Maksimov and Kolovsky, Equation (3)

Average Accuracy: 71.2% → 86.4%

Time: 23.3s

Precision: binary64

Cost: 20616

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq -6 \cdot 10^{+274}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1.1 \cdot 10^{+232}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \left(-2 \cdot J\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \end{array} \]

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))))
   (if (<= U -6e+274)
     U
     (if (<= U -1.1e+232)
       (- U)
       (* (* 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));
	double tmp;
	if (U <= -6e+274) {
		tmp = U;
	} else if (U <= -1.1e+232) {
		tmp = -U;
	} else {
		tmp = (t_0 * (-2.0 * J)) * hypot(1.0, (U / (2.0 * (J * t_0))));
	}
	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 (U <= -6e+274) {
		tmp = U;
	} else if (U <= -1.1e+232) {
		tmp = -U;
	} else {
		tmp = (t_0 * (-2.0 * J)) * Math.hypot(1.0, (U / (2.0 * (J * t_0))));
	}
	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 U <= -6e+274:
		tmp = U
	elif U <= -1.1e+232:
		tmp = -U
	else:
		tmp = (t_0 * (-2.0 * J)) * math.hypot(1.0, (U / (2.0 * (J * t_0))))
	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 (U <= -6e+274)
		tmp = U;
	elseif (U <= -1.1e+232)
		tmp = Float64(-U);
	else
		tmp = Float64(Float64(t_0 * Float64(-2.0 * J)) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))));
	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 (U <= -6e+274)
		tmp = U;
	elseif (U <= -1.1e+232)
		tmp = -U;
	else
		tmp = (t_0 * (-2.0 * J)) * hypot(1.0, (U / (2.0 * (J * t_0))));
	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[LessEqual[U, -6e+274], U, If[LessEqual[U, -1.1e+232], (-U), 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. Split input into 3 regimes

2. if U < -5.99999999999999991e274

   1. Initial program 22.7%

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

   2. Simplified 45.3%

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

      [Start] 22.7	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 22.7	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 22.7	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 45.3	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 45.3	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 45.3	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in U around -inf 43.0%

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

   if -5.99999999999999991e274 < U < -1.1e232

   1. Initial program 32.5%

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

   2. Simplified 55.7%

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

      [Start] 32.5	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 32.5	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 32.5	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 55.7	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 55.7	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 55.7	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in J around 0 40.6%

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

   4. Simplified 40.6%

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

      [Start] 40.6	\[ -1 \cdot U \]
      mul-1-neg [=>] 40.6	\[ \color{blue}{-U} \]

   if -1.1e232 < U

   1. Initial program 74.2%

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

   2. Simplified 89.4%

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

      [Start] 74.2	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 74.2	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 89.4	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 89.4	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -6 \cdot 10^{+274}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1.1 \cdot 10^{+232}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.2%
Cost	20616

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq -2.15 \cdot 10^{+273}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1.5 \cdot 10^{+227}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	86.4%
Cost	20616

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq -8 \cdot 10^{+274}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1.45 \cdot 10^{+234}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \]

Alternative 3
Accuracy	72.5%
Cost	14092

\[\begin{array}{l} t_0 := \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{if}\;J \leq -4.5 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -5.8 \cdot 10^{-225}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{elif}\;J \leq 8.2 \cdot 10^{-277}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	66.9%
Cost	7569

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;K \leq -1.86 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq -3.6 \cdot 10^{+51}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -0.00016 \lor \neg \left(K \leq 2.9 \cdot 10^{-16}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \end{array} \]

Alternative 5
Accuracy	58.3%
Cost	7376

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -5.2 \cdot 10^{-85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -5.5 \cdot 10^{-216}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -9.2 \cdot 10^{-258}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.1 \cdot 10^{-17}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	39.2%
Cost	1620

\[\begin{array}{l} \mathbf{if}\;U \leq -1.6 \cdot 10^{+272}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1 \cdot 10^{+134}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -6.5 \cdot 10^{+48}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq -7 \cdot 10^{-92}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.12 \cdot 10^{+21}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(1 + 0.125 \cdot \left(\frac{U}{J} \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 7
Accuracy	26.9%
Cost	920

\[\begin{array}{l} \mathbf{if}\;K \leq -2.15 \cdot 10^{+122}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -2.2 \cdot 10^{-75}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -3.35 \cdot 10^{-205}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 1.6 \cdot 10^{-299}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 5.9 \cdot 10^{-217}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 6.5 \cdot 10^{-103}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 8
Accuracy	39.2%
Cost	852

\[\begin{array}{l} \mathbf{if}\;U \leq -1.5 \cdot 10^{+272}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -2.3 \cdot 10^{+134}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -2.4 \cdot 10^{+49}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq -2.9 \cdot 10^{-95}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 8.2 \cdot 10^{+21}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 9
Accuracy	27.0%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023141 
(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)))))