Maksimov and Kolovsky, Equation (4)

Average Error: 17.2 → 0.1

Time: 11.2s

Precision: binary64

Cost: 13504

Math

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

↓

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

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

Derivation

1. Initial program 17.2

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

2. Simplified 17.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
   Proof
   (fma.f64 (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) (*.f64 J (cos.f64 (/.f64 K 2))) U): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) (*.f64 J (cos.f64 (/.f64 K 2)))) U)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) J) (cos.f64 (/.f64 K 2)))) U): 1 points increase in error, 3 points decrease in error
   (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.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

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.6
Cost	13376

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

Alternative 2
Error	10.2
Cost	7504

\[\begin{array}{l} t_0 := J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell + \ell\right)\right)\\ t_1 := U + J \cdot \left(\ell + \ell\right)\\ \mathbf{if}\;U \leq -2.1 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq -5.7 \cdot 10^{-214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 2.45 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 3.05 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \end{array} \]

Alternative 3
Error	10.2
Cost	7504

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + J \cdot \left(\ell + \ell\right)\\ \mathbf{if}\;U \leq -2.2 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq -1.55 \cdot 10^{-214}:\\ \;\;\;\;J \cdot \left(t_0 \cdot \left(\ell + \ell\right)\right)\\ \mathbf{elif}\;U \leq 4.4 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 3.05 \cdot 10^{-145}:\\ \;\;\;\;\ell \cdot \left(t_0 \cdot \left(J + J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \end{array} \]

Alternative 4
Error	10.2
Cost	7504

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + J \cdot \left(\ell + \ell\right)\\ \mathbf{if}\;U \leq -1.1 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq -5.1 \cdot 10^{-213}:\\ \;\;\;\;t_0 \cdot \left(J \cdot \left(2 \cdot \ell\right)\right)\\ \mathbf{elif}\;U \leq 3.1 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 3.05 \cdot 10^{-145}:\\ \;\;\;\;\ell \cdot \left(t_0 \cdot \left(J + J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \end{array} \]

Alternative 5
Error	0.6
Cost	7104

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

Alternative 6
Error	0.6
Cost	7104

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

Alternative 7
Error	8.9
Cost	6720

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

Alternative 8
Error	19.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;U \leq 1.05 \cdot 10^{-242}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{-166}:\\ \;\;\;\;\ell \cdot \left(J + J\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 9
Error	8.9
Cost	448

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

Alternative 10
Error	62.0
Cost	64

\[0 \]

Alternative 11
Error	18.5
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022337 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))