Maksimov and Kolovsky, Equation (4)

Average Error: 17.6 → 0.1

Time: 13.2s

Precision: binary64

Cost: 13504

Math

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

↓

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

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 (* 2.0 (sinh l))) (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 * (2.0 * sinh(l))) * 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 * (2.0d0 * sinh(l))) * 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 * (2.0 * Math.sinh(l))) * 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 * (2.0 * math.sinh(l))) * 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(2.0 * sinh(l))) * 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 * (2.0 * sinh(l))) * 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[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 17.6

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

2. Taylor expanded in l around -inf 17.6

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

3. Simplified 0.1

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

   [Start] 17.6	\[ \left(J \cdot \left(e^{\ell} - e^{-1 \cdot \ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   mul-1-neg [=>] 17.6	\[ \left(J \cdot \left(e^{\ell} - e^{\color{blue}{-\ell}}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   /-rgt-identity [<=] 17.6	\[ \left(J \cdot \color{blue}{\frac{e^{\ell} - e^{-\ell}}{1}}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   metadata-eval [<=] 17.6	\[ \left(J \cdot \frac{e^{\ell} - e^{-\ell}}{\color{blue}{\frac{2}{2}}}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   associate-/l* [<=] 17.6	\[ \left(J \cdot \color{blue}{\frac{\left(e^{\ell} - e^{-\ell}\right) \cdot 2}{2}}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   *-commutative [=>] 17.6	\[ \left(J \cdot \frac{\color{blue}{2 \cdot \left(e^{\ell} - e^{-\ell}\right)}}{2}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   associate-*r/ [<=] 17.6	\[ \left(J \cdot \color{blue}{\left(2 \cdot \frac{e^{\ell} - e^{-\ell}}{2}\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   sinh-def [<=] 0.1	\[ \left(J \cdot \left(2 \cdot \color{blue}{\sinh \ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.6
Cost	7488

\[U + \cos \left(\frac{K}{2}\right) \cdot \frac{J \cdot 2}{\ell \cdot -0.16666666666666666 + \frac{1}{\ell}} \]

Alternative 2
Error	9.1
Cost	7241

\[\begin{array}{l} \mathbf{if}\;J \leq -2 \cdot 10^{+175} \lor \neg \left(J \leq 3.7 \cdot 10^{+253}\right):\\ \;\;\;\;2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 3
Error	9.1
Cost	7240

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -2 \cdot 10^{+175}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(J \cdot t_0\right)\right)\\ \mathbf{elif}\;J \leq 1.6 \cdot 10^{+252}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\ell \cdot t_0\right)\right)\\ \end{array} \]

Alternative 4
Error	0.7
Cost	7104

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

Alternative 5
Error	0.8
Cost	7104

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

Alternative 6
Error	8.6
Cost	6848

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

Alternative 7
Error	8.9
Cost	6720

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

Alternative 8
Error	19.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;J \leq -1.06 \cdot 10^{+158}:\\ \;\;\;\;2 \cdot \left(J \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 9
Error	9.0
Cost	448

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

Alternative 10
Error	18.9
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023047 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))