Maksimov and Kolovsky, Equation (4)

Average Error: 17.4 → 0.1

Time: 12.8s

Precision: binary64

Cost: 13504

Math

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

↓

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

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

Derivation

1. Initial program 17.4

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

2. Taylor expanded in l around -inf 17.4

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

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] 17.4	\[ \left(J \cdot \left(e^{\ell} - e^{-1 \cdot \ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   mul-1-neg [=>] 17.4	\[ \left(J \cdot \left(e^{\ell} - e^{\color{blue}{-\ell}}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   *-lft-identity [<=] 17.4	\[ \left(J \cdot \left(\color{blue}{1 \cdot e^{\ell}} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   *-lft-identity [<=] 17.4	\[ \left(J \cdot \left(1 \cdot e^{\ell} - \color{blue}{1 \cdot e^{-\ell}}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   distribute-lft-out-- [=>] 17.4	\[ \left(J \cdot \color{blue}{\left(1 \cdot \left(e^{\ell} - e^{-\ell}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   metadata-eval [<=] 17.4	\[ \left(J \cdot \left(\color{blue}{\frac{2}{2}} \cdot \left(e^{\ell} - e^{-\ell}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   associate-/r/ [<=] 17.4	\[ \left(J \cdot \color{blue}{\frac{2}{\frac{2}{e^{\ell} - e^{-\ell}}}}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   associate-/l* [<=] 17.4	\[ \left(J \cdot \color{blue}{\frac{2 \cdot \left(e^{\ell} - e^{-\ell}\right)}{2}}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   associate-*r/ [<=] 17.4	\[ \left(J \cdot \color{blue}{\left(2 \cdot \frac{e^{\ell} - e^{-\ell}}{2}\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. Applied egg-rr 16.2

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(K \cdot 0.5\right)\right)} - 1\right)} + U \]

5. Simplified 0.1

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

   [Start] 16.2	\[ \left(e^{\mathsf{log1p}\left(\left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(K \cdot 0.5\right)\right)} - 1\right) + U \]
   expm1-def [=>] 8.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]
   expm1-log1p [=>] 0.1	\[ \color{blue}{\left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(K \cdot 0.5\right)} + U \]
   associate-*l* [=>] 0.1	\[ \color{blue}{J \cdot \left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]
   associate-*r* [<=] 0.1	\[ J \cdot \color{blue}{\left(2 \cdot \left(\sinh \ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]
   *-commutative [=>] 0.1	\[ J \cdot \left(2 \cdot \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot \sinh \ell\right)}\right) + U \]
   *-commutative [=>] 0.1	\[ J \cdot \left(2 \cdot \left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \sinh \ell\right)\right) + U \]

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.6
Cost	13376

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

Alternative 2
Error	0.6
Cost	7104

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

Alternative 3
Error	0.6
Cost	7104

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

Alternative 4
Error	8.5
Cost	6848

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

Alternative 5
Error	8.8
Cost	6720

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

Alternative 6
Error	19.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;J \leq 4.8 \cdot 10^{+169}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.6 \cdot 10^{+223}:\\ \;\;\;\;2 \cdot \left(J \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 7
Error	8.8
Cost	448

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

Alternative 8
Error	18.7
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023012 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))