Maksimov and Kolovsky, Equation (3)

Average Error: 17.9 → 8.2

Time: 19.4s

Precision: binary64

Cost: 20352

Math

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

↓

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

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))))
   (* (* (* -2.0 J) t_0) (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 ((-2.0 * J) * t_0) * 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 ((-2.0 * J) * t_0) * 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 ((-2.0 * J) * t_0) * 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(Float64(-2.0 * J) * t_0) * 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 = ((-2.0 * J) * t_0) * 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[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $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 17.9

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

2. Simplified 8.2

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

   [Start] 17.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   unpow2 [=>] 17.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
   hypot-1-def [=>] 8.2	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
   associate-*l* [=>] 8.2	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

3. Final simplification 8.2

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

Alternatives

Alternative 1
Error	8.2
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	8.3
Cost	20352

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

Alternative 3
Error	18.1
Cost	13960

\[\begin{array}{l} \mathbf{if}\;U \leq -1.7 \cdot 10^{+127}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \mathbf{elif}\;U \leq 5.8 \cdot 10^{+236}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{elif}\;U \leq 4.5 \cdot 10^{+296}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 4
Error	20.5
Cost	7569

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;K \leq -9.5 \cdot 10^{+237}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq -1.45 \cdot 10^{+211}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -3.6 \cdot 10^{-5} \lor \neg \left(K \leq 4.4 \cdot 10^{+17}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \end{array} \]

Alternative 5
Error	26.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;U \leq -3.6 \cdot 10^{+104}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 7 \cdot 10^{+135}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{+239}:\\ \;\;\;\;-2 \cdot \frac{J}{\frac{U}{J}} - U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 6
Error	38.7
Cost	1360

\[\begin{array}{l} \mathbf{if}\;U \leq -8.4 \cdot 10^{+104}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -2.3 \cdot 10^{+68}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.35 \cdot 10^{-73}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq 1160000000:\\ \;\;\;\;J \cdot \left(-2 \cdot \left(U \cdot \frac{-0.5}{J} - \frac{J}{U}\right)\right)\\ \mathbf{elif}\;U \leq 4.2 \cdot 10^{+67}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 3.7 \cdot 10^{+129}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+241}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 7
Error	46.9
Cost	1052

\[\begin{array}{l} \mathbf{if}\;K \leq -1.45 \cdot 10^{+278}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -2.9 \cdot 10^{+77}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -1.28 \cdot 10^{+33}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -7.2 \cdot 10^{-18}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -2.6 \cdot 10^{-228}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 1.25 \cdot 10^{-150}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 4.6 \cdot 10^{+139}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 8
Error	38.6
Cost	1052

\[\begin{array}{l} \mathbf{if}\;U \leq -3.3 \cdot 10^{+103}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.4 \cdot 10^{+68}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.2 \cdot 10^{-73}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq 2900000000:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.5 \cdot 10^{+67}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 3.2 \cdot 10^{+129}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 10^{+241}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 9
Error	46.8
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023002 
(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)))))