Maksimov and Kolovsky, Equation (4)

Average Error: 17.1 → 0.1

Time: 13.2s

Precision: binary64

Cost: 19776

Math

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

↓

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

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
 (fma (* 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 fma((J * 2.0), (sinh(l) * 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 fma(Float64(J * 2.0), Float64(sinh(l) * cos(Float64(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 * 2.0), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 17.1

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

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. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13504

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

Alternative 2
Error	8.9
Cost	7240

\[\begin{array}{l} \mathbf{if}\;J \leq 2.2391028067941813 \cdot 10^{+111}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{elif}\;J \leq 2.532952835702173 \cdot 10^{+140}:\\ \;\;\;\;\ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) + 2 \cdot \ell\right)\\ \end{array} \]

Alternative 3
Error	8.9
Cost	7240

\[\begin{array}{l} \mathbf{if}\;J \leq 2.2391028067941813 \cdot 10^{+111}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{elif}\;J \leq 2.532952835702173 \cdot 10^{+140}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \left(J \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) + 2 \cdot \ell\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7104

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

Alternative 5
Error	8.5
Cost	6848

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

Alternative 6
Error	8.7
Cost	960

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

Alternative 7
Error	17.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;U \leq -3.6946638497656255 \cdot 10^{-264}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.859746936840123 \cdot 10^{-262}:\\ \;\;\;\;2 \cdot \left(J \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 8
Error	8.8
Cost	448

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

Alternative 9
Error	18.4
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022291 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))