Maksimov and Kolovsky, Equation (4)

Average Error: 73.0% → 99.9%

Time: 16.4s

Precision: binary64

Cost: 13504.00

Math

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

↓

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

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

Derivation

1. Initial program 73.0

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

2. Taylor expanded in J around 0 73.0

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

3. Simplified 99.9

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

   [Start] 73.0	\[ \cos \left(0.5 \cdot K\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U \]
   *-commutative [=>] 73.0	\[ \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]
   associate-*l* [=>] 73.0	\[ \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]
   /-rgt-identity [<=] 73.0	\[ \color{blue}{\frac{e^{\ell} - e^{-\ell}}{1}} \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
   metadata-eval [<=] 73.0	\[ \frac{e^{\ell} - e^{-\ell}}{\color{blue}{\frac{2}{2}}} \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
   associate-/l* [<=] 73.0	\[ \color{blue}{\frac{\left(e^{\ell} - e^{-\ell}\right) \cdot 2}{2}} \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
   associate-*l/ [<=] 73.0	\[ \color{blue}{\left(\frac{e^{\ell} - e^{-\ell}}{2} \cdot 2\right)} \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
   sinh-def [<=] 99.9	\[ \left(\color{blue}{\sinh \ell} \cdot 2\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
   *-commutative [<=] 99.9	\[ \color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
   *-commutative [<=] 99.9	\[ \left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right)} + U \]
   associate-*l* [<=] 99.9	\[ \color{blue}{\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J} + U \]
   *-commutative [<=] 99.9	\[ \left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J + U \]
   associate-*l* [=>] 99.9	\[ \color{blue}{\left(2 \cdot \left(\sinh \ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)} \cdot J + U \]
   associate-*l* [=>] 99.9	\[ \color{blue}{2 \cdot \left(\left(\sinh \ell \cdot \cos \left(K \cdot 0.5\right)\right) \cdot J\right)} + U \]
   *-commutative [=>] 99.9	\[ 2 \cdot \color{blue}{\left(J \cdot \left(\sinh \ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]
   *-commutative [=>] 99.9	\[ 2 \cdot \left(J \cdot \left(\sinh \ell \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right)\right) + U \]
   *-commutative [=>] 99.9	\[ 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \sinh \ell\right)}\right) + U \]

4. Final simplification 99.9

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

Alternatives

Alternative 1
Error	86.8%
Cost	7108.00

\[\begin{array}{l} \mathbf{if}\;J \leq 1.8 \cdot 10^{+215}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)\right)\\ \end{array} \]

Alternative 2
Error	86.8%
Cost	7108.00

\[\begin{array}{l} \mathbf{if}\;J \leq 2.9 \cdot 10^{+215}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\\ \end{array} \]

Alternative 3
Error	98.9%
Cost	7104.00

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

Alternative 4
Error	86.8%
Cost	6848.00

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

Alternative 5
Error	86.4%
Cost	448.00

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

Alternative 6
Error	70.9%
Cost	64.00

\[U \]

Reproduce

herbie shell --seed 2023093 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))