Maksimov and Kolovsky, Equation (4)

Average Error: 16.8 → 0.1

Time: 10.0s

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 16.8

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

2. Taylor expanded in l around -inf 16.8

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

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
   (*.f64 2 (sinh.f64 l)): 0 points increase in error, 0 points decrease in error
   (*.f64 2 (Rewrite=> sinh-def_binary64 (/.f64 (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) 2))): 117 points increase in error, 138 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) 2)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-/l*_binary64 (/.f64 2 (/.f64 2 (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))))): 3 points increase in error, 2 points decrease in error
   (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 2 2) (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))): 2 points increase in error, 3 points decrease in error
   (*.f64 (Rewrite=> metadata-eval 1) (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> *-lft-identity_binary64 (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (exp.f64 l) (exp.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 l)))): 0 points increase in error, 0 points decrease in error

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	8.9
Cost	7240

\[\begin{array}{l} t_0 := U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \mathbf{if}\;U \leq -5.4 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 7.5 \cdot 10^{-302}:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	8.9
Cost	7240

\[\begin{array}{l} t_0 := U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \mathbf{if}\;U \leq -6.2 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 1.25 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	8.9
Cost	7240

\[\begin{array}{l} t_0 := U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \mathbf{if}\;U \leq -3.5 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 7.8 \cdot 10^{-302}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(2 \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.7
Cost	7104

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

Alternative 5
Error	8.6
Cost	6848

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

Alternative 6
Error	8.9
Cost	6720

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

Alternative 7
Error	17.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;U \leq -5.5 \cdot 10^{-278}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.15 \cdot 10^{-291}:\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 8
Error	8.9
Cost	448

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

Alternative 9
Error	61.7
Cost	64

\[-3 \]

Alternative 10
Error	18.1
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022338 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))