error	18.3
Cost	584

\[\begin{array}{l} t_0 := \ell \cdot \left(J + J\right)\\ \mathbf{if}\;J \leq -2.3 \cdot 10^{+165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.15 \cdot 10^{+232}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	8.8
Cost	448

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

Alternative 5
Error	18.1
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023068 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))