Maksimov and Kolovsky, Equation (3)

Average Error: 18.4 → 8.0

Time: 18.8s

Precision: binary64

Cost: 20352

Math

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(t_0 \cdot \left(-2 \cdot J\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right) \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (* (* -2.0 J) (cos (/ K 2.0)))
  (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))

↓

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* t_0 (* -2.0 J)) (hypot 1.0 (/ U (* 2.0 (* J t_0)))))))

C

double code(double J, double K, double U) {
	return ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + pow((U / ((2.0 * J) * cos((K / 2.0)))), 2.0)));
}

↓

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return (t_0 * (-2.0 * J)) * hypot(1.0, (U / (2.0 * (J * t_0))));
}

Java

public static double code(double J, double K, double U) {
	return ((-2.0 * J) * Math.cos((K / 2.0))) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * Math.cos((K / 2.0)))), 2.0)));
}

↓

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return (t_0 * (-2.0 * J)) * Math.hypot(1.0, (U / (2.0 * (J * t_0))));
}

Python

def code(J, K, U):
	return ((-2.0 * J) * math.cos((K / 2.0))) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * math.cos((K / 2.0)))), 2.0)))

↓

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return (t_0 * (-2.0 * J)) * math.hypot(1.0, (U / (2.0 * (J * t_0))))

Julia

function code(J, K, U)
	return Float64(Float64(Float64(-2.0 * J) * cos(Float64(K / 2.0))) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * cos(Float64(K / 2.0)))) ^ 2.0))))
end

↓

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(t_0 * Float64(-2.0 * J)) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))))
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + ((U / ((2.0 * J) * cos((K / 2.0)))) ^ 2.0)));
end

↓

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = (t_0 * (-2.0 * J)) * hypot(1.0, (U / (2.0 * (J * t_0))));
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(2.0 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 18.4

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

2. Simplified 8.0

   \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]
   Proof

   [Start] 18.4	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   unpow2 [=>] 18.4	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
   hypot-1-def [=>] 8.0	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
   associate-*l* [=>] 8.0	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

3. Final simplification 8.0

   \[\leadsto \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]

Alternatives

Alternative 1
Error	8.0
Cost	20352

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ -2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right) \end{array} \]

Alternative 2
Error	8.0
Cost	20352

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\right) \cdot \left(-2 \cdot J\right) \end{array} \]

Alternative 3
Error	17.0
Cost	13961

\[\begin{array}{l} \mathbf{if}\;J \leq -1 \cdot 10^{-283} \lor \neg \left(J \leq 9.2 \cdot 10^{-209}\right):\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 4
Error	28.1
Cost	7640

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{if}\;U \leq -1.15 \cdot 10^{+147}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 6.4 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 4.5 \cdot 10^{-61}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 8 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 6.9 \cdot 10^{+132}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 6 \cdot 10^{+202}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 5
Error	20.1
Cost	7568

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \frac{t_0}{\frac{-0.5}{J}}\\ \mathbf{if}\;K \leq -3.5 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;K \leq -9.5 \cdot 10^{+98}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -95000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;K \leq 7 \cdot 10^{-13}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-2 \cdot J\right)\\ \end{array} \]

Alternative 6
Error	46.6
Cost	1052

\[\begin{array}{l} \mathbf{if}\;K \leq -8.8 \cdot 10^{+232}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -1.62 \cdot 10^{-170}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -1.02 \cdot 10^{-287}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 2.55 \cdot 10^{-216}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 1.3 \cdot 10^{-21}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 3.5 \cdot 10^{+161}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 3.3 \cdot 10^{+205}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 7
Error	39.0
Cost	852

\[\begin{array}{l} \mathbf{if}\;J \leq -7.6 \cdot 10^{-134}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 1.15 \cdot 10^{-208}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.9 \cdot 10^{-83}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{-51}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 16500000:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 8
Error	46.4
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023041 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))