Maksimov and Kolovsky, Equation (4)

Average Error: 16.7 → 0.1

Time: 17.2s

Precision: binary64

Cost: 19776

Math

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

↓

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

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

Derivation

1. Initial program 16.7

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

2. Simplified 16.7

   \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]
   Proof
   (fma.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K 2)) U): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K 2))) U)): 1 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.1

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

4. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(2 \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
   Proof
   (fma.f64 J (*.f64 (*.f64 2 (sinh.f64 l)) (cos.f64 (/.f64 K 2))) U): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 J (*.f64 (*.f64 2 (sinh.f64 l)) (cos.f64 (/.f64 K 2)))) U)): 2 points increase in error, 1 points decrease in error
   (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 J (*.f64 2 (sinh.f64 l))) (cos.f64 (/.f64 K 2)))) U): 7 points increase in error, 6 points decrease in error

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13504

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

Alternative 2
Error	8.9
Cost	7240

\[\begin{array}{l} \mathbf{if}\;U \leq -6.5 \cdot 10^{-218}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \mathbf{elif}\;U \leq 1.6 \cdot 10^{-196}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \end{array} \]

Alternative 3
Error	8.9
Cost	7240

\[\begin{array}{l} \mathbf{if}\;U \leq -1.72 \cdot 10^{-217}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{-194}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7104

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

Alternative 5
Error	8.3
Cost	6848

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

Alternative 6
Error	8.5
Cost	832

\[U + \frac{J}{\ell \cdot -0.08333333333333333 + 0.5 \cdot \frac{1}{\ell}} \]

Alternative 7
Error	8.6
Cost	448

\[U + \frac{J}{\frac{0.5}{\ell}} \]

Alternative 8
Error	8.6
Cost	448

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

Alternative 9
Error	18.0
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022318 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))