Maksimov and Kolovsky, Equation (3)

Average Error: 28.25% → 13.22%

Time: 20.7s

Precision: binary64

Cost: 20617

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -1.6 \cdot 10^{-268} \lor \neg \left(J \leq 4.5 \cdot 10^{-244}\right):\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

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))))
   (if (or (<= J -1.6e-268) (not (<= J 4.5e-244)))
     (* (* (* J -2.0) t_0) (hypot 1.0 (/ U (* 2.0 (* J t_0)))))
     (- U))))

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));
	double tmp;
	if ((J <= -1.6e-268) || !(J <= 4.5e-244)) {
		tmp = ((J * -2.0) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	} else {
		tmp = -U;
	}
	return tmp;
}

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));
	double tmp;
	if ((J <= -1.6e-268) || !(J <= 4.5e-244)) {
		tmp = ((J * -2.0) * t_0) * Math.hypot(1.0, (U / (2.0 * (J * t_0))));
	} else {
		tmp = -U;
	}
	return tmp;
}

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))
	tmp = 0
	if (J <= -1.6e-268) or not (J <= 4.5e-244):
		tmp = ((J * -2.0) * t_0) * math.hypot(1.0, (U / (2.0 * (J * t_0))))
	else:
		tmp = -U
	return tmp

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))
	tmp = 0.0
	if ((J <= -1.6e-268) || !(J <= 4.5e-244))
		tmp = Float64(Float64(Float64(J * -2.0) * t_0) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))));
	else
		tmp = Float64(-U);
	end
	return tmp
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_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((J <= -1.6e-268) || ~((J <= 4.5e-244)))
		tmp = ((J * -2.0) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	else
		tmp = -U;
	end
	tmp_2 = tmp;
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]}, If[Or[LessEqual[J, -1.6e-268], N[Not[LessEqual[J, 4.5e-244]], $MachinePrecision]], N[(N[(N[(J * -2.0), $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], (-U)]]

Derivation

1. Split input into 2 regimes

2. if J < -1.5999999999999999e-268 or 4.5000000000000002e-244 < J

   1. Initial program 24.27

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

   2. Simplified 9.6

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

      [Start] 24.27	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 24.27	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 9.6	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 9.6	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if -1.5999999999999999e-268 < J < 4.5000000000000002e-244

   1. Initial program 70.4

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

   2. Simplified 46.07

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

      [Start] 70.4	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 70.43	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 70.43	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 46.07	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 46.07	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 46.07	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in J around 0 51.51

      \[\leadsto \color{blue}{-1 \cdot U} \]

   4. Simplified 51.51

      \[\leadsto \color{blue}{-U} \]
      Proof

      [Start] 51.51	\[ -1 \cdot U \]
      mul-1-neg [=>] 51.51	\[ \color{blue}{-U} \]

3. Recombined 2 regimes into one program.

4. Final simplification 13.22

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.6 \cdot 10^{-268} \lor \neg \left(J \leq 4.5 \cdot 10^{-244}\right):\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternatives

Alternative 1
Error	13.22%
Cost	20617

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -1.1 \cdot 10^{-268} \lor \neg \left(J \leq 5.5 \cdot 10^{-246}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 2
Error	13.2%
Cost	20617

\[\begin{array}{l} \mathbf{if}\;J \leq -2.32 \cdot 10^{-268} \lor \neg \left(J \leq 3.1 \cdot 10^{-245}\right):\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{\frac{U}{2}}{J}}{\cos \left(K \cdot 0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 3
Error	13.42%
Cost	20616

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -4 \cdot 10^{-274}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\right)\\ \mathbf{elif}\;J \leq 1.9 \cdot 10^{-237}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	27.66%
Cost	14357

\[\begin{array}{l} \mathbf{if}\;U \leq -2.2 \cdot 10^{+212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -6.2 \cdot 10^{+184}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.5 \cdot 10^{+72} \lor \neg \left(U \leq 4.7 \cdot 10^{+111}\right) \land U \leq 2.4 \cdot 10^{+168}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 5
Error	38.18%
Cost	7964

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ t_1 := \left(J \cdot -2\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{if}\;U \leq -1.52 \cdot 10^{+212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.05 \cdot 10^{+184}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -5.5 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq -2.3 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq -5 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 3.9 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 1.22 \cdot 10^{+219}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 6
Error	41.6%
Cost	7376

\[\begin{array}{l} \mathbf{if}\;U \leq -2 \cdot 10^{+212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.06 \cdot 10^{+184}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -5.5 \cdot 10^{+144}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.12 \cdot 10^{+24}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 7
Error	73.94%
Cost	788

\[\begin{array}{l} \mathbf{if}\;U \leq -2.15 \cdot 10^{+212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.5 \cdot 10^{+184}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1.05 \cdot 10^{+122}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.42 \cdot 10^{-39}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 5.5 \cdot 10^{+19}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 8
Error	61.08%
Cost	720

\[\begin{array}{l} \mathbf{if}\;U \leq -1.9 \cdot 10^{+212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -5.8 \cdot 10^{+184}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -4.8 \cdot 10^{+144}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 9 \cdot 10^{+20}:\\ \;\;\;\;J \cdot -2\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 9
Error	73.31%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023103 
(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)))))